1,1,221,0,4.753930," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4,x)","a \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True))","A",0
2,1,199,0,4.456685," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3,x)","a \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True))","C",0
3,1,133,0,3.288830," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2,x)","a \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))","A",0
4,1,110,0,3.157396," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x,x)","a \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))","C",0
5,1,19,0,1.615909," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2),x)","\begin{cases} \frac{- \sqrt{- a^{2} x^{2} + 1} + \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-sqrt(-a**2*x**2 + 1) + asin(a*x))/a, Ne(a, 0)), (x, True))","A",0
6,1,70,0,5.922341," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x,x)","a \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))","B",0
7,1,65,0,2.369589," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2,x)","a \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))","C",0
8,1,136,0,3.039023," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3,x)","a \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))","C",0
9,1,185,0,3.205882," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4,x)","a \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))","C",0
10,1,258,0,4.437265," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**5,x)","a \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + \begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}"," ",0,"a*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
11,1,39,0,0.105314," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3,x)","- \frac{x^{4}}{4} - \frac{2 x^{3}}{3 a} - \frac{x^{2}}{a^{2}} - \frac{2 x}{a^{3}} - \frac{2 \log{\left(a x - 1 \right)}}{a^{4}}"," ",0,"-x**4/4 - 2*x**3/(3*a) - x**2/a**2 - 2*x/a**3 - 2*log(a*x - 1)/a**4","A",0
12,1,29,0,0.100623," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2,x)","- \frac{x^{3}}{3} - \frac{x^{2}}{a} - \frac{2 x}{a^{2}} - \frac{2 \log{\left(a x - 1 \right)}}{a^{3}}"," ",0,"-x**3/3 - x**2/a - 2*x/a**2 - 2*log(a*x - 1)/a**3","A",0
13,1,22,0,0.094670," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x,x)","- \frac{x^{2}}{2} - \frac{2 x}{a} - \frac{2 \log{\left(a x - 1 \right)}}{a^{2}}"," ",0,"-x**2/2 - 2*x/a - 2*log(a*x - 1)/a**2","A",0
14,1,12,0,0.083607," ","integrate((a*x+1)**2/(-a**2*x**2+1),x)","- x - \frac{2 \log{\left(a x - 1 \right)}}{a}"," ",0,"-x - 2*log(a*x - 1)/a","A",0
15,1,10,0,0.121200," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x,x)","\log{\left(x \right)} - 2 \log{\left(x - \frac{1}{a} \right)}"," ",0,"log(x) - 2*log(x - 1/a)","A",0
16,1,17,0,0.143141," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2,x)","- 2 a \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{1}{x}"," ",0,"-2*a*(-log(x) + log(x - 1/a)) - 1/x","A",0
17,1,27,0,0.162781," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3,x)","- 2 a^{2} \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{4 a x + 1}{2 x^{2}}"," ",0,"-2*a**2*(-log(x) + log(x - 1/a)) - (4*a*x + 1)/(2*x**2)","A",0
18,1,36,0,0.182464," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**4,x)","- 2 a^{3} \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{6 a^{2} x^{2} + 3 a x + 1}{3 x^{3}}"," ",0,"-2*a**3*(-log(x) + log(x - 1/a)) - (6*a**2*x**2 + 3*a*x + 1)/(3*x**3)","A",0
19,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2,x)","\int \frac{x^{2} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
20,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x,x)","\int \frac{x \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
21,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
22,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
23,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
24,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
25,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\left(a x + 1\right)^{3}}{x^{4} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
26,1,49,0,0.169617," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*x**3,x)","\frac{x^{4}}{4} - \frac{4}{a^{5} x - a^{4}} + \frac{4 x^{3}}{3 a} + \frac{4 x^{2}}{a^{2}} + \frac{12 x}{a^{3}} + \frac{16 \log{\left(a x - 1 \right)}}{a^{4}}"," ",0,"x**4/4 - 4/(a**5*x - a**4) + 4*x**3/(3*a) + 4*x**2/a**2 + 12*x/a**3 + 16*log(a*x - 1)/a**4","A",0
27,1,39,0,0.158592," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*x**2,x)","\frac{x^{3}}{3} - \frac{4}{a^{4} x - a^{3}} + \frac{2 x^{2}}{a} + \frac{8 x}{a^{2}} + \frac{12 \log{\left(a x - 1 \right)}}{a^{3}}"," ",0,"x**3/3 - 4/(a**4*x - a**3) + 2*x**2/a + 8*x/a**2 + 12*log(a*x - 1)/a**3","A",0
28,1,31,0,0.146885," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*x,x)","\frac{x^{2}}{2} - \frac{4}{a^{3} x - a^{2}} + \frac{4 x}{a} + \frac{8 \log{\left(a x - 1 \right)}}{a^{2}}"," ",0,"x**2/2 - 4/(a**3*x - a**2) + 4*x/a + 8*log(a*x - 1)/a**2","A",0
29,1,19,0,0.131085," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2,x)","x - \frac{4}{a^{2} x - a} + \frac{4 \log{\left(a x - 1 \right)}}{a}"," ",0,"x - 4/(a**2*x - a) + 4*log(a*x - 1)/a","A",0
30,1,8,0,0.160271," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/x,x)","\log{\left(x \right)} - \frac{4}{a x - 1}"," ",0,"log(x) - 4/(a*x - 1)","A",0
31,1,26,0,0.225282," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/x**2,x)","4 a \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right) + \frac{- 5 a x + 1}{a x^{2} - x}"," ",0,"4*a*(log(x) - log(x - 1/a)) + (-5*a*x + 1)/(a*x**2 - x)","A",0
32,1,41,0,0.253552," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/x**3,x)","8 a^{2} \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right) + \frac{- 16 a^{2} x^{2} + 7 a x + 1}{2 a x^{3} - 2 x^{2}}"," ",0,"8*a**2*(log(x) - log(x - 1/a)) + (-16*a**2*x**2 + 7*a*x + 1)/(2*a*x**3 - 2*x**2)","A",0
33,1,49,0,0.274540," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/x**4,x)","12 a^{3} \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right) + \frac{- 36 a^{3} x^{3} + 18 a^{2} x^{2} + 5 a x + 1}{3 a x^{4} - 3 x^{3}}"," ",0,"12*a**3*(log(x) - log(x - 1/a)) + (-36*a**3*x**3 + 18*a**2*x**2 + 5*a*x + 1)/(3*a*x**4 - 3*x**3)","A",0
34,0,0,0,0.000000," ","integrate(x**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**3*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
35,0,0,0,0.000000," ","integrate(x**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
36,0,0,0,0.000000," ","integrate(x/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
37,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
38,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x*(a*x + 1)), x)","F",0
39,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)), x)","F",0
40,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**3,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{3} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)), x)","F",0
41,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**4,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{4} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)), x)","F",0
42,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**5,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{5} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(x**5*(a*x + 1)), x)","F",0
43,1,37,0,0.105526," ","integrate(x**3/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{x^{4}}{4} + \frac{2 x^{3}}{3 a} - \frac{x^{2}}{a^{2}} + \frac{2 x}{a^{3}} - \frac{2 \log{\left(a x + 1 \right)}}{a^{4}}"," ",0,"-x**4/4 + 2*x**3/(3*a) - x**2/a**2 + 2*x/a**3 - 2*log(a*x + 1)/a**4","A",0
44,1,27,0,0.101010," ","integrate(x**2/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{x^{3}}{3} + \frac{x^{2}}{a} - \frac{2 x}{a^{2}} + \frac{2 \log{\left(a x + 1 \right)}}{a^{3}}"," ",0,"-x**3/3 + x**2/a - 2*x/a**2 + 2*log(a*x + 1)/a**3","A",0
45,1,20,0,0.092630," ","integrate(x/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{x^{2}}{2} + \frac{2 x}{a} - \frac{2 \log{\left(a x + 1 \right)}}{a^{2}}"," ",0,"-x**2/2 + 2*x/a - 2*log(a*x + 1)/a**2","A",0
46,1,10,0,0.085095," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1),x)","- x + \frac{2 \log{\left(a x + 1 \right)}}{a}"," ",0,"-x + 2*log(a*x + 1)/a","A",0
47,1,10,0,0.119448," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/x,x)","\log{\left(x \right)} - 2 \log{\left(x + \frac{1}{a} \right)}"," ",0,"log(x) - 2*log(x + 1/a)","A",0
48,1,17,0,0.139173," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/x**2,x)","- 2 a \left(\log{\left(x \right)} - \log{\left(x + \frac{1}{a} \right)}\right) - \frac{1}{x}"," ",0,"-2*a*(log(x) - log(x + 1/a)) - 1/x","A",0
49,1,27,0,0.156740," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)","- 2 a^{2} \left(- \log{\left(x \right)} + \log{\left(x + \frac{1}{a} \right)}\right) - \frac{- 4 a x + 1}{2 x^{2}}"," ",0,"-2*a**2*(-log(x) + log(x + 1/a)) - (-4*a*x + 1)/(2*x**2)","A",0
50,1,36,0,0.176972," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)","- 2 a^{3} \left(\log{\left(x \right)} - \log{\left(x + \frac{1}{a} \right)}\right) - \frac{6 a^{2} x^{2} - 3 a x + 1}{3 x^{3}}"," ",0,"-2*a**3*(log(x) - log(x + 1/a)) - (6*a**2*x**2 - 3*a*x + 1)/(3*x**3)","A",0
51,0,0,0,0.000000," ","integrate(x**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
52,0,0,0,0.000000," ","integrate(x**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
53,0,0,0,0.000000," ","integrate(x/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
54,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
55,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x*(a*x + 1)**3), x)","F",0
56,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{2} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)**3), x)","F",0
57,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{3} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**3*(a*x + 1)**3), x)","F",0
58,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{4} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**4*(a*x + 1)**3), x)","F",0
59,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{5} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(x**5*(a*x + 1)**3), x)","F",0
60,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**m,x)","\int x^{m} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(x**m*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
61,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**2,x)","\int x^{2} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(x**2*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
62,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x,x)","\int x \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(x*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
63,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
64,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x, x)","F",0
65,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{2}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**2, x)","F",0
66,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**3,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{3}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**3, x)","F",0
67,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**4,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{4}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**4, x)","F",0
68,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**5,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{5}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**5, x)","F",0
69,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**6,x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{6}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/x**6, x)","F",0
70,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)*x**m,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
71,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)*x**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
72,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)*x**2,x)","\int x^{2} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}\, dx"," ",0,"Integral(x**2*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
73,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)*x,x)","\int x \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}\, dx"," ",0,"Integral(x*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
74,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
75,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}{x}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x, x)","F",0
76,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**2,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}{x^{2}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x**2, x)","F",0
77,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**3,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}{x^{3}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x**3, x)","F",0
78,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**4,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}{x^{4}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x**4, x)","F",0
79,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**5,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}{x^{5}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)/x**5, x)","F",0
80,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)*x**m,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
81,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)*x**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
82,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)*x**2,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
83,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)*x,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
84,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
85,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x,x)","\int \frac{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}{x}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2)/x, x)","F",0
86,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**2,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
87,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
88,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
89,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**5,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
90,0,0,0,0.000000," ","integrate(x**m/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \frac{x^{m}}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(x**m/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
91,0,0,0,0.000000," ","integrate(x**3/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \frac{x^{3}}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(x**3/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
92,0,0,0,0.000000," ","integrate(x**2/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \frac{x^{2}}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(x**2/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
93,0,0,0,0.000000," ","integrate(x/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \frac{x}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(x/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
94,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)","\int \frac{1}{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)","F",0
95,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x,x)","\int \frac{1}{x \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/(x*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)","F",0
96,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2,x)","\int \frac{1}{x^{2} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/(x**2*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)","F",0
97,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**3,x)","\int \frac{1}{x^{3} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/(x**3*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)","F",0
98,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**4,x)","\int \frac{1}{x^{4} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/(x**4*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)","F",0
99,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**5,x)","\int \frac{1}{x^{5} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, dx"," ",0,"Integral(1/(x**5*sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))), x)","F",0
100,0,0,0,0.000000," ","integrate(x**m/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \frac{x^{m}}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**m/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
101,0,0,0,0.000000," ","integrate(x**3/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \frac{x^{3}}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
102,0,0,0,0.000000," ","integrate(x**2/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \frac{x^{2}}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
103,0,0,0,0.000000," ","integrate(x/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \frac{x}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)","F",0
104,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2),x)","\int \frac{1}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(-3/2), x)","F",0
105,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x,x)","\int \frac{1}{x \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(1/(x*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)","F",0
106,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**2,x)","\int \frac{1}{x^{2} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(1/(x**2*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)","F",0
107,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**3,x)","\int \frac{1}{x^{3} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(1/(x**3*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)","F",0
108,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**4,x)","\int \frac{1}{x^{4} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(1/(x**4*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)","F",0
109,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(3/2)/x**5,x)","\int \frac{1}{x^{5} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(1/(x**5*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2)), x)","F",0
110,-1,0,0,0.000000," ","integrate(x**m/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
111,0,0,0,0.000000," ","integrate(x**3/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\int \frac{x^{3}}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**3/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2), x)","F",0
112,0,0,0,0.000000," ","integrate(x**2/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\int \frac{x^{2}}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**2/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2), x)","F",0
113,0,0,0,0.000000," ","integrate(x/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\int \frac{x}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x/((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2), x)","F",0
114,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2),x)","\int \frac{1}{\left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(-5/2), x)","F",0
115,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x,x)","\int \frac{1}{x \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(1/(x*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2)), x)","F",0
116,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**2,x)","\int \frac{1}{x^{2} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(1/(x**2*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2)), x)","F",0
117,0,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**3,x)","\int \frac{1}{x^{3} \left(\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(1/(x**3*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(5/2)), x)","F",0
118,-1,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
119,-1,0,0,0.000000," ","integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(5/2)/x**5,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
120,-1,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)*x**m,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
121,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)*x**2,x)","\int x^{2} \sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}\, dx"," ",0,"Integral(x**2*((x + 1)/sqrt(1 - x**2))**(1/3), x)","F",0
122,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)*x,x)","\int x \sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}\, dx"," ",0,"Integral(x*((x + 1)/sqrt(1 - x**2))**(1/3), x)","F",0
123,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3),x)","\int \sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(1/3), x)","F",0
124,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)/x,x)","\int \frac{\sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}}{x}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(1/3)/x, x)","F",0
125,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)/x**2,x)","\int \frac{\sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}}{x^{2}}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(1/3)/x**2, x)","F",0
126,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(1/3)/x**3,x)","\int \frac{\sqrt[3]{\frac{x + 1}{\sqrt{1 - x^{2}}}}}{x^{3}}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(1/3)/x**3, x)","F",0
127,-1,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)*x**m,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
128,-1,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)*x**2,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
129,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)*x,x)","\int x \left(\frac{x + 1}{\sqrt{1 - x^{2}}}\right)^{\frac{2}{3}}\, dx"," ",0,"Integral(x*((x + 1)/sqrt(1 - x**2))**(2/3), x)","F",0
130,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3),x)","\int \left(\frac{x + 1}{\sqrt{1 - x^{2}}}\right)^{\frac{2}{3}}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(2/3), x)","F",0
131,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)/x,x)","\int \frac{\left(\frac{x + 1}{\sqrt{1 - x^{2}}}\right)^{\frac{2}{3}}}{x}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(2/3)/x, x)","F",0
132,0,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)/x**2,x)","\int \frac{\left(\frac{x + 1}{\sqrt{1 - x^{2}}}\right)^{\frac{2}{3}}}{x^{2}}\, dx"," ",0,"Integral(((x + 1)/sqrt(1 - x**2))**(2/3)/x**2, x)","F",0
133,-1,0,0,0.000000," ","integrate(((1+x)/(-x**2+1)**(1/2))**(2/3)/x**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
134,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)*x**m,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
135,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)*x**2,x)","\int x^{2} \sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(x**2*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4), x)","F",0
136,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)*x,x)","\int x \sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(x*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4), x)","F",0
137,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4),x)","\int \sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4), x)","F",0
138,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)/x,x)","\int \frac{\sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4)/x, x)","F",0
139,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)/x**2,x)","\int \frac{\sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{2}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4)/x**2, x)","F",0
140,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)/x**3,x)","\int \frac{\sqrt[4]{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{x^{3}}\, dx"," ",0,"Integral(((a*x + 1)/sqrt(-a**2*x**2 + 1))**(1/4)/x**3, x)","F",0
141,0,0,0,0.000000," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*x**m,x)","\int \frac{x^{m} \left(a x + 1\right)^{2}}{\left(a x - 1\right)^{2}}\, dx"," ",0,"Integral(x**m*(a*x + 1)**2/(a*x - 1)**2, x)","F",0
142,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**m,x)","\int \frac{x^{m} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**m*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
143,1,99,0,3.621475," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m,x)","\frac{a m x^{2} x^{m} \Phi\left(a x, 1, m + 2\right) \Gamma\left(m + 2\right)}{\Gamma\left(m + 3\right)} + \frac{2 a x^{2} x^{m} \Phi\left(a x, 1, m + 2\right) \Gamma\left(m + 2\right)}{\Gamma\left(m + 3\right)} + \frac{m x x^{m} \Phi\left(a x, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)} + \frac{x x^{m} \Phi\left(a x, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)}"," ",0,"a*m*x**2*x**m*lerchphi(a*x, 1, m + 2)*gamma(m + 2)/gamma(m + 3) + 2*a*x**2*x**m*lerchphi(a*x, 1, m + 2)*gamma(m + 2)/gamma(m + 3) + m*x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma(m + 2) + x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma(m + 2)","B",0
144,1,97,0,3.219066," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m,x)","\frac{a x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"a*x**2*x**m*gamma(m/2 + 1)*hyper((1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 2)) + x*x**m*gamma(m/2 + 1/2)*hyper((1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3/2))","C",0
145,0,0,0,0.000000," ","integrate(x**m/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
146,1,119,0,3.172010," ","integrate(x**m/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{a m x^{2} x^{m} \Phi\left(a x e^{i \pi}, 1, m + 2\right) \Gamma\left(m + 2\right)}{\Gamma\left(m + 3\right)} - \frac{2 a x^{2} x^{m} \Phi\left(a x e^{i \pi}, 1, m + 2\right) \Gamma\left(m + 2\right)}{\Gamma\left(m + 3\right)} + \frac{m x x^{m} \Phi\left(a x e^{i \pi}, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)} + \frac{x x^{m} \Phi\left(a x e^{i \pi}, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)}"," ",0,"-a*m*x**2*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 2)*gamma(m + 2)/gamma(m + 3) - 2*a*x**2*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 2)*gamma(m + 2)/gamma(m + 3) + m*x*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 1)*gamma(m + 1)/gamma(m + 2) + x*x**m*lerchphi(a*x*exp_polar(I*pi), 1, m + 1)*gamma(m + 1)/gamma(m + 2)","C",0
147,0,0,0,0.000000," ","integrate(x**m/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{m} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**m*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
148,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m,x)","\int x^{m} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**m*exp(n*atanh(a*x)), x)","F",0
149,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3,x)","\int x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**3*exp(n*atanh(a*x)), x)","F",0
150,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2,x)","\int x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**2*exp(n*atanh(a*x)), x)","F",0
151,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x,x)","\int x e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x*exp(n*atanh(a*x)), x)","F",0
152,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x)),x)","\int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x)), x)","F",0
153,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x,x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/x, x)","F",0
154,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2,x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/x**2, x)","F",0
155,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**3,x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{3}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/x**3, x)","F",0
156,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**4,x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{4}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/x**4, x)","F",0
157,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**p,x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{p} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
158,1,226,0,8.979171," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4,x)","\begin{cases} \frac{3 c^{4} \sqrt{- a^{2} x^{2} + 1} + 2 c^{4} \left(\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + 2 c^{4} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 3 c^{4} \left(\begin{cases} \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} + \frac{2 \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{4} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((3*c**4*sqrt(-a**2*x**2 + 1) + 2*c**4*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) + 2*c**4*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) - 3*c**4*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x)/8, (a*x > -1) & (a*x < 1))) + c**4*Piecewise((-(-a**2*x**2 + 1)**(5/2)/5 + 2*(-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**4*asin(a*x))/a, Ne(a, 0)), (c**4*x, True))","A",0
159,1,134,0,6.370691," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3,x)","\begin{cases} \frac{2 c^{3} \sqrt{- a^{2} x^{2} + 1} + 2 c^{3} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{3} \left(\begin{cases} \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{3} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{3} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((2*c**3*sqrt(-a**2*x**2 + 1) + 2*c**3*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) - c**3*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x)/8, (a*x > -1) & (a*x < 1))) + c**3*asin(a*x))/a, Ne(a, 0)), (c**3*x, True))","A",0
160,1,102,0,5.384325," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2,x)","\begin{cases} \frac{c^{2} \sqrt{- a^{2} x^{2} + 1} - c^{2} \left(\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{2} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{2} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{2} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((c**2*sqrt(-a**2*x**2 + 1) - c**2*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) + c**2*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**2*asin(a*x))/a, Ne(a, 0)), (c**2*x, True))","A",0
161,1,37,0,3.644541," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c),x)","\begin{cases} \frac{c \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c x & \text{otherwise} \end{cases}"," ",0,"Piecewise((c*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1)))/a, Ne(a, 0)), (c*x, True))","A",0
162,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
163,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
164,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
165,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**4,x)","\frac{\int \frac{a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
166,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**5,x)","- \frac{\int \frac{a x}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}}"," ",0,"-(Integral(a*x/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x**2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x**2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**5","F",0
167,1,126,0,0.955076," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**p,x)","\begin{cases} c^{p} x & \text{for}\: a = 0 \\\frac{a x \log{\left(x - \frac{1}{a} \right)}}{a^{2} c x - a c} - \frac{\log{\left(x - \frac{1}{a} \right)}}{a^{2} c x - a c} - \frac{2}{a^{2} c x - a c} & \text{for}\: p = -1 \\- x - \frac{2 \log{\left(x - \frac{1}{a} \right)}}{a} & \text{for}\: p = 0 \\- \frac{a p x \left(- a c x + c\right)^{p}}{a p^{2} + a p} - \frac{p \left(- a c x + c\right)^{p}}{a p^{2} + a p} - \frac{2 \left(- a c x + c\right)^{p}}{a p^{2} + a p} & \text{otherwise} \end{cases}"," ",0,"Piecewise((c**p*x, Eq(a, 0)), (a*x*log(x - 1/a)/(a**2*c*x - a*c) - log(x - 1/a)/(a**2*c*x - a*c) - 2/(a**2*c*x - a*c), Eq(p, -1)), (-x - 2*log(x - 1/a)/a, Eq(p, 0)), (-a*p*x*(-a*c*x + c)**p/(a*p**2 + a*p) - p*(-a*c*x + c)**p/(a*p**2 + a*p) - 2*(-a*c*x + c)**p/(a*p**2 + a*p), True))","A",0
168,1,66,0,0.088509," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**5,x)","\frac{a^{5} c^{5} x^{6}}{6} - \frac{3 a^{4} c^{5} x^{5}}{5} + \frac{a^{3} c^{5} x^{4}}{2} + \frac{2 a^{2} c^{5} x^{3}}{3} - \frac{3 a c^{5} x^{2}}{2} + c^{5} x"," ",0,"a**5*c**5*x**6/6 - 3*a**4*c**5*x**5/5 + a**3*c**5*x**4/2 + 2*a**2*c**5*x**3/3 - 3*a*c**5*x**2/2 + c**5*x","B",0
169,1,36,0,0.078486," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**4,x)","- \frac{a^{4} c^{4} x^{5}}{5} + \frac{a^{3} c^{4} x^{4}}{2} - a c^{4} x^{2} + c^{4} x"," ",0,"-a**4*c**4*x**5/5 + a**3*c**4*x**4/2 - a*c**4*x**2 + c**4*x","A",0
170,1,37,0,0.076871," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**3,x)","\frac{a^{3} c^{3} x^{4}}{4} - \frac{a^{2} c^{3} x^{3}}{3} - \frac{a c^{3} x^{2}}{2} + c^{3} x"," ",0,"a**3*c**3*x**4/4 - a**2*c**3*x**3/3 - a*c**3*x**2/2 + c**3*x","A",0
171,1,15,0,0.067812," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**2,x)","- \frac{a^{2} c^{2} x^{3}}{3} + c^{2} x"," ",0,"-a**2*c**2*x**3/3 + c**2*x","A",0
172,1,10,0,0.062514," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c),x)","\frac{a c x^{2}}{2} + c x"," ",0,"a*c*x**2/2 + c*x","A",0
173,1,20,0,0.137577," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c),x)","- \frac{2}{a^{2} c x - a c} + \frac{\log{\left(a x - 1 \right)}}{a c}"," ",0,"-2/(a**2*c*x - a*c) + log(a*x - 1)/(a*c)","A",0
174,1,22,0,0.181014," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**2,x)","\frac{x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}}"," ",0,"x/(a**2*c**2*x**2 - 2*a*c**2*x + c**2)","B",0
175,1,48,0,0.233257," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**3,x)","\frac{- 3 a x - 1}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}}"," ",0,"(-3*a*x - 1)/(6*a**4*c**3*x**3 - 18*a**3*c**3*x**2 + 18*a**2*c**3*x - 6*a*c**3)","A",0
176,1,61,0,0.290293," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**4,x)","- \frac{- 2 a x - 1}{6 a^{5} c^{4} x^{4} - 24 a^{4} c^{4} x^{3} + 36 a^{3} c^{4} x^{2} - 24 a^{2} c^{4} x + 6 a c^{4}}"," ",0,"-(-2*a*x - 1)/(6*a**5*c**4*x**4 - 24*a**4*c**4*x**3 + 36*a**3*c**4*x**2 - 24*a**2*c**4*x + 6*a*c**4)","B",0
177,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**p,x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
178,1,459,0,22.992594," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**4,x)","- a^{5} c^{4} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + a^{4} c^{4} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - 2 a^{2} c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) - a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right)"," ",0,"-a**5*c**4*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + a**4*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + 2*a**3*c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) - 2*a**2*c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) - a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))","A",0
179,1,301,0,8.537335," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**3,x)","a^{4} c^{3} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - 2 a^{2} c^{3} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right)"," ",0,"a**4*c**3*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 2*a**2*c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))","A",0
180,1,221,0,14.794200," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**2,x)","- a^{3} c^{2} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right)"," ",0,"-a**3*c**2*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) - a**2*c**2*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))","A",0
181,1,165,0,10.289445," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c),x)","a^{2} c \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right)"," ",0,"a**2*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))","A",0
182,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c),x)","- \frac{\int \frac{3 a x}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(3*a*x/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
183,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**2,x)","\frac{\int \frac{3 a x}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(3*a*x/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
184,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**3,x)","- \frac{\int \frac{3 a x}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(3*a*x/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
185,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**4,x)","\frac{\int \frac{3 a x}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(3*a*x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
186,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**5,x)","- \frac{\int \frac{3 a x}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 5 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}}"," ",0,"-(Integral(3*a*x/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**5","F",0
187,1,541,0,3.052628," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**p,x)","\begin{cases} c^{p} x & \text{for}\: a = 0 \\- \frac{a^{2} x^{2} \log{\left(x - \frac{1}{a} \right)}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{2 a x \log{\left(x - \frac{1}{a} \right)}}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{4 a x}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac{\log{\left(x - \frac{1}{a} \right)}}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac{2}{a^{3} c x^{2} - 2 a^{2} c x + a c} & \text{for}\: p = -1 \\\frac{a^{2} x^{2}}{a^{2} x - a} + \frac{4 a x \log{\left(x - \frac{1}{a} \right)}}{a^{2} x - a} - \frac{a x}{a^{2} x - a} - \frac{4 \log{\left(x - \frac{1}{a} \right)}}{a^{2} x - a} - \frac{4}{a^{2} x - a} & \text{for}\: p = 0 \\- \frac{a c x^{2}}{2} - 3 c x - \frac{4 c \log{\left(x - \frac{1}{a} \right)}}{a} & \text{for}\: p = 1 \\\frac{a^{2} p^{2} x^{2} \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac{a^{2} p x^{2} \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{2 a p^{2} x \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{2 a p x \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} - \frac{4 a x \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{p^{2} \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{3 p \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} + \frac{4 \left(- a c x + c\right)^{p}}{a^{2} p^{3} x - a^{2} p x - a p^{3} + a p} & \text{otherwise} \end{cases}"," ",0,"Piecewise((c**p*x, Eq(a, 0)), (-a**2*x**2*log(x - 1/a)/(a**3*c*x**2 - 2*a**2*c*x + a*c) + 2*a*x*log(x - 1/a)/(a**3*c*x**2 - 2*a**2*c*x + a*c) + 4*a*x/(a**3*c*x**2 - 2*a**2*c*x + a*c) - log(x - 1/a)/(a**3*c*x**2 - 2*a**2*c*x + a*c) - 2/(a**3*c*x**2 - 2*a**2*c*x + a*c), Eq(p, -1)), (a**2*x**2/(a**2*x - a) + 4*a*x*log(x - 1/a)/(a**2*x - a) - a*x/(a**2*x - a) - 4*log(x - 1/a)/(a**2*x - a) - 4/(a**2*x - a), Eq(p, 0)), (-a*c*x**2/2 - 3*c*x - 4*c*log(x - 1/a)/a, Eq(p, 1)), (a**2*p**2*x**2*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) - a**2*p*x**2*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) + 2*a*p**2*x*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) + 2*a*p*x*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) - 4*a*x*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) + p**2*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) + 3*p*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p) + 4*(-a*c*x + c)**p/(a**2*p**3*x - a**2*p*x - a*p**3 + a*p), True))","A",0
188,1,63,0,0.100059," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**5,x)","- \frac{a^{5} c^{5} x^{6}}{6} + \frac{a^{4} c^{5} x^{5}}{5} + \frac{a^{3} c^{5} x^{4}}{2} - \frac{2 a^{2} c^{5} x^{3}}{3} - \frac{a c^{5} x^{2}}{2} + c^{5} x"," ",0,"-a**5*c**5*x**6/6 + a**4*c**5*x**5/5 + a**3*c**5*x**4/2 - 2*a**2*c**5*x**3/3 - a*c**5*x**2/2 + c**5*x","A",0
189,1,29,0,0.082405," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**4,x)","\frac{a^{4} c^{4} x^{5}}{5} - \frac{2 a^{2} c^{4} x^{3}}{3} + c^{4} x"," ",0,"a**4*c**4*x**5/5 - 2*a**2*c**4*x**3/3 + c**4*x","A",0
190,1,37,0,0.087069," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**3,x)","- \frac{a^{3} c^{3} x^{4}}{4} - \frac{a^{2} c^{3} x^{3}}{3} + \frac{a c^{3} x^{2}}{2} + c^{3} x"," ",0,"-a**3*c**3*x**4/4 - a**2*c**3*x**3/3 + a*c**3*x**2/2 + c**3*x","A",0
191,1,24,0,0.080409," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c)**2,x)","\frac{a^{2} c^{2} x^{3}}{3} + a c^{2} x^{2} + c^{2} x"," ",0,"a**2*c**2*x**3/3 + a*c**2*x**2 + c**2*x","A",0
192,1,26,0,0.126837," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a*c*x+c),x)","- \frac{a c x^{2}}{2} - 3 c x - \frac{4 c \log{\left(a x - 1 \right)}}{a}"," ",0,"-a*c*x**2/2 - 3*c*x - 4*c*log(a*x - 1)/a","A",0
193,1,37,0,0.223288," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a*c*x+c),x)","- \frac{- 4 a x + 2}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac{\log{\left(a x - 1 \right)}}{a c}"," ",0,"-(-4*a*x + 2)/(a**3*c*x**2 - 2*a**2*c*x + a*c) - log(a*x - 1)/(a*c)","A",0
194,1,51,0,0.264505," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a*c*x+c)**2,x)","\frac{- 3 a^{2} x^{2} - 1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}}"," ",0,"(-3*a**2*x**2 - 1)/(3*a**4*c**2*x**3 - 9*a**3*c**2*x**2 + 9*a**2*c**2*x - 3*a*c**2)","B",0
195,1,70,0,0.325175," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a*c*x+c)**3,x)","- \frac{- 3 a^{2} x^{2} - 2 a x - 1}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}}"," ",0,"-(-3*a**2*x**2 - 2*a*x - 1)/(6*a**5*c**3*x**4 - 24*a**4*c**3*x**3 + 36*a**3*c**3*x**2 - 24*a**2*c**3*x + 6*a*c**3)","A",0
196,1,80,0,0.378880," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a*c*x+c)**4,x)","\frac{- 5 a^{2} x^{2} - 5 a x - 2}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}}"," ",0,"(-5*a**2*x**2 - 5*a*x - 2)/(15*a**6*c**4*x**5 - 75*a**5*c**4*x**4 + 150*a**4*c**4*x**3 - 150*a**3*c**4*x**2 + 75*a**2*c**4*x - 15*a*c**4)","A",0
197,0,0,0,0.000000," ","integrate((-a*c*x+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{p} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
198,0,0,0,0.000000," ","integrate((-a*c*x+c)**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","- c^{3} \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a x + 1}\right)\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \left(- \frac{3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\right)\, dx + \int \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx\right)"," ",0,"-c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(-3*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(a**3*x**3*sqrt(-a**2*x**2 + 1)/(a*x + 1), x))","F",0
199,0,0,0,0.000000," ","integrate((-a*c*x+c)**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","c^{2} \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \left(- \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\right)\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx\right)"," ",0,"c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x + 1), x))","F",0
200,0,0,0,0.000000," ","integrate((-a*c*x+c)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","- c \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a x + 1}\right)\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a x + 1}\, dx\right)"," ",0,"-c*(Integral(-sqrt(-a**2*x**2 + 1)/(a*x + 1), x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a*x + 1), x))","F",0
201,1,44,0,4.558996," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c),x)","\frac{\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}}{c}"," ",0,"Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))/c","A",0
202,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**2,x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}}"," ",0,"Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 - a**2*x**2 - a*x + 1), x)/c**2","F",0
203,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**3,x)","- \frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}}"," ",0,"-Integral(sqrt(-a**2*x**2 + 1)/(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1), x)/c**3","F",0
204,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**4,x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx}{c^{4}}"," ",0,"Integral(sqrt(-a**2*x**2 + 1)/(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 + 2*a**2*x**2 - 3*a*x + 1), x)/c**4","F",0
205,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**5,x)","- \frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{6} x^{6} - 4 a^{5} x^{5} + 5 a^{4} x^{4} - 5 a^{2} x^{2} + 4 a x - 1}\, dx}{c^{5}}"," ",0,"-Integral(sqrt(-a**2*x**2 + 1)/(a**6*x**6 - 4*a**5*x**5 + 5*a**4*x**4 - 5*a**2*x**2 + 4*a*x - 1), x)/c**5","F",0
206,0,0,0,0.000000," ","integrate((-a*c*x+c)**p/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\left(- a c x + c\right)^{p}}{a x + 1}\right)\, dx - \int \frac{a x \left(- a c x + c\right)^{p}}{a x + 1}\, dx"," ",0,"-Integral(-(-a*c*x + c)**p/(a*x + 1), x) - Integral(a*x*(-a*c*x + c)**p/(a*x + 1), x)","F",0
207,1,68,0,0.171974," ","integrate((-a*c*x+c)**4/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{a^{4} c^{4} x^{5}}{5} + \frac{3 a^{3} c^{4} x^{4}}{2} - \frac{16 a^{2} c^{4} x^{3}}{3} + 13 a c^{4} x^{2} - 31 c^{4} x + \frac{32 c^{4} \log{\left(a x + 1 \right)}}{a}"," ",0,"-a**4*c**4*x**5/5 + 3*a**3*c**4*x**4/2 - 16*a**2*c**4*x**3/3 + 13*a*c**4*x**2 - 31*c**4*x + 32*c**4*log(a*x + 1)/a","A",0
208,1,56,0,0.151070," ","integrate((-a*c*x+c)**3/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{a^{3} c^{3} x^{4}}{4} - \frac{5 a^{2} c^{3} x^{3}}{3} + \frac{11 a c^{3} x^{2}}{2} - 15 c^{3} x + \frac{16 c^{3} \log{\left(a x + 1 \right)}}{a}"," ",0,"a**3*c**3*x**4/4 - 5*a**2*c**3*x**3/3 + 11*a*c**3*x**2/2 - 15*c**3*x + 16*c**3*log(a*x + 1)/a","A",0
209,1,41,0,0.132004," ","integrate((-a*c*x+c)**2/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{a^{2} c^{2} x^{3}}{3} + 2 a c^{2} x^{2} - 7 c^{2} x + \frac{8 c^{2} \log{\left(a x + 1 \right)}}{a}"," ",0,"-a**2*c**2*x**3/3 + 2*a*c**2*x**2 - 7*c**2*x + 8*c**2*log(a*x + 1)/a","A",0
210,1,24,0,0.113780," ","integrate((-a*c*x+c)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{a c x^{2}}{2} - 3 c x + \frac{4 c \log{\left(a x + 1 \right)}}{a}"," ",0,"a*c*x**2/2 - 3*c*x + 4*c*log(a*x + 1)/a","A",0
211,1,10,0,0.067812," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c),x)","\frac{\log{\left(a c x + c \right)}}{a c}"," ",0,"log(a*c*x + c)/(a*c)","A",0
212,1,22,0,0.144360," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**2,x)","- \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{2} - \frac{\log{\left(x + \frac{1}{a} \right)}}{2}}{a c^{2}}"," ",0,"-(log(x - 1/a)/2 - log(x + 1/a)/2)/(a*c**2)","B",0
213,1,39,0,0.221549," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**3,x)","- \frac{1}{2 a^{2} c^{3} x - 2 a c^{3}} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{4} + \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a c^{3}}"," ",0,"-1/(2*a**2*c**3*x - 2*a*c**3) + (-log(x - 1/a)/4 + log(x + 1/a)/4)/(a*c**3)","A",0
214,1,56,0,0.281248," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**4,x)","- \frac{a x - 2}{4 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 4 a c^{4}} - \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a c^{4}}"," ",0,"-(a*x - 2)/(4*a**3*c**4*x**2 - 8*a**2*c**4*x + 4*a*c**4) - (log(x - 1/a)/8 - log(x + 1/a)/8)/(a*c**4)","A",0
215,1,76,0,0.371963," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**5,x)","\frac{- 3 a^{2} x^{2} + 9 a x - 10}{24 a^{4} c^{5} x^{3} - 72 a^{3} c^{5} x^{2} + 72 a^{2} c^{5} x - 24 a c^{5}} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{16} + \frac{\log{\left(x + \frac{1}{a} \right)}}{16}}{a c^{5}}"," ",0,"(-3*a**2*x**2 + 9*a*x - 10)/(24*a**4*c**5*x**3 - 72*a**3*c**5*x**2 + 72*a**2*c**5*x - 24*a*c**5) + (-log(x - 1/a)/16 + log(x + 1/a)/16)/(a*c**5)","A",0
216,0,0,0,0.000000," ","integrate((-a*c*x+c)**p/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{p} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
217,0,0,0,0.000000," ","integrate((-a*c*x+c)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- c^{3} \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \left(- \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \frac{3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx\right)"," ",0,"-c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(3*a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**5*x**5*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))","F",0
218,0,0,0,0.000000," ","integrate((-a*c*x+c)**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","c^{2} \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx\right)"," ",0,"c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))","F",0
219,0,0,0,0.000000," ","integrate((-a*c*x+c)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- c \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx\right)"," ",0,"-c*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))","F",0
220,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c),x)","- \frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right)\, dx}{c}"," ",0,"-(Integral(sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x))/c","F",0
221,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**2,x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right)\, dx}{c^{2}}"," ",0,"(Integral(sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x))/c**2","F",0
222,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**3,x)","- \frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\right)\, dx}{c^{3}}"," ",0,"-(Integral(sqrt(-a**2*x**2 + 1)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x))/c**3","F",0
223,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**4,x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\right)\, dx}{c^{4}}"," ",0,"(Integral(sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 + 3*a**3*x**3 - 3*a**2*x**2 - a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 + 3*a**3*x**3 - 3*a**2*x**2 - a*x + 1), x))/c**4","F",0
224,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**5,x)","- \frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\right)\, dx}{c^{5}}"," ",0,"-(Integral(sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6 + 6*a**5*x**5 - 6*a**3*x**3 + 2*a**2*x**2 + 2*a*x - 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6 + 6*a**5*x**5 - 6*a**3*x**3 + 2*a**2*x**2 + 2*a*x - 1), x))/c**5","F",0
225,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**6,x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\, dx + \int \left(- \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\right)\, dx}{c^{6}}"," ",0,"(Integral(sqrt(-a**2*x**2 + 1)/(a**9*x**9 - 3*a**8*x**8 + 8*a**6*x**6 - 6*a**5*x**5 - 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**9*x**9 - 3*a**8*x**8 + 8*a**6*x**6 - 6*a**5*x**5 - 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x + 1), x))/c**6","F",0
226,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(9/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{9}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(9/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
227,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(7/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(7/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
228,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(5/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(5/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
229,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(3/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
230,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
231,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(1/2),x)","\int \frac{a x + 1}{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
232,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(3/2),x)","\int \frac{a x + 1}{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/((-c*(a*x - 1))**(3/2)*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
233,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(5/2),x)","\int \frac{a x + 1}{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/((-c*(a*x - 1))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
234,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(7/2),x)","\int \frac{a x + 1}{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/((-c*(a*x - 1))**(7/2)*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
235,1,172,0,31.788838," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(7/2),x)","c^{3} \left(\begin{cases} \sqrt{c} x & \text{for}\: a = 0 \\0 & \text{for}\: c = 0 \\- \frac{2 \left(- a c x + c\right)^{\frac{3}{2}}}{3 a c} & \text{otherwise} \end{cases}\right) - \frac{2 c \left(- \frac{c \left(- a c x + c\right)^{\frac{3}{2}}}{3} + \frac{\left(- a c x + c\right)^{\frac{5}{2}}}{5}\right)}{a} + \frac{2 \left(\frac{c^{2} \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{2 c \left(- a c x + c\right)^{\frac{5}{2}}}{5} + \frac{\left(- a c x + c\right)^{\frac{7}{2}}}{7}\right)}{a} + \frac{2 \left(- \frac{c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left(- a c x + c\right)^{\frac{5}{2}}}{5} - \frac{3 c \left(- a c x + c\right)^{\frac{7}{2}}}{7} + \frac{\left(- a c x + c\right)^{\frac{9}{2}}}{9}\right)}{a c}"," ",0,"c**3*Piecewise((sqrt(c)*x, Eq(a, 0)), (0, Eq(c, 0)), (-2*(-a*c*x + c)**(3/2)/(3*a*c), True)) - 2*c*(-c*(-a*c*x + c)**(3/2)/3 + (-a*c*x + c)**(5/2)/5)/a + 2*(c**2*(-a*c*x + c)**(3/2)/3 - 2*c*(-a*c*x + c)**(5/2)/5 + (-a*c*x + c)**(7/2)/7)/a + 2*(-c**3*(-a*c*x + c)**(3/2)/3 + 3*c**2*(-a*c*x + c)**(5/2)/5 - 3*c*(-a*c*x + c)**(7/2)/7 + (-a*c*x + c)**(9/2)/9)/(a*c)","A",0
236,1,76,0,14.058930," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(5/2),x)","c^{2} \left(\begin{cases} \sqrt{c} x & \text{for}\: a = 0 \\0 & \text{for}\: c = 0 \\- \frac{2 \left(- a c x + c\right)^{\frac{3}{2}}}{3 a c} & \text{otherwise} \end{cases}\right) + \frac{2 \left(\frac{c^{2} \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{2 c \left(- a c x + c\right)^{\frac{5}{2}}}{5} + \frac{\left(- a c x + c\right)^{\frac{7}{2}}}{7}\right)}{a c}"," ",0,"c**2*Piecewise((sqrt(c)*x, Eq(a, 0)), (0, Eq(c, 0)), (-2*(-a*c*x + c)**(3/2)/(3*a*c), True)) + 2*(c**2*(-a*c*x + c)**(3/2)/3 - 2*c*(-a*c*x + c)**(5/2)/5 + (-a*c*x + c)**(7/2)/7)/(a*c)","A",0
237,1,58,0,14.307842," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(3/2),x)","c \left(\begin{cases} \sqrt{c} x & \text{for}\: a = 0 \\0 & \text{for}\: c = 0 \\- \frac{2 \left(- a c x + c\right)^{\frac{3}{2}}}{3 a c} & \text{otherwise} \end{cases}\right) + \frac{2 \left(- \frac{c \left(- a c x + c\right)^{\frac{3}{2}}}{3} + \frac{\left(- a c x + c\right)^{\frac{5}{2}}}{5}\right)}{a c}"," ",0,"c*Piecewise((sqrt(c)*x, Eq(a, 0)), (0, Eq(c, 0)), (-2*(-a*c*x + c)**(3/2)/(3*a*c), True)) + 2*(-c*(-a*c*x + c)**(3/2)/3 + (-a*c*x + c)**(5/2)/5)/(a*c)","A",0
238,1,31,0,4.684394," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2),x)","- \frac{2 \left(2 c \sqrt{- a c x + c} - \frac{\left(- a c x + c\right)^{\frac{3}{2}}}{3}\right)}{a c}"," ",0,"-2*(2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c)","A",0
239,1,48,0,36.400679," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**(1/2),x)","\begin{cases} \frac{\frac{2}{\sqrt{- a c x + c}} - \frac{2 \left(- \frac{c}{\sqrt{- a c x + c}} - \sqrt{- a c x + c}\right)}{c}}{a} & \text{for}\: a \neq 0 \\\frac{x}{\sqrt{c}} & \text{otherwise} \end{cases}"," ",0,"Piecewise(((2/sqrt(-a*c*x + c) - 2*(-c/sqrt(-a*c*x + c) - sqrt(-a*c*x + c))/c)/a, Ne(a, 0)), (x/sqrt(c), True))","A",0
240,1,29,0,41.879944," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**(3/2),x)","\frac{4}{3 a \left(- a c x + c\right)^{\frac{3}{2}}} - \frac{2}{a c \sqrt{- a c x + c}}"," ",0,"4/(3*a*(-a*c*x + c)**(3/2)) - 2/(a*c*sqrt(-a*c*x + c))","A",0
241,1,31,0,92.029084," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**(5/2),x)","\frac{4}{5 a \left(- a c x + c\right)^{\frac{5}{2}}} - \frac{2}{3 a c \left(- a c x + c\right)^{\frac{3}{2}}}"," ",0,"4/(5*a*(-a*c*x + c)**(5/2)) - 2/(3*a*c*(-a*c*x + c)**(3/2))","A",0
242,1,31,0,61.976876," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a*c*x+c)**(7/2),x)","\frac{4}{7 a \left(- a c x + c\right)^{\frac{7}{2}}} - \frac{2}{5 a c \left(- a c x + c\right)^{\frac{5}{2}}}"," ",0,"4/(7*a*(-a*c*x + c)**(7/2)) - 2/(5*a*c*(-a*c*x + c)**(5/2))","A",0
243,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(9/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{9}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(9/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
244,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(7/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
245,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(5/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(5/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
246,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(3/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
247,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
248,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(1/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
249,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(3/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-c*(a*x - 1))**(3/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
250,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(5/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-c*(a*x - 1))**(5/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
251,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(7/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-c*(a*x - 1))**(7/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
252,0,0,0,0.000000," ","integrate((-a*c*x+c)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{9}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(a*x - 1))**(9/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
253,0,0,0,0.000000," ","integrate((-a*c*x+c)**(7/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(a*x - 1))**(7/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
254,0,0,0,0.000000," ","integrate((-a*c*x+c)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(a*x - 1))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
255,0,0,0,0.000000," ","integrate((-a*c*x+c)**(3/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(a*x - 1))**(3/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
256,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
257,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(-c*(a*x - 1))*(a*x + 1)), x)","F",0
258,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(3/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1))**(3/2)*(a*x + 1)), x)","F",0
259,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(5/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1))**(5/2)*(a*x + 1)), x)","F",0
260,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**(7/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1))**(7/2)*(a*x + 1)), x)","F",0
261,1,128,0,128.958352," ","integrate((-a*c*x+c)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{32 \sqrt{2} c^{4} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{a \sqrt{- c}} + \frac{32 c^{3} \sqrt{- a c x + c}}{a} + \frac{16 c^{2} \left(- a c x + c\right)^{\frac{3}{2}}}{3 a} + \frac{8 c \left(- a c x + c\right)^{\frac{5}{2}}}{5 a} + \frac{4 \left(- a c x + c\right)^{\frac{7}{2}}}{7 a} + \frac{2 \left(- a c x + c\right)^{\frac{9}{2}}}{9 a c}"," ",0,"32*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(a*sqrt(-c)) + 32*c**3*sqrt(-a*c*x + c)/a + 16*c**2*(-a*c*x + c)**(3/2)/(3*a) + 8*c*(-a*c*x + c)**(5/2)/(5*a) + 4*(-a*c*x + c)**(7/2)/(7*a) + 2*(-a*c*x + c)**(9/2)/(9*a*c)","A",0
262,1,109,0,76.553556," ","integrate((-a*c*x+c)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{16 \sqrt{2} c^{3} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{a \sqrt{- c}} + \frac{16 c^{2} \sqrt{- a c x + c}}{a} + \frac{8 c \left(- a c x + c\right)^{\frac{3}{2}}}{3 a} + \frac{4 \left(- a c x + c\right)^{\frac{5}{2}}}{5 a} + \frac{2 \left(- a c x + c\right)^{\frac{7}{2}}}{7 a c}"," ",0,"16*sqrt(2)*c**3*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(a*sqrt(-c)) + 16*c**2*sqrt(-a*c*x + c)/a + 8*c*(-a*c*x + c)**(3/2)/(3*a) + 4*(-a*c*x + c)**(5/2)/(5*a) + 2*(-a*c*x + c)**(7/2)/(7*a*c)","A",0
263,1,90,0,61.077888," ","integrate((-a*c*x+c)**(3/2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{8 \sqrt{2} c^{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{a \sqrt{- c}} + \frac{8 c \sqrt{- a c x + c}}{a} + \frac{4 \left(- a c x + c\right)^{\frac{3}{2}}}{3 a} + \frac{2 \left(- a c x + c\right)^{\frac{5}{2}}}{5 a c}"," ",0,"8*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(a*sqrt(-c)) + 8*c*sqrt(-a*c*x + c)/a + 4*(-a*c*x + c)**(3/2)/(3*a) + 2*(-a*c*x + c)**(5/2)/(5*a*c)","A",0
264,1,75,0,8.099006," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{2 \left(- \frac{2 \sqrt{2} c^{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 c \sqrt{- a c x + c} - \frac{\left(- a c x + c\right)^{\frac{3}{2}}}{3}\right)}{a c}"," ",0,"-2*(-2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c)","A",0
265,1,58,0,34.187420," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(1/2),x)","\frac{2 \sqrt{- a c x + c}}{a c} + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{\sqrt{- \frac{1}{c}} \sqrt{- a c x + c}} \right)}}{a c \sqrt{- \frac{1}{c}}}"," ",0,"2*sqrt(-a*c*x + c)/(a*c) + 2*sqrt(2)*atan(sqrt(2)/(sqrt(-1/c)*sqrt(-a*c*x + c)))/(a*c*sqrt(-1/c))","A",0
266,1,39,0,53.995666," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(3/2),x)","\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{a c \sqrt{- c}}"," ",0,"sqrt(2)*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(a*c*sqrt(-c))","A",0
267,1,60,0,31.298403," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(5/2),x)","\frac{1}{a c^{2} \sqrt{- a c x + c}} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{2 a c^{2} \sqrt{- c}}"," ",0,"1/(a*c**2*sqrt(-a*c*x + c)) + sqrt(2)*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(2*a*c**2*sqrt(-c))","A",0
268,1,80,0,66.364090," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(7/2),x)","\frac{1}{3 a c^{2} \left(- a c x + c\right)^{\frac{3}{2}}} + \frac{1}{2 a c^{3} \sqrt{- a c x + c}} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{4 a c^{3} \sqrt{- c}}"," ",0,"1/(3*a*c**2*(-a*c*x + c)**(3/2)) + 1/(2*a*c**3*sqrt(-a*c*x + c)) + sqrt(2)*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(4*a*c**3*sqrt(-c))","A",0
269,1,99,0,40.713395," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(9/2),x)","\frac{1}{5 a c^{2} \left(- a c x + c\right)^{\frac{5}{2}}} + \frac{1}{6 a c^{3} \left(- a c x + c\right)^{\frac{3}{2}}} + \frac{1}{4 a c^{4} \sqrt{- a c x + c}} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{8 a c^{4} \sqrt{- c}}"," ",0,"1/(5*a*c**2*(-a*c*x + c)**(5/2)) + 1/(6*a*c**3*(-a*c*x + c)**(3/2)) + 1/(4*a*c**4*sqrt(-a*c*x + c)) + sqrt(2)*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/(8*a*c**4*sqrt(-c))","A",0
270,0,0,0,0.000000," ","integrate((-a*c*x+c)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(5/2)*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
271,0,0,0,0.000000," ","integrate((-a*c*x+c)**(3/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-c*(a*x - 1))**(3/2)*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
272,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
273,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(1/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(sqrt(-c*(a*x - 1))*(a*x + 1)**3), x)","F",0
274,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1))**(3/2)*(a*x + 1)**3), x)","F",0
275,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(5/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1))**(5/2)*(a*x + 1)**3), x)","F",0
276,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(7/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1))**(7/2)*(a*x + 1)**3), x)","F",0
277,-1,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**(9/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
278,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
279,-2,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(5/2),x)","\text{Exception raised: HeuristicGCDFailed}"," ",0,"Exception raised: HeuristicGCDFailed","F(-2)",0
280,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
281,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(1/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
282,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/sqrt(-c*(a*x - 1)), x)","F",0
283,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(-c*(a*x - 1))**(3/2), x)","F",0
284,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
285,-2,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**(7/2),x)","\text{Exception raised: HeuristicGCDFailed}"," ",0,"Exception raised: HeuristicGCDFailed","F(-2)",0
286,1,192,0,6.525193," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4*(-a*c*x+c),x)","c \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right)"," ",0,"c*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True))","A",0
287,1,66,0,0.698676," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a*c*x+c),x)","\begin{cases} \frac{c x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{c x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 c \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{c x^{4}}{4} & \text{otherwise} \end{cases}"," ",0,"Piecewise((c*x**4*sqrt(-a**2*x**2 + 1)/5 - c*x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*c*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (c*x**4/4, True))","A",0
288,1,150,0,4.879361," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c),x)","c \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"c*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True))","A",0
289,1,42,0,0.406088," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c),x)","\begin{cases} \frac{c x^{2} \sqrt{- a^{2} x^{2} + 1}}{3} - \frac{c \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} & \text{for}\: a \neq 0 \\\frac{c x^{2}}{2} & \text{otherwise} \end{cases}"," ",0,"Piecewise((c*x**2*sqrt(-a**2*x**2 + 1)/3 - c*sqrt(-a**2*x**2 + 1)/(3*a**2), Ne(a, 0)), (c*x**2/2, True))","A",0
290,1,37,0,3.593931," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c),x)","\begin{cases} \frac{c \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c x & \text{otherwise} \end{cases}"," ",0,"Piecewise((c*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1)))/a, Ne(a, 0)), (c*x, True))","A",0
291,1,66,0,10.265305," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)/x,x)","\frac{a^{2} c \left(\begin{cases} - x^{2} & \text{for}\: a^{2} = 0 \\\frac{2 \sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right)}{2} - \frac{c \left(- \log{\left(-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)} + \log{\left(1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)}\right)}{2}"," ",0,"a**2*c*Piecewise((-x**2, Eq(a**2, 0)), (2*sqrt(-a**2*x**2 + 1)/a**2, True))/2 - c*(-log(-1 + 1/sqrt(-a**2*x**2 + 1)) + log(1 + 1/sqrt(-a**2*x**2 + 1)))/2","A",0
292,1,88,0,3.425095," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)/x**2,x)","c \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)"," ",0,"c*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))","A",0
293,1,73,0,11.450212," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)/x**3,x)","- a^{2} c \left(\frac{\log{\left(\sqrt{- a^{2} x^{2} + 1} - 1 \right)}}{4} - \frac{\log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)}}{4} - \frac{1}{4 \left(\sqrt{- a^{2} x^{2} + 1} + 1\right)} - \frac{1}{4 \left(\sqrt{- a^{2} x^{2} + 1} - 1\right)}\right)"," ",0,"-a**2*c*(log(sqrt(-a**2*x**2 + 1) - 1)/4 - log(sqrt(-a**2*x**2 + 1) + 1)/4 - 1/(4*(sqrt(-a**2*x**2 + 1) + 1)) - 1/(4*(sqrt(-a**2*x**2 + 1) - 1)))","A",0
294,1,90,0,3.233777," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)/x**4,x)","c \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"c*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))","A",0
295,1,486,0,10.356539," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a*c*x+c)**2,x)","a^{3} c^{2} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) - a**2*c**2*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) - a*c**2*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + c**2*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True))","A",0
296,1,374,0,7.524225," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c)**2,x)","a^{3} c^{2} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) - a**2*c**2*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - a*c**2*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + c**2*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))","C",0
297,1,330,0,7.168649," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c)**2,x)","a^{3} c^{2} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - a**2*c**2*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) - a*c**2*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))","A",0
298,1,102,0,5.298064," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2,x)","\begin{cases} \frac{c^{2} \sqrt{- a^{2} x^{2} + 1} - c^{2} \left(\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{2} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{2} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{2} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((c**2*sqrt(-a**2*x**2 + 1) - c**2*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) + c**2*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**2*asin(a*x))/a, Ne(a, 0)), (c**2*x, True))","A",0
299,1,201,0,11.933968," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x,x)","a^{3} c^{2} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) - a**2*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - a*c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))","C",0
300,1,153,0,4.596965," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**2,x)","a^{3} c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - a c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - a**2*c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - a*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))","C",0
301,1,226,0,5.714498," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**3,x)","a^{3} c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - a**2*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - a*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))","C",0
302,1,270,0,6.260323," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**4,x)","a^{3} c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - a**2*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - a*c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))","C",0
303,1,415,0,7.981840," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**5,x)","a^{3} c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - a**2*c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) - a*c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + c**2*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
304,1,522,0,8.614102," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**6,x)","a^{3} c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) - a**2*c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) - a*c**2*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c**2*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))","C",0
305,1,644,0,11.609933," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**7,x)","a^{3} c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{5 a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} + \frac{5 a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{5 a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{5 i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} - \frac{5 i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{5 i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) - a**2*c**2*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) - a*c**2*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True)) + c**2*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))","C",0
306,1,512,0,11.220663," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a*c*x+c)**3,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7 a^{2}} - \frac{6 x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{35 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{35 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-x**6*sqrt(-a**2*x**2 + 1)/(7*a**2) - 6*x**4*sqrt(-a**2*x**2 + 1)/(35*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(35*a**6) - 16*sqrt(-a**2*x**2 + 1)/(35*a**8), Ne(a, 0)), (x**8/8, True)) + 2*a**3*c**3*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) - 2*a*c**3*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + c**3*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True))","A",0
307,1,423,0,9.476196," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c)**3,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) + 2*a**3*c**3*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) - 2*a*c**3*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))","C",0
308,1,355,0,7.712163," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c)**3,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 2*a**3*c**3*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 2*a*c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))","A",0
309,1,134,0,6.395735," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3,x)","\begin{cases} \frac{2 c^{3} \sqrt{- a^{2} x^{2} + 1} + 2 c^{3} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{3} \left(\begin{cases} \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{3} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{3} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((2*c**3*sqrt(-a**2*x**2 + 1) + 2*c**3*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) - c**3*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x)/8, (a*x > -1) & (a*x < 1))) + c**3*asin(a*x))/a, Ne(a, 0)), (c**3*x, True))","A",0
310,1,226,0,12.535445," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**3*c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) - 2*a*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))","C",0
311,1,199,0,5.716232," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**2,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a**3*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 2*a*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))","C",0
312,1,228,0,5.006361," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**3,x)","- a^{4} c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + 2*a**3*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 2*a*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))","C",0
313,1,279,0,5.849478," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**4,x)","- a^{4} c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*a**3*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - 2*a*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))","C",0
314,1,347,0,7.337480," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**5,x)","- a^{4} c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**3*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - 2*a*c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
315,1,476,0,8.257361," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**3/x**6,x)","- a^{4} c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 2*a**3*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) - 2*a*c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c**3*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))","C",0
316,1,842,0,18.306427," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a*c*x+c)**4,x)","a^{5} c^{4} \left(\begin{cases} - \frac{i x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{7}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{7 i x^{5}}{192 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{35 i x^{3}}{384 a^{6} \sqrt{a^{2} x^{2} - 1}} + \frac{35 i x}{128 a^{8} \sqrt{a^{2} x^{2} - 1}} - \frac{35 i \operatorname{acosh}{\left(a x \right)}}{128 a^{9}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{7}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{7 x^{5}}{192 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{35 x^{3}}{384 a^{6} \sqrt{- a^{2} x^{2} + 1}} - \frac{35 x}{128 a^{8} \sqrt{- a^{2} x^{2} + 1}} + \frac{35 \operatorname{asin}{\left(a x \right)}}{128 a^{9}} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7 a^{2}} - \frac{6 x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{35 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{35 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-I*x**9/(8*sqrt(a**2*x**2 - 1)) - I*x**7/(48*a**2*sqrt(a**2*x**2 - 1)) - 7*I*x**5/(192*a**4*sqrt(a**2*x**2 - 1)) - 35*I*x**3/(384*a**6*sqrt(a**2*x**2 - 1)) + 35*I*x/(128*a**8*sqrt(a**2*x**2 - 1)) - 35*I*acosh(a*x)/(128*a**9), Abs(a**2*x**2) > 1), (x**9/(8*sqrt(-a**2*x**2 + 1)) + x**7/(48*a**2*sqrt(-a**2*x**2 + 1)) + 7*x**5/(192*a**4*sqrt(-a**2*x**2 + 1)) + 35*x**3/(384*a**6*sqrt(-a**2*x**2 + 1)) - 35*x/(128*a**8*sqrt(-a**2*x**2 + 1)) + 35*asin(a*x)/(128*a**9), True)) - 3*a**4*c**4*Piecewise((-x**6*sqrt(-a**2*x**2 + 1)/(7*a**2) - 6*x**4*sqrt(-a**2*x**2 + 1)/(35*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(35*a**6) - 16*sqrt(-a**2*x**2 + 1)/(35*a**8), Ne(a, 0)), (x**8/8, True)) + 2*a**3*c**4*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) + 2*a**2*c**4*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) - 3*a*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True))","A",0
317,1,683,0,13.185005," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c)**4,x)","a^{5} c^{4} \left(\begin{cases} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7 a^{2}} - \frac{6 x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{35 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{35 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-x**6*sqrt(-a**2*x**2 + 1)/(7*a**2) - 6*x**4*sqrt(-a**2*x**2 + 1)/(35*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(35*a**6) - 16*sqrt(-a**2*x**2 + 1)/(35*a**8), Ne(a, 0)), (x**8/8, True)) - 3*a**4*c**4*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) + 2*a**3*c**4*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 2*a**2*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 3*a*c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))","C",0
318,1,617,0,12.485102," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c)**4,x)","a^{5} c^{4} \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) - 3*a**4*c**4*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 2*a**3*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + 2*a**2*c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) - 3*a*c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))","A",0
319,1,226,0,8.976626," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4,x)","\begin{cases} \frac{3 c^{4} \sqrt{- a^{2} x^{2} + 1} + 2 c^{4} \left(\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + 2 c^{4} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 3 c^{4} \left(\begin{cases} \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} + \frac{2 \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{4} \operatorname{asin}{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((3*c**4*sqrt(-a**2*x**2 + 1) + 2*c**4*Piecewise((-a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) + 2*c**4*Piecewise(((-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) - 3*c**4*Piecewise((a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 - a*x*sqrt(-a**2*x**2 + 1)/2 + 3*asin(a*x)/8, (a*x > -1) & (a*x < 1))) + c**4*Piecewise((-(-a**2*x**2 + 1)**(5/2)/5 + 2*(-a**2*x**2 + 1)**(3/2)/3 - sqrt(-a**2*x**2 + 1), (a*x > -1) & (a*x < 1))) + c**4*asin(a*x))/a, Ne(a, 0)), (c**4*x, True))","A",0
320,1,420,0,19.056719," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x,x)","a^{5} c^{4} \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) - 3*a**4*c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**3*c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a**2*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*a*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))","C",0
321,1,306,0,6.682940," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**2,x)","a^{5} c^{4} \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) - 3*a**4*c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 2*a**3*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + 2*a**2*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 3*a*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))","C",0
322,1,357,0,7.720618," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**3,x)","a^{5} c^{4} \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) - 3*a**4*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + 2*a**3*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*a**2*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - 3*a*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))","C",0
323,1,359,0,6.938130," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**4,x)","a^{5} c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*a**4*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*a**3*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**2*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - 3*a*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))","C",0
324,1,505,0,9.355706," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**5,x)","a^{5} c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 3*a**4*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 2*a**3*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 2*a**2*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) - 3*a*c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
325,1,607,0,10.102058," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**6,x)","a^{5} c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - 3*a**4*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 2*a**3*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + 2*a**2*c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) - 3*a*c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c**4*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))","C",0
326,1,801,0,13.507748," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**4/x**7,x)","a^{5} c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{4} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} - \frac{5 a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} + \frac{5 a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{5 a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{5 i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} - \frac{5 i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{5 i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) - 3*a**4*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + 2*a**3*c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + 2*a**2*c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) - 3*a*c**4*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True)) + c**4*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))","C",0
327,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c),x)","- \frac{\int \frac{x^{4}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(x**4/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
328,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a*c*x+c),x)","- \frac{\int \frac{x^{3}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(x**3/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
329,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a*c*x+c),x)","- \frac{\int \frac{x^{2}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(x**2/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
330,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a*c*x+c),x)","- \frac{\int \frac{x}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(x/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
331,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
332,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x**2*sqrt(-a**2*x**2 + 1) - x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x**2*sqrt(-a**2*x**2 + 1) - x*sqrt(-a**2*x**2 + 1)), x))/c","F",0
333,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x))/c","F",0
334,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x))/c","F",0
335,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a*c*x+c),x)","- \frac{\int \frac{a x}{a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"-(Integral(a*x/(a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x))/c","F",0
336,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c)**2,x)","\frac{\int \frac{x^{4}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**4/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
337,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a*c*x+c)**2,x)","\frac{\int \frac{x^{3}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**3/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
338,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a*c*x+c)**2,x)","\frac{\int \frac{x^{2}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**2/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
339,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a*c*x+c)**2,x)","\frac{\int \frac{x}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
340,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
341,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{2} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{2} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
342,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**4*sqrt(-a**2*x**2 + 1) - 2*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**4*sqrt(-a**2*x**2 + 1) - 2*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
343,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**5*sqrt(-a**2*x**2 + 1) - 2*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**5*sqrt(-a**2*x**2 + 1) - 2*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
344,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a*c*x+c)**2,x)","\frac{\int \frac{a x}{a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} - 2 a x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**2*x**6*sqrt(-a**2*x**2 + 1) - 2*a*x**5*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**2*x**6*sqrt(-a**2*x**2 + 1) - 2*a*x**5*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
345,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c)**3,x)","- \frac{\int \frac{x^{4}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(x**4/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
346,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a*c*x+c)**3,x)","- \frac{\int \frac{x^{3}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(x**3/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
347,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a*c*x+c)**3,x)","- \frac{\int \frac{x^{2}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(x**2/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
348,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a*c*x+c)**3,x)","- \frac{\int \frac{x}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(x/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
349,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
350,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + 3*a*x**2*sqrt(-a**2*x**2 + 1) - x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + 3*a*x**2*sqrt(-a**2*x**2 + 1) - x*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
351,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**5*sqrt(-a**2*x**2 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + 3*a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**5*sqrt(-a**2*x**2 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + 3*a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
352,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + 3*a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + 3*a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
353,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a*c*x+c)**3,x)","- \frac{\int \frac{a x}{a^{3} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{3} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a x^{5} \sqrt{- a^{2} x^{2} + 1} - x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"-(Integral(a*x/(a**3*x**7*sqrt(-a**2*x**2 + 1) - 3*a**2*x**6*sqrt(-a**2*x**2 + 1) + 3*a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**7*sqrt(-a**2*x**2 + 1) - 3*a**2*x**6*sqrt(-a**2*x**2 + 1) + 3*a*x**5*sqrt(-a**2*x**2 + 1) - x**4*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
354,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a*c*x+c)**4,x)","\frac{\int \frac{x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{6}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
355,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c)**4,x)","\frac{\int \frac{x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
356,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a*c*x+c)**4,x)","\frac{\int \frac{x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
357,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a*c*x+c)**4,x)","\frac{\int \frac{x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
358,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a*c*x+c)**4,x)","\frac{\int \frac{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
359,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**4,x)","\frac{\int \frac{a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
360,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a*c*x+c)**4,x)","\frac{\int \frac{a x}{a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 4*a**3*x**4*sqrt(-a**2*x**2 + 1) + 6*a**2*x**3*sqrt(-a**2*x**2 + 1) - 4*a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 4*a**3*x**4*sqrt(-a**2*x**2 + 1) + 6*a**2*x**3*sqrt(-a**2*x**2 + 1) - 4*a*x**2*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
361,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a*c*x+c)**4,x)","\frac{\int \frac{a x}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 4*a**3*x**5*sqrt(-a**2*x**2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2 + 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 4*a**3*x**5*sqrt(-a**2*x**2 + 1) + 6*a**2*x**4*sqrt(-a**2*x**2 + 1) - 4*a*x**3*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
362,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a*c*x+c)**4,x)","\frac{\int \frac{a x}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 4*a**3*x**6*sqrt(-a**2*x**2 + 1) + 6*a**2*x**5*sqrt(-a**2*x**2 + 1) - 4*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 4*a**3*x**6*sqrt(-a**2*x**2 + 1) + 6*a**2*x**5*sqrt(-a**2*x**2 + 1) - 4*a*x**4*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
363,1,37,0,0.321081," ","integrate((1+x)**2/(-x**2+1)**(1/2)*x,x)","- \frac{x^{2} \sqrt{1 - x^{2}}}{3} - x \sqrt{1 - x^{2}} - \frac{5 \sqrt{1 - x^{2}}}{3} + \operatorname{asin}{\left(x \right)}"," ",0,"-x**2*sqrt(1 - x**2)/3 - x*sqrt(1 - x**2) - 5*sqrt(1 - x**2)/3 + asin(x)","A",0
364,1,27,0,0.183707," ","integrate((1+x)**2/(-x**2+1)**(1/2),x)","- \frac{x \sqrt{1 - x^{2}}}{2} - 2 \sqrt{1 - x^{2}} + \frac{3 \operatorname{asin}{\left(x \right)}}{2}"," ",0,"-x*sqrt(1 - x**2)/2 - 2*sqrt(1 - x**2) + 3*asin(x)/2","A",0
365,1,54,0,0.603748," ","integrate((1+x)**3/(-x**2+1)**(1/2)*x,x)","- \frac{x^{3} \sqrt{1 - x^{2}}}{4} - x^{2} \sqrt{1 - x^{2}} - \frac{15 x \sqrt{1 - x^{2}}}{8} - 3 \sqrt{1 - x^{2}} + \frac{15 \operatorname{asin}{\left(x \right)}}{8}"," ",0,"-x**3*sqrt(1 - x**2)/4 - x**2*sqrt(1 - x**2) - 15*x*sqrt(1 - x**2)/8 - 3*sqrt(1 - x**2) + 15*asin(x)/8","A",0
366,1,44,0,0.331156," ","integrate((1+x)**3/(-x**2+1)**(1/2),x)","- \frac{x^{2} \sqrt{1 - x^{2}}}{3} - \frac{3 x \sqrt{1 - x^{2}}}{2} - \frac{11 \sqrt{1 - x^{2}}}{3} + \frac{5 \operatorname{asin}{\left(x \right)}}{2}"," ",0,"-x**2*sqrt(1 - x**2)/3 - 3*x*sqrt(1 - x**2)/2 - 11*sqrt(1 - x**2)/3 + 5*asin(x)/2","A",0
367,1,8,0,0.116631," ","integrate(x/(-x**2+1)**(1/2),x)","- \sqrt{1 - x^{2}}"," ",0,"-sqrt(1 - x**2)","A",0
368,1,2,0,0.117977," ","integrate(1/(-x**2+1)**(1/2),x)","\operatorname{asin}{\left(x \right)}"," ",0,"asin(x)","A",0
369,0,0,0,0.000000," ","integrate(1/(1+x)/(-x**2+1)**(1/2)*x,x)","\int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(x + 1\right)}\, dx"," ",0,"Integral(x/(sqrt(-(x - 1)*(x + 1))*(x + 1)), x)","F",0
370,0,0,0,0.000000," ","integrate(1/(1+x)/(-x**2+1)**(1/2),x)","\int \frac{1}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(x + 1\right)}\, dx"," ",0,"Integral(1/(sqrt(-(x - 1)*(x + 1))*(x + 1)), x)","F",0
371,0,0,0,0.000000," ","integrate((1+x)**(5/2)/(-x**2+1)**(1/2)*x,x)","\int \frac{x \left(x + 1\right)^{\frac{5}{2}}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*(x + 1)**(5/2)/sqrt(-(x - 1)*(x + 1)), x)","F",0
372,0,0,0,0.000000," ","integrate((1+x)**(5/2)/(-x**2+1)**(1/2),x)","\int \frac{\left(x + 1\right)^{\frac{5}{2}}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral((x + 1)**(5/2)/sqrt(-(x - 1)*(x + 1)), x)","F",0
373,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(3/2)*x,x)","\int \frac{x \left(1 - x\right)^{\frac{3}{2}} \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*(1 - x)**(3/2)*(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
374,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(3/2),x)","\int \frac{\left(1 - x\right)^{\frac{3}{2}} \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral((1 - x)**(3/2)*(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
375,0,0,0,0.000000," ","integrate((1+x)**(3/2)/(-x**2+1)**(1/2)*x,x)","\int \frac{x \left(x + 1\right)^{\frac{3}{2}}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*(x + 1)**(3/2)/sqrt(-(x - 1)*(x + 1)), x)","F",0
376,0,0,0,0.000000," ","integrate((1+x)**(3/2)/(-x**2+1)**(1/2),x)","\int \frac{\left(x + 1\right)^{\frac{3}{2}}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral((x + 1)**(3/2)/sqrt(-(x - 1)*(x + 1)), x)","F",0
377,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(1/2)*x,x)","\int \frac{x \sqrt{1 - x} \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(1 - x)*(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
378,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(1/2),x)","\int \frac{\sqrt{1 - x} \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(sqrt(1 - x)*(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
379,0,0,0,0.000000," ","integrate((1+x)**(1/2)/(-x**2+1)**(1/2)*x,x)","\int \frac{x \sqrt{x + 1}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
380,0,0,0,0.000000," ","integrate((1+x)**(1/2)/(-x**2+1)**(1/2),x)","\int \frac{\sqrt{x + 1}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(sqrt(x + 1)/sqrt(-(x - 1)*(x + 1)), x)","F",0
381,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*x/(1-x)**(1/2),x)","\int \frac{x \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{1 - x}}\, dx"," ",0,"Integral(x*(x + 1)/(sqrt(-(x - 1)*(x + 1))*sqrt(1 - x)), x)","F",0
382,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)/(1-x)**(1/2),x)","\int \frac{x + 1}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{1 - x}}\, dx"," ",0,"Integral((x + 1)/(sqrt(-(x - 1)*(x + 1))*sqrt(1 - x)), x)","F",0
383,0,0,0,0.000000," ","integrate(1/(1+x)**(1/2)/(-x**2+1)**(1/2)*x,x)","\int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{x + 1}}\, dx"," ",0,"Integral(x/(sqrt(-(x - 1)*(x + 1))*sqrt(x + 1)), x)","F",0
384,0,0,0,0.000000," ","integrate(1/(1+x)**(1/2)/(-x**2+1)**(1/2),x)","\int \frac{1}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{x + 1}}\, dx"," ",0,"Integral(1/(sqrt(-(x - 1)*(x + 1))*sqrt(x + 1)), x)","F",0
385,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*x/(1-x)**(3/2),x)","\int \frac{x \left(x + 1\right)}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(1 - x\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*(x + 1)/(sqrt(-(x - 1)*(x + 1))*(1 - x)**(3/2)), x)","F",0
386,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)/(1-x)**(3/2),x)","\int \frac{x + 1}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(1 - x\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((x + 1)/(sqrt(-(x - 1)*(x + 1))*(1 - x)**(3/2)), x)","F",0
387,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a*c*x+c)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
388,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a*c*x+c)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
389,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a*c*x+c)**(1/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
390,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
391,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
392,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
393,1,83,0,10.430439," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a*c*x+c)**(1/2),x)","\frac{2 \left(- 2 c^{4} \sqrt{- a c x + c} + \frac{7 c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{9 c^{2} \left(- a c x + c\right)^{\frac{5}{2}}}{5} + \frac{5 c \left(- a c x + c\right)^{\frac{7}{2}}}{7} - \frac{\left(- a c x + c\right)^{\frac{9}{2}}}{9}\right)}{a^{4} c^{4}}"," ",0,"2*(-2*c**4*sqrt(-a*c*x + c) + 7*c**3*(-a*c*x + c)**(3/2)/3 - 9*c**2*(-a*c*x + c)**(5/2)/5 + 5*c*(-a*c*x + c)**(7/2)/7 - (-a*c*x + c)**(9/2)/9)/(a**4*c**4)","A",0
394,1,68,0,8.675461," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a*c*x+c)**(1/2),x)","- \frac{2 \left(2 c^{3} \sqrt{- a c x + c} - \frac{5 c^{2} \left(- a c x + c\right)^{\frac{3}{2}}}{3} + \frac{4 c \left(- a c x + c\right)^{\frac{5}{2}}}{5} - \frac{\left(- a c x + c\right)^{\frac{7}{2}}}{7}\right)}{a^{3} c^{3}}"," ",0,"-2*(2*c**3*sqrt(-a*c*x + c) - 5*c**2*(-a*c*x + c)**(3/2)/3 + 4*c*(-a*c*x + c)**(5/2)/5 - (-a*c*x + c)**(7/2)/7)/(a**3*c**3)","A",0
395,1,48,0,7.407005," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a*c*x+c)**(1/2),x)","\frac{2 \left(- 2 c^{2} \sqrt{- a c x + c} + c \left(- a c x + c\right)^{\frac{3}{2}} - \frac{\left(- a c x + c\right)^{\frac{5}{2}}}{5}\right)}{a^{2} c^{2}}"," ",0,"2*(-2*c**2*sqrt(-a*c*x + c) + c*(-a*c*x + c)**(3/2) - (-a*c*x + c)**(5/2)/5)/(a**2*c**2)","A",0
396,1,31,0,4.624673," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2),x)","- \frac{2 \left(2 c \sqrt{- a c x + c} - \frac{\left(- a c x + c\right)^{\frac{3}{2}}}{3}\right)}{a c}"," ",0,"-2*(2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c)","A",0
397,1,39,0,7.282305," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2)/x,x)","\frac{2 c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - 2 \sqrt{- a c x + c}"," ",0,"2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 2*sqrt(-a*c*x + c)","A",0
398,1,119,0,19.586556," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2)/x**2,x)","\frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} + \frac{2 a c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{\sqrt{- a c x + c}}{x}"," ",0,"a*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 - a*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 + 2*a*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - sqrt(-a*c*x + c)/x","B",0
399,1,270,0,36.998701," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2)/x**3,x)","- \frac{10 a^{2} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{6 a^{2} c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} - \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} + \frac{a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{a \sqrt{- a c x + c}}{x}"," ",0,"-10*a**2*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 6*a**2*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 3*a**2*c**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 - 3*a**2*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 + a**2*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 - a**2*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 - a*sqrt(-a*c*x + c)/x","B",0
400,1,439,0,55.920497," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2)/x**4,x)","\frac{66 a^{3} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} - \frac{80 a^{3} c^{5} \left(- a c x + c\right)^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{30 a^{3} c^{4} \left(- a c x + c\right)^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} - \frac{10 a^{3} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} - \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} + \frac{6 a^{3} c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{3 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} - \frac{3 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8}"," ",0,"66*a**3*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) - 80*a**3*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 30*a**3*c**4*(-a*c*x + c)**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) - 10*a**3*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 5*a**3*c**4*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 5*a**3*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 + 6*a**3*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 3*a**3*c**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 - 3*a**3*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8","B",0
401,1,639,0,78.973187," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a*c*x+c)**(1/2)/x**5,x)","- \frac{558 a^{4} c^{8} \sqrt{- a c x + c}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} + \frac{1022 a^{4} c^{7} \left(- a c x + c\right)^{\frac{3}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} - \frac{770 a^{4} c^{6} \left(- a c x + c\right)^{\frac{5}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} + \frac{66 a^{4} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{210 a^{4} c^{5} \left(- a c x + c\right)^{\frac{7}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} - \frac{80 a^{4} c^{5} \left(- a c x + c\right)^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left(- c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right)}}{128} - \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left(c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right)}}{128} + \frac{30 a^{4} c^{4} \left(- a c x + c\right)^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{5 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} - \frac{5 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16}"," ",0,"-558*a**4*c**8*sqrt(-a*c*x + c)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) + 1022*a**4*c**7*(-a*c*x + c)**(3/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) - 770*a**4*c**6*(-a*c*x + c)**(5/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) + 66*a**4*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 210*a**4*c**5*(-a*c*x + c)**(7/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) - 80*a**4*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 35*a**4*c**5*sqrt(c**(-9))*log(-c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c))/128 - 35*a**4*c**5*sqrt(c**(-9))*log(c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c))/128 + 30*a**4*c**4*(-a*c*x + c)**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 5*a**4*c**4*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 5*a**4*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16","B",0
402,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a*c*x+c)**(1/2),x)","\int \frac{x^{3} \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
403,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(-a*c*x+c)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
404,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(-a*c*x+c)**(1/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
405,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
406,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
407,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
408,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
409,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(a x + 1\right)^{3}}{x^{4} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
410,-1,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**5,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
411,0,0,0,0.000000," ","integrate(x**m*(-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
412,0,0,0,0.000000," ","integrate(x**2*(-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
413,0,0,0,0.000000," ","integrate(x*(-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
414,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
415,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x*(a*x + 1)), x)","F",0
416,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)), x)","F",0
417,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{3} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)), x)","F",0
418,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{4} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*sqrt(-(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)), x)","F",0
419,1,126,0,18.582167," ","integrate(x**3*(-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{2 \left(- \frac{2 \sqrt{2} c^{5} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 c^{4} \sqrt{- a c x + c} - \frac{c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{c^{2} \left(- a c x + c\right)^{\frac{5}{2}}}{5} + \frac{c \left(- a c x + c\right)^{\frac{7}{2}}}{7} - \frac{\left(- a c x + c\right)^{\frac{9}{2}}}{9}\right)}{a^{4} c^{4}}"," ",0,"2*(-2*sqrt(2)*c**5*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c**4*sqrt(-a*c*x + c) - c**3*(-a*c*x + c)**(3/2)/3 - c**2*(-a*c*x + c)**(5/2)/5 + c*(-a*c*x + c)**(7/2)/7 - (-a*c*x + c)**(9/2)/9)/(a**4*c**4)","A",0
420,1,97,0,14.870816," ","integrate(x**2*(-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{2 \left(- \frac{2 \sqrt{2} c^{4} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 c^{3} \sqrt{- a c x + c} - \frac{c^{2} \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{\left(- a c x + c\right)^{\frac{7}{2}}}{7}\right)}{a^{3} c^{3}}"," ",0,"-2*(-2*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c**3*sqrt(-a*c*x + c) - c**2*(-a*c*x + c)**(3/2)/3 - (-a*c*x + c)**(7/2)/7)/(a**3*c**3)","A",0
421,1,94,0,11.868116," ","integrate(x*(-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{2 \left(- \frac{2 \sqrt{2} c^{3} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 c^{2} \sqrt{- a c x + c} - \frac{c \left(- a c x + c\right)^{\frac{3}{2}}}{3} - \frac{\left(- a c x + c\right)^{\frac{5}{2}}}{5}\right)}{a^{2} c^{2}}"," ",0,"2*(-2*sqrt(2)*c**3*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c**2*sqrt(-a*c*x + c) - c*(-a*c*x + c)**(3/2)/3 - (-a*c*x + c)**(5/2)/5)/(a**2*c**2)","A",0
422,1,75,0,8.083781," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{2 \left(- \frac{2 \sqrt{2} c^{2} \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 c \sqrt{- a c x + c} - \frac{\left(- a c x + c\right)^{\frac{3}{2}}}{3}\right)}{a c}"," ",0,"-2*(-2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c)","A",0
423,1,80,0,10.109259," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x,x)","\frac{2 c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{4 \sqrt{2} c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 \sqrt{- a c x + c}"," ",0,"2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 2*sqrt(-a*c*x + c)","A",0
424,1,162,0,14.770246," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**2,x)","\frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{a c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{6 a c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} + \frac{4 \sqrt{2} a c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{\sqrt{- a c x + c}}{x}"," ",0,"a*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 - a*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 - 6*a*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 4*sqrt(2)*a*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - sqrt(-a*c*x + c)/x","B",0
425,1,352,0,25.662262," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)","- \frac{10 a^{2} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{6 a^{2} c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} - \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} - \frac{3 a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} + \frac{3 a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)}}{2} + \frac{8 a^{2} c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{4 \sqrt{2} a^{2} c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} + \frac{3 a \sqrt{- a c x + c}}{x}"," ",0,"-10*a**2*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 6*a**2*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 3*a**2*c**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 - 3*a**2*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 - 3*a**2*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 + 3*a**2*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c))/2 + 8*a**2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*a**2*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 3*a*sqrt(-a*c*x + c)/x","B",0
426,1,614,0,29.743019," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)","\frac{66 a^{3} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} - \frac{80 a^{3} c^{5} \left(- a c x + c\right)^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{30 a^{3} c^{4} \left(- a c x + c\right)^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{30 a^{3} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} - \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} - \frac{18 a^{3} c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} - \frac{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} + \frac{9 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{8} + 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)} - 2 a^{3} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)} - \frac{8 a^{3} c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} + \frac{4 \sqrt{2} a^{3} c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{4 a^{2} \sqrt{- a c x + c}}{x}"," ",0,"66*a**3*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) - 80*a**3*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 30*a**3*c**4*(-a*c*x + c)**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 30*a**3*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 5*a**3*c**4*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 5*a**3*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 18*a**3*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) - 9*a**3*c**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 + 9*a**3*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 + 2*a**3*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) - 2*a**3*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) - 8*a**3*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 4*sqrt(2)*a**3*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 4*a**2*sqrt(-a*c*x + c)/x","B",0
427,1,991,0,49.265608," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)","- \frac{558 a^{4} c^{8} \sqrt{- a c x + c}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} + \frac{1022 a^{4} c^{7} \left(- a c x + c\right)^{\frac{3}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} - \frac{770 a^{4} c^{6} \left(- a c x + c\right)^{\frac{5}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} - \frac{198 a^{4} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{210 a^{4} c^{5} \left(- a c x + c\right)^{\frac{7}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left(- a c x + c\right)^{2} - 1536 c^{5} \left(- a c x + c\right)^{3} + 384 c^{4} \left(- a c x + c\right)^{4}} + \frac{240 a^{4} c^{5} \left(- a c x + c\right)^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} + \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left(- c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right)}}{128} - \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left(c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right)}}{128} - \frac{90 a^{4} c^{4} \left(- a c x + c\right)^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left(- a c x + c\right)^{2} + 48 c^{3} \left(- a c x + c\right)^{3}} - \frac{40 a^{4} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} - \frac{15 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} + \frac{15 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left(c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right)}}{16} + \frac{24 a^{4} c^{3} \left(- a c x + c\right)^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left(- a c x + c\right)^{2}} + \frac{3 a^{4} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{2} - \frac{3 a^{4} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left(c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right)}}{2} - 2 a^{4} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)} + 2 a^{4} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left(c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right)} + \frac{8 a^{4} c \operatorname{atan}{\left(\frac{\sqrt{- a c x + c}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{4 \sqrt{2} a^{4} c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} + \frac{4 a^{3} \sqrt{- a c x + c}}{x}"," ",0,"-558*a**4*c**8*sqrt(-a*c*x + c)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) + 1022*a**4*c**7*(-a*c*x + c)**(3/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) - 770*a**4*c**6*(-a*c*x + c)**(5/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) - 198*a**4*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 210*a**4*c**5*(-a*c*x + c)**(7/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) + 240*a**4*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 35*a**4*c**5*sqrt(c**(-9))*log(-c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c))/128 - 35*a**4*c**5*sqrt(c**(-9))*log(c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c))/128 - 90*a**4*c**4*(-a*c*x + c)**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) - 40*a**4*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) - 15*a**4*c**4*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 + 15*a**4*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 + 24*a**4*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 3*a**4*c**3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/2 - 3*a**4*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/2 - 2*a**4*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) + 2*a**4*c**2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) + 8*a**4*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*a**4*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 4*a**3*sqrt(-a*c*x + c)/x","B",0
428,0,0,0,0.000000," ","integrate(x**3*(-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{3} \sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
429,0,0,0,0.000000," ","integrate(x**2*(-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
430,0,0,0,0.000000," ","integrate(x*(-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
431,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
432,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x*(a*x + 1)**3), x)","F",0
433,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{2} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)**3), x)","F",0
434,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{3} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**3*(a*x + 1)**3), x)","F",0
435,0,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\sqrt{- c \left(a x - 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{4} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**4*(a*x + 1)**3), x)","F",0
436,-1,0,0,0.000000," ","integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
437,0,0,0,0.000000," ","integrate((-a*c*x+c)**p/exp(2*p*atanh(a*x)),x)","\int \left(- c \left(a x - 1\right)\right)^{p} e^{- 2 p \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*exp(-2*p*atanh(a*x)), x)","F",0
438,0,0,0,0.000000," ","integrate(exp(2*p*atanh(a*x))*(-a*c*x+c)**p,x)","\begin{cases} \frac{x}{c} & \text{for}\: a = 0 \wedge p = -1 \\c^{p} x & \text{for}\: a = 0 \\- \frac{\int \frac{1}{a x e^{2 \operatorname{atanh}{\left(a x \right)}} - e^{2 \operatorname{atanh}{\left(a x \right)}}}\, dx}{c} & \text{for}\: p = -1 \\\frac{a x \left(- a c x + c\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}}{a p + a} + \frac{\left(- a c x + c\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}}{a p + a} & \text{otherwise} \end{cases}"," ",0,"Piecewise((x/c, Eq(a, 0) & Eq(p, -1)), (c**p*x, Eq(a, 0)), (-Integral(1/(a*x*exp(2*atanh(a*x)) - exp(2*atanh(a*x))), x)/c, Eq(p, -1)), (a*x*(-a*c*x + c)**p*exp(2*p*atanh(a*x))/(a*p + a) + (-a*c*x + c)**p*exp(2*p*atanh(a*x))/(a*p + a), True))","F",0
439,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**p,x)","\int \left(- c \left(a x - 1\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(a*x - 1))**p*exp(n*atanh(a*x)), x)","F",0
440,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**3,x)","- c^{3} \left(\int 3 a x e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- 3 a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx + \int a^{3} x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx\right)"," ",0,"-c**3*(Integral(3*a*x*exp(n*atanh(a*x)), x) + Integral(-3*a**2*x**2*exp(n*atanh(a*x)), x) + Integral(a**3*x**3*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x)), x))","F",0
441,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c)**2,x)","c^{2} \left(\int \left(- 2 a x e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx + \int a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx\right)"," ",0,"c**2*(Integral(-2*a*x*exp(n*atanh(a*x)), x) + Integral(a**2*x**2*exp(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x)), x))","F",0
442,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a*c*x+c),x)","- c \left(\int a x e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx\right)"," ",0,"-c*(Integral(a*x*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x)), x))","F",0
443,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c),x)","- \frac{\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a x - 1}\, dx}{c}"," ",0,"-Integral(exp(n*atanh(a*x))/(a*x - 1), x)/c","F",0
444,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**2,x)","\begin{cases} \text{NaN} & \text{for}\: a = \frac{1}{x} \wedge c = 0 \wedge n = -2 \\\tilde{\infty} x e^{\infty n} & \text{for}\: a = \frac{1}{x} \\\tilde{\infty} \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\- \frac{a x \operatorname{atanh}{\left(a x \right)}}{a^{2} c^{2} x e^{2 \operatorname{atanh}{\left(a x \right)}} - a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{\operatorname{atanh}{\left(a x \right)}}{a^{2} c^{2} x e^{2 \operatorname{atanh}{\left(a x \right)}} - a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -2 \\- \frac{a x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} - \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} & \text{otherwise} \end{cases}"," ",0,"Piecewise((nan, Eq(c, 0) & Eq(n, -2) & Eq(a, 1/x)), (zoo*x*exp(oo*n), Eq(a, 1/x)), (zoo*Integral(exp(n*atanh(a*x)), x), Eq(c, 0)), (-a*x*atanh(a*x)/(a**2*c**2*x*exp(2*atanh(a*x)) - a*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(a**2*c**2*x*exp(2*atanh(a*x)) - a*c**2*exp(2*atanh(a*x))), Eq(n, -2)), (-a*x*exp(n*atanh(a*x))/(a**2*c**2*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2) - exp(n*atanh(a*x))/(a**2*c**2*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2), True))","F",0
445,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**3,x)","\begin{cases} \tilde{\infty} \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\\frac{a^{2} x^{2} \operatorname{atanh}{\left(a x \right)}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{4 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{4 \operatorname{atanh}{\left(a x \right)}}} + \frac{2 a x \operatorname{atanh}{\left(a x \right)}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{4 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{4 \operatorname{atanh}{\left(a x \right)}}} - \frac{a x}{2 a^{3} c^{3} x^{2} e^{4 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{4 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{4 \operatorname{atanh}{\left(a x \right)}}} + \frac{\operatorname{atanh}{\left(a x \right)}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{4 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{4 \operatorname{atanh}{\left(a x \right)}}} - \frac{1}{2 a^{3} c^{3} x^{2} e^{4 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{4 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{4 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -4 \\- \frac{a^{2} x^{2} \operatorname{atanh}{\left(a x \right)}}{2 a^{3} c^{3} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{2 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{a x}{2 a^{3} c^{3} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{2 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{\operatorname{atanh}{\left(a x \right)}}{2 a^{3} c^{3} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{2 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{1}{2 a^{3} c^{3} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{3} x e^{2 \operatorname{atanh}{\left(a x \right)}} + 2 a c^{3} e^{2 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -2 \\- \frac{a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac{a n x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac{2 a x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac{n e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac{3 e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} & \text{otherwise} \end{cases}"," ",0,"Piecewise((zoo*Integral(exp(n*atanh(a*x)), x), Eq(c, 0)), (a**2*x**2*atanh(a*x)/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) + 2*a*x*atanh(a*x)/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) - a*x/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) + atanh(a*x)/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) - 1/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))), Eq(n, -4)), (-a**2*x**2*atanh(a*x)/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))) + a*x/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))) + atanh(a*x)/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))) + 1/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))), Eq(n, -2)), (-a**2*x**2*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3) + a*n*x*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3) + 2*a*x*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3) + n*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3) + 3*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3), True))","F",0
446,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a*c*x+c)**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
447,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**p,x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
448,1,357,0,11.933448," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**4,x)","a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \frac{2 c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{2 c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{3 c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 2*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 - 3*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4","A",0
449,1,228,0,7.375378," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**3,x)","a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 2 c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \frac{2 c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}}"," ",0,"a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 2*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 - c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3","A",0
450,1,151,0,7.261444," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**2,x)","a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2","A",0
451,1,61,0,20.408468," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x),x)","\begin{cases} \frac{- c \sqrt{- a^{2} x^{2} + 1} + \frac{c \left(- \log{\left(-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)} + \log{\left(1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)}\right)}{2}}{a} & \text{for}\: a \neq 0 \\c x + \tilde{\infty} c \log{\left(x \right)} & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-c*sqrt(-a**2*x**2 + 1) + c*(-log(-1 + 1/sqrt(-a**2*x**2 + 1)) + log(1 + 1/sqrt(-a**2*x**2 + 1)))/2)/a, Ne(a, 0)), (c*x + zoo*c*log(x), True))","A",0
452,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x),x)","\frac{a \left(\int \frac{x}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c}"," ",0,"a*(Integral(x/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
453,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**2,x)","\frac{a^{2} \left(\int \frac{x^{2}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{2}}"," ",0,"a**2*(Integral(x**2/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
454,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**3,x)","\frac{a^{3} \left(\int \frac{x^{3}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{3}}"," ",0,"a**3*(Integral(x**3/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
455,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**4,x)","\frac{a^{4} \left(\int \frac{x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{4}}"," ",0,"a**4*(Integral(x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
456,1,274,0,7.925877," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**p,x)","- a \left(\begin{cases} \frac{0^{p} x}{a} + \frac{0^{p} \log{\left(a x - 1 \right)}}{a^{2}} - \frac{a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(2 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 2 - p \\ 3 - p \end{matrix}\middle| {a x} \right)}}{\Gamma\left(3 - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a x}\right| > 1 \\\frac{0^{p} x}{a} + \frac{0^{p} \log{\left(- a x + 1 \right)}}{a^{2}} - \frac{a^{- p} c^{p} p x^{2} x^{- p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(2 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 2 - p \\ 3 - p \end{matrix}\middle| {a x} \right)}}{\Gamma\left(3 - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{0^{p} \log{\left(a x - 1 \right)}}{a} - \frac{a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a x} \right)}}{\Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a x}\right| > 1 \\\frac{0^{p} \log{\left(- a x + 1 \right)}}{a} - \frac{a^{- p} c^{p} p x x^{- p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a x} \right)}}{\Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}"," ",0,"-a*Piecewise((0**p*x/a + 0**p*log(a*x - 1)/a**2 - a**(-p)*c**p*p*x**2*x**(-p)*exp(I*pi*p)*gamma(p)*gamma(2 - p)*hyper((1 - p, 2 - p), (3 - p,), a*x)/(gamma(3 - p)*gamma(p + 1)), Abs(a*x) > 1), (0**p*x/a + 0**p*log(-a*x + 1)/a**2 - a**(-p)*c**p*p*x**2*x**(-p)*exp(I*pi*p)*gamma(p)*gamma(2 - p)*hyper((1 - p, 2 - p), (3 - p,), a*x)/(gamma(3 - p)*gamma(p + 1)), True)) - Piecewise((0**p*log(a*x - 1)/a - a**(-p)*c**p*p*x*x**(-p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a*x)/(gamma(2 - p)*gamma(p + 1)), Abs(a*x) > 1), (0**p*log(-a*x + 1)/a - a**(-p)*c**p*p*x*x**(-p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a*x)/(gamma(2 - p)*gamma(p + 1)), True))","C",0
457,1,63,0,0.281478," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**5,x)","\frac{- a^{5} c^{5} x + 3 a^{4} c^{5} \log{\left(x \right)} - \frac{- 8 a^{3} c^{5} x^{3} - 4 a^{2} c^{5} x^{2} + 4 a c^{5} x - c^{5}}{4 x^{4}}}{a^{5}}"," ",0,"(-a**5*c**5*x + 3*a**4*c**5*log(x) - (-8*a**3*c**5*x**3 - 4*a**2*c**5*x**2 + 4*a*c**5*x - c**5)/(4*x**4))/a**5","A",0
458,1,39,0,0.194680," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**4,x)","\frac{- a^{4} c^{4} x + 2 a^{3} c^{4} \log{\left(x \right)} - \frac{- 3 a c^{4} x + c^{4}}{3 x^{3}}}{a^{4}}"," ",0,"(-a**4*c**4*x + 2*a**3*c**4*log(x) - (-3*a*c**4*x + c**4)/(3*x**3))/a**4","A",0
459,1,37,0,0.169755," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**3,x)","\frac{- a^{3} c^{3} x + a^{2} c^{3} \log{\left(x \right)} - \frac{2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}}"," ",0,"(-a**3*c**3*x + a**2*c**3*log(x) - (2*a*c**3*x - c**3)/(2*x**2))/a**3","A",0
460,1,17,0,0.095680," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**2,x)","\frac{- a^{2} c^{2} x - \frac{c^{2}}{x}}{a^{2}}"," ",0,"(-a**2*c**2*x - c**2/x)/a**2","A",0
461,1,12,0,0.089873," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x),x)","\frac{- a c x - c \log{\left(x \right)}}{a}"," ",0,"(-a*c*x - c*log(x))/a","A",0
462,1,26,0,0.166442," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x),x)","\frac{2}{a^{2} c x - a c} - \frac{x}{c} - \frac{3 \log{\left(a x - 1 \right)}}{a c}"," ",0,"2/(a**2*c*x - a*c) - x/c - 3*log(a*x - 1)/(a*c)","A",0
463,1,51,0,0.253114," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**2,x)","- \frac{- 5 a x + 4}{a^{3} c^{2} x^{2} - 2 a^{2} c^{2} x + a c^{2}} - \frac{x}{c^{2}} - \frac{4 \log{\left(a x - 1 \right)}}{a c^{2}}"," ",0,"-(-5*a*x + 4)/(a**3*c**2*x**2 - 2*a**2*c**2*x + a*c**2) - x/c**2 - 4*log(a*x - 1)/(a*c**2)","A",0
464,1,75,0,0.352140," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**3,x)","- \frac{- 54 a^{2} x^{2} + 87 a x - 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} - \frac{x}{c^{3}} - \frac{5 \log{\left(a x - 1 \right)}}{a c^{3}}"," ",0,"-(-54*a**2*x**2 + 87*a*x - 37)/(6*a**4*c**3*x**3 - 18*a**3*c**3*x**2 + 18*a**2*c**3*x - 6*a*c**3) - x/c**3 - 5*log(a*x - 1)/(a*c**3)","A",0
465,1,95,0,0.449798," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**4,x)","- \frac{- 28 a^{3} x^{3} + 68 a^{2} x^{2} - 58 a x + 17}{2 a^{5} c^{4} x^{4} - 8 a^{4} c^{4} x^{3} + 12 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 2 a c^{4}} - \frac{x}{c^{4}} - \frac{6 \log{\left(a x - 1 \right)}}{a c^{4}}"," ",0,"-(-28*a**3*x**3 + 68*a**2*x**2 - 58*a*x + 17)/(2*a**5*c**4*x**4 - 8*a**4*c**4*x**3 + 12*a**3*c**4*x**2 - 8*a**2*c**4*x + 2*a*c**4) - x/c**4 - 6*log(a*x - 1)/(a*c**4)","A",0
466,1,354,0,18.873718," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**4,x)","- a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \frac{2 c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{2 c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"-a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 2*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 - c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4","A",0
467,1,104,0,35.547093," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**3,x)","\frac{2 c^{3} \sqrt{- a^{2} x^{2} + 1} + \frac{3 c^{3} \log{\left(-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)}}{2} - \frac{3 c^{3} \log{\left(1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}} \right)}}{2} + \frac{c^{3}}{2 \left(1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}}\right)} + \frac{c^{3}}{2 \left(-1 + \frac{1}{\sqrt{- a^{2} x^{2} + 1}}\right)}}{2 a}"," ",0,"(2*c**3*sqrt(-a**2*x**2 + 1) + 3*c**3*log(-1 + 1/sqrt(-a**2*x**2 + 1))/2 - 3*c**3*log(1 + 1/sqrt(-a**2*x**2 + 1))/2 + c**3/(2*(1 + 1/sqrt(-a**2*x**2 + 1))) + c**3/(2*(-1 + 1/sqrt(-a**2*x**2 + 1))))/(2*a)","A",0
468,1,151,0,13.945527," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**2,x)","- a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \frac{c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"-a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2","A",0
469,1,104,0,10.248160," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x),x)","- a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 2 c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a}"," ",0,"-a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 2*c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a","A",0
470,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x),x)","\frac{a \left(\int \frac{x}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{2}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{3}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{4}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c}"," ",0,"a*(Integral(x/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**3/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**4/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
471,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**2,x)","\frac{a^{2} \left(\int \frac{x^{2}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{3}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{4}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{5}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{2}}"," ",0,"a**2*(Integral(x**2/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**3/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**4/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**5/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
472,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**3,x)","\frac{a^{3} \left(\int \frac{x^{3}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{4}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{5}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{6}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{3}}"," ",0,"a**3*(Integral(x**3/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**4/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**5/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
473,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**4,x)","\frac{a^{4} \left(\int \frac{x^{4}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{5}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{6}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{7}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{4}}"," ",0,"a**4*(Integral(x**4/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**5/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
474,0,0,0,0.000000," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**p,x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} \left(a x + 1\right)^{2}}{\left(a x - 1\right)^{2}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*(a*x + 1)**2/(a*x - 1)**2, x)","F",0
475,1,63,0,0.283890," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**5,x)","\frac{a^{5} c^{5} x - a^{4} c^{5} \log{\left(x \right)} + \frac{24 a^{3} c^{5} x^{3} - 12 a^{2} c^{5} x^{2} - 4 a c^{5} x + 3 c^{5}}{12 x^{4}}}{a^{5}}"," ",0,"(a**5*c**5*x - a**4*c**5*log(x) + (24*a**3*c**5*x**3 - 12*a**2*c**5*x**2 - 4*a*c**5*x + 3*c**5)/(12*x**4))/a**5","A",0
476,1,31,0,0.158074," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**4,x)","\frac{a^{4} c^{4} x + \frac{6 a^{2} c^{4} x^{2} - c^{4}}{3 x^{3}}}{a^{4}}"," ",0,"(a**4*c**4*x + (6*a**2*c**4*x**2 - c**4)/(3*x**3))/a**4","A",0
477,1,37,0,0.176799," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**3,x)","\frac{a^{3} c^{3} x + a^{2} c^{3} \log{\left(x \right)} + \frac{2 a c^{3} x + c^{3}}{2 x^{2}}}{a^{3}}"," ",0,"(a**3*c**3*x + a**2*c**3*log(x) + (2*a*c**3*x + c**3)/(2*x**2))/a**3","A",0
478,1,26,0,0.130360," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x)**2,x)","\frac{a^{2} c^{2} x + 2 a c^{2} \log{\left(x \right)} - \frac{c^{2}}{x}}{a^{2}}"," ",0,"(a**2*c**2*x + 2*a*c**2*log(x) - c**2/x)/a**2","A",0
479,1,17,0,0.225928," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a/x),x)","c x + \frac{c \left(- \log{\left(x \right)} + 4 \log{\left(x - \frac{1}{a} \right)}\right)}{a}"," ",0,"c*x + c*(-log(x) + 4*log(x - 1/a))/a","A",0
480,1,41,0,0.245493," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a/x),x)","\frac{- 8 a x + 6}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{x}{c} + \frac{5 \log{\left(a x - 1 \right)}}{a c}"," ",0,"(-8*a*x + 6)/(a**3*c*x**2 - 2*a**2*c*x + a*c) + x/c + 5*log(a*x - 1)/(a*c)","A",0
481,1,73,0,0.344367," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a/x)**2,x)","\frac{- 39 a^{2} x^{2} + 60 a x - 25}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} + \frac{x}{c^{2}} + \frac{6 \log{\left(a x - 1 \right)}}{a c^{2}}"," ",0,"(-39*a**2*x**2 + 60*a*x - 25)/(3*a**4*c**2*x**3 - 9*a**3*c**2*x**2 + 9*a**2*c**2*x - 3*a*c**2) + x/c**2 + 6*log(a*x - 1)/(a*c**2)","A",0
482,1,94,0,0.440259," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a/x)**3,x)","\frac{- 114 a^{3} x^{3} + 267 a^{2} x^{2} - 224 a x + 65}{6 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 36 a^{3} c^{3} x^{2} - 24 a^{2} c^{3} x + 6 a c^{3}} + \frac{x}{c^{3}} + \frac{7 \log{\left(a x - 1 \right)}}{a c^{3}}"," ",0,"(-114*a**3*x**3 + 267*a**2*x**2 - 224*a*x + 65)/(6*a**5*c**3*x**4 - 24*a**4*c**3*x**3 + 36*a**3*c**3*x**2 - 24*a**2*c**3*x + 6*a*c**3) + x/c**3 + 7*log(a*x - 1)/(a*c**3)","A",0
483,1,114,0,0.549685," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a/x)**4,x)","\frac{- 390 a^{4} x^{4} + 1230 a^{3} x^{3} - 1555 a^{2} x^{2} + 905 a x - 202}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} + \frac{x}{c^{4}} + \frac{8 \log{\left(a x - 1 \right)}}{a c^{4}}"," ",0,"(-390*a**4*x**4 + 1230*a**3*x**3 - 1555*a**2*x**2 + 905*a*x - 202)/(15*a**6*c**4*x**5 - 75*a**5*c**4*x**4 + 150*a**4*c**4*x**3 - 150*a**3*c**4*x**2 + 75*a**2*c**4*x - 15*a*c**4) + x/c**4 + 8*log(a*x - 1)/(a*c**4)","A",0
484,0,0,0,0.000000," ","integrate((c-c/a/x)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
485,0,0,0,0.000000," ","integrate((c-c/a/x)**4/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{4} \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left(- \frac{4 a x \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right)\, dx + \int \frac{6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx + \int \left(- \frac{4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\right)\, dx + \int \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a x^{5} + x^{4}}\, dx\right)}{a^{4}}"," ",0,"c**4*(Integral(sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(-4*a*x*sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(6*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(-4*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x) + Integral(a**4*x**4*sqrt(-a**2*x**2 + 1)/(a*x**5 + x**4), x))/a**4","F",0
486,0,0,0,0.000000," ","integrate((c-c/a/x)**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{3} \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\right)\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\, dx + \int \left(- \frac{3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\right)\, dx + \int \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a x^{4} + x^{3}}\, dx\right)}{a^{3}}"," ",0,"c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a*x**4 + x**3), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a*x**4 + x**3), x) + Integral(-3*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x**4 + x**3), x) + Integral(a**3*x**3*sqrt(-a**2*x**2 + 1)/(a*x**4 + x**3), x))/a**3","F",0
487,0,0,0,0.000000," ","integrate((c-c/a/x)**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{2} \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\, dx + \int \left(- \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\right)\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a x^{3} + x^{2}}\, dx\right)}{a^{2}}"," ",0,"c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a*x**3 + x**2), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a*x**3 + x**2), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a*x**3 + x**2), x))/a**2","F",0
488,0,0,0,0.000000," ","integrate((c-c/a/x)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a x^{2} + x}\right)\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a x^{2} + x}\, dx\right)}{a}"," ",0,"c*(Integral(-sqrt(-a**2*x**2 + 1)/(a*x**2 + x), x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a*x**2 + x), x))/a","F",0
489,1,53,0,9.017907," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x),x)","a \left(\begin{cases} \tilde{\infty} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) & \text{for}\: c = 0 \\- \frac{x^{2}}{2 c} & \text{for}\: a^{2} = 0 \\\frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2} c} & \text{otherwise} \end{cases}\right)"," ",0,"a*Piecewise((zoo*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)), Eq(c, 0)), (-x**2/(2*c), Eq(a**2, 0)), (sqrt(-a**2*x**2 + 1)/(a**2*c), True))","A",0
490,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**2,x)","\frac{a^{2} \int \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}}"," ",0,"a**2*Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 - a**2*x**2 - a*x + 1), x)/c**2","F",0
491,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**3,x)","\frac{a^{3} \int \frac{x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}}"," ",0,"a**3*Integral(x**3*sqrt(-a**2*x**2 + 1)/(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1), x)/c**3","F",0
492,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**4,x)","\frac{a^{4} \int \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx}{c^{4}}"," ",0,"a**4*Integral(x**4*sqrt(-a**2*x**2 + 1)/(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 + 2*a**2*x**2 - 3*a*x + 1), x)/c**4","F",0
493,0,0,0,0.000000," ","integrate((c-c/a/x)**p/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\left(c - \frac{c}{a x}\right)^{p}}{a x + 1}\right)\, dx - \int \frac{a x \left(c - \frac{c}{a x}\right)^{p}}{a x + 1}\, dx"," ",0,"-Integral(-(c - c/(a*x))**p/(a*x + 1), x) - Integral(a*x*(c - c/(a*x))**p/(a*x + 1), x)","F",0
494,1,58,0,0.472323," ","integrate((c-c/a/x)**4/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{4} x - \frac{2 c^{4} \left(13 \log{\left(x \right)} - 16 \log{\left(x + \frac{1}{a} \right)}\right)}{a} - \frac{48 a^{2} c^{4} x^{2} - 9 a c^{4} x + c^{4}}{3 a^{4} x^{3}}"," ",0,"-c**4*x - 2*c**4*(13*log(x) - 16*log(x + 1/a))/a - (48*a**2*c**4*x**2 - 9*a*c**4*x + c**4)/(3*a**4*x**3)","A",0
495,1,44,0,0.358315," ","integrate((c-c/a/x)**3/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{3} x - \frac{c^{3} \left(11 \log{\left(x \right)} - 16 \log{\left(x + \frac{1}{a} \right)}\right)}{a} - \frac{10 a c^{3} x - c^{3}}{2 a^{3} x^{2}}"," ",0,"-c**3*x - c**3*(11*log(x) - 16*log(x + 1/a))/a - (10*a*c**3*x - c**3)/(2*a**3*x**2)","A",0
496,1,32,0,0.285201," ","integrate((c-c/a/x)**2/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{2} x - \frac{4 c^{2} \left(\log{\left(x \right)} - 2 \log{\left(x + \frac{1}{a} \right)}\right)}{a} - \frac{c^{2}}{a^{2} x}"," ",0,"-c**2*x - 4*c**2*(log(x) - 2*log(x + 1/a))/a - c**2/(a**2*x)","A",0
497,1,19,0,0.220957," ","integrate((c-c/a/x)/(a*x+1)**2*(-a**2*x**2+1),x)","- c x - \frac{c \left(\log{\left(x \right)} - 4 \log{\left(x + \frac{1}{a} \right)}\right)}{a}"," ",0,"-c*x - c*(log(x) - 4*log(x + 1/a))/a","A",0
498,1,19,0,0.110746," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x),x)","- a \left(\frac{x}{a c} - \frac{\log{\left(a x + 1 \right)}}{a^{2} c}\right)"," ",0,"-a*(x/(a*c) - log(a*x + 1)/(a**2*c))","A",0
499,1,36,0,0.153802," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**2,x)","- a^{2} \left(\frac{x}{a^{2} c^{2}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{2} - \frac{\log{\left(x + \frac{1}{a} \right)}}{2}}{a^{3} c^{2}}\right)"," ",0,"-a**2*(x/(a**2*c**2) + (log(x - 1/a)/2 - log(x + 1/a)/2)/(a**3*c**2))","B",0
500,1,58,0,0.308337," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**3,x)","- a^{3} \left(- \frac{1}{2 a^{5} c^{3} x - 2 a^{4} c^{3}} + \frac{x}{a^{3} c^{3}} + \frac{\frac{5 \log{\left(x - \frac{1}{a} \right)}}{4} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{4} c^{3}}\right)"," ",0,"-a**3*(-1/(2*a**5*c**3*x - 2*a**4*c**3) + x/(a**3*c**3) + (5*log(x - 1/a)/4 - log(x + 1/a)/4)/(a**4*c**3))","A",0
501,1,75,0,0.393955," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**4,x)","- a^{4} \left(\frac{- 7 a x + 6}{4 a^{7} c^{4} x^{2} - 8 a^{6} c^{4} x + 4 a^{5} c^{4}} + \frac{x}{a^{4} c^{4}} + \frac{\frac{17 \log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{5} c^{4}}\right)"," ",0,"-a**4*((-7*a*x + 6)/(4*a**7*c**4*x**2 - 8*a**6*c**4*x + 4*a**5*c**4) + x/(a**4*c**4) + (17*log(x - 1/a)/8 - log(x + 1/a)/8)/(a**5*c**4))","A",0
502,0,0,0,0.000000," ","integrate((c-c/a/x)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\frac{c^{3} \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right)\, dx + \int \frac{3 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\, dx + \int \left(- \frac{2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right)\, dx + \int \left(- \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right)\, dx + \int \frac{3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\, dx + \int \left(- \frac{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{6} + 3 a^{2} x^{5} + 3 a x^{4} + x^{3}}\right)\, dx\right)}{a^{3}}"," ",0,"c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(-2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(3*a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x) + Integral(-a**5*x**5*sqrt(-a**2*x**2 + 1)/(a**3*x**6 + 3*a**2*x**5 + 3*a*x**4 + x**3), x))/a**3","F",0
503,0,0,0,0.000000," ","integrate((c-c/a/x)**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\frac{c^{2} \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left(- \frac{2 a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right)\, dx + \int \frac{2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\, dx + \int \left(- \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{5} + 3 a^{2} x^{4} + 3 a x^{3} + x^{2}}\right)\, dx\right)}{a^{2}}"," ",0,"c**2*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(-2*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x) + Integral(-a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**5 + 3*a**2*x**4 + 3*a*x**3 + x**2), x))/a**2","F",0
504,0,0,0,0.000000," ","integrate((c-c/a/x)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\frac{c \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right)\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\, dx + \int \left(- \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{4} + 3 a^{2} x^{3} + 3 a x^{2} + x}\right)\, dx\right)}{a}"," ",0,"c*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x**2 + x), x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x**2 + x), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x**2 + x), x) + Integral(-a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**3*x**4 + 3*a**2*x**3 + 3*a*x**2 + x), x))/a","F",0
505,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x),x)","\frac{a \left(\int \frac{x \sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right)\, dx\right)}{c}"," ",0,"a*(Integral(x*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x) + Integral(-a**2*x**3*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x))/c","F",0
506,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**2,x)","\frac{a^{2} \left(\int \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx + \int \left(- \frac{a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\right)\, dx\right)}{c^{2}}"," ",0,"a**2*(Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x) + Integral(-a**2*x**4*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x))/c**2","F",0
507,1,34,0,23.324379," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**3,x)","- \frac{2 \left(\frac{\sqrt{- a^{2} x^{2} + 1}}{2 c^{3}} + \frac{1}{2 c^{3} \sqrt{- a^{2} x^{2} + 1}}\right)}{a}"," ",0,"-2*(sqrt(-a**2*x**2 + 1)/(2*c**3) + 1/(2*c**3*sqrt(-a**2*x**2 + 1)))/a","A",0
508,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**4,x)","\frac{a^{4} \left(\int \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\, dx + \int \left(- \frac{a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\right)\, dx\right)}{c^{4}}"," ",0,"a**4*(Integral(x**4*sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 + 3*a**3*x**3 - 3*a**2*x**2 - a*x + 1), x) + Integral(-a**2*x**6*sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 + 3*a**3*x**3 - 3*a**2*x**2 - a*x + 1), x))/c**4","F",0
509,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**5,x)","\frac{a^{5} \left(\int \frac{x^{5} \sqrt{- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{7} \sqrt{- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\right)\, dx\right)}{c^{5}}"," ",0,"a**5*(Integral(x**5*sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6 + 6*a**5*x**5 - 6*a**3*x**3 + 2*a**2*x**2 + 2*a*x - 1), x) + Integral(-a**2*x**7*sqrt(-a**2*x**2 + 1)/(a**8*x**8 - 2*a**7*x**7 - 2*a**6*x**6 + 6*a**5*x**5 - 6*a**3*x**3 + 2*a**2*x**2 + 2*a*x - 1), x))/c**5","F",0
510,-1,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(9/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
511,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(7/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(7/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
512,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(5/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
513,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(3/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
514,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
515,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(1/2),x)","\int \frac{a x + 1}{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
516,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(3/2),x)","\int \frac{a x + 1}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/((-c*(-1 + 1/(a*x)))**(3/2)*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
517,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(5/2),x)","\int \frac{a x + 1}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/((-c*(-1 + 1/(a*x)))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
518,1,2421,0,31.185888," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(9/2),x)","- c^{4} \left(\begin{cases} \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{\sqrt{c} \operatorname{acosh}{\left(\sqrt{a} \sqrt{x} \right)}}{a} - \frac{\sqrt{c} \sqrt{x}}{\sqrt{a} \sqrt{a x - 1}} & \text{for}\: \left|{a x}\right| > 1 \\\frac{i \sqrt{c} \operatorname{asin}{\left(\sqrt{a} \sqrt{x} \right)}}{a} + \frac{i \sqrt{c} \sqrt{x} \sqrt{- a x + 1}}{\sqrt{a}} & \text{otherwise} \end{cases}\right) - \frac{4 c^{5} \operatorname{atan}{\left(\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right)}}{a \sqrt{- c}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a x}}}{a} - \frac{2 c^{4} \left(\begin{cases} \frac{4 i a^{\frac{11}{2}} \sqrt{c} x^{\frac{7}{2}}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 i a^{\frac{9}{2}} \sqrt{c} x^{\frac{5}{2}}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 i a^{5} \sqrt{c} x^{3} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{2 i a^{4} \sqrt{c} x^{2} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{8 i a^{3} \sqrt{c} x \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{6 i a^{2} \sqrt{c} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} & \text{for}\: \left|{a x}\right| > 1 \\- \frac{4 i a^{\frac{11}{2}} \sqrt{c} x^{\frac{7}{2}}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{4 i a^{\frac{9}{2}} \sqrt{c} x^{\frac{5}{2}}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 a^{5} \sqrt{c} x^{3} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{2 a^{4} \sqrt{c} x^{2} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{8 a^{3} \sqrt{c} x \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{6 a^{2} \sqrt{c} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{4} \left(\begin{cases} \frac{16 i a^{\frac{19}{2}} \sqrt{c} x^{\frac{13}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{48 i a^{\frac{17}{2}} \sqrt{c} x^{\frac{11}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{48 i a^{\frac{15}{2}} \sqrt{c} x^{\frac{9}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{16 i a^{\frac{13}{2}} \sqrt{c} x^{\frac{7}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{16 i a^{9} \sqrt{c} x^{6} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{40 i a^{8} \sqrt{c} x^{5} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{30 i a^{7} \sqrt{c} x^{4} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{40 i a^{6} \sqrt{c} x^{3} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{100 i a^{5} \sqrt{c} x^{2} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{96 i a^{4} \sqrt{c} x \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{30 i a^{3} \sqrt{c} \sqrt{a x - 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} & \text{for}\: \left|{a x}\right| > 1 \\\frac{16 i a^{\frac{19}{2}} \sqrt{c} x^{\frac{13}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{48 i a^{\frac{17}{2}} \sqrt{c} x^{\frac{11}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{48 i a^{\frac{15}{2}} \sqrt{c} x^{\frac{9}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{16 i a^{\frac{13}{2}} \sqrt{c} x^{\frac{7}{2}}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{16 a^{9} \sqrt{c} x^{6} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{40 a^{8} \sqrt{c} x^{5} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{30 a^{7} \sqrt{c} x^{4} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{40 a^{6} \sqrt{c} x^{3} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{100 a^{5} \sqrt{c} x^{2} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} - \frac{96 a^{4} \sqrt{c} x \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} + \frac{30 a^{3} \sqrt{c} \sqrt{- a x + 1}}{- 105 i a^{\frac{13}{2}} x^{\frac{13}{2}} + 315 i a^{\frac{11}{2}} x^{\frac{11}{2}} - 315 i a^{\frac{9}{2}} x^{\frac{9}{2}} + 105 i a^{\frac{7}{2}} x^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"-c**4*Piecewise((sqrt(a)*sqrt(c)*x**(3/2)/sqrt(a*x - 1) - sqrt(c)*acosh(sqrt(a)*sqrt(x))/a - sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(a*x - 1)), Abs(a*x) > 1), (I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)*sqrt(-a*x + 1)/sqrt(a), True)) - 4*c**5*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 4*c**4*sqrt(c - c/(a*x))/a - 2*c**4*Piecewise((4*I*a**(11/2)*sqrt(c)*x**(7/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**(9/2)*sqrt(c)*x**(5/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**5*sqrt(c)*x**3*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) + 2*I*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) + 8*I*a**3*sqrt(c)*x*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 6*I*a**2*sqrt(c)*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*I*a**(11/2)*sqrt(c)*x**(7/2)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 4*I*a**(9/2)*sqrt(c)*x**(5/2)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 4*a**5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 2*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 8*a**3*sqrt(c)*x*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 6*a**2*sqrt(c)*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)), True))/a**3 + c**4*Piecewise((16*I*a**(19/2)*sqrt(c)*x**(13/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 48*I*a**(17/2)*sqrt(c)*x**(11/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 48*I*a**(15/2)*sqrt(c)*x**(9/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16*I*a**(13/2)*sqrt(c)*x**(7/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16*I*a**9*sqrt(c)*x**6*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 40*I*a**8*sqrt(c)*x**5*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 30*I*a**7*sqrt(c)*x**4*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 40*I*a**6*sqrt(c)*x**3*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 100*I*a**5*sqrt(c)*x**2*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 96*I*a**4*sqrt(c)*x*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 30*I*a**3*sqrt(c)*sqrt(a*x - 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)), Abs(a*x) > 1), (16*I*a**(19/2)*sqrt(c)*x**(13/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 48*I*a**(17/2)*sqrt(c)*x**(11/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 48*I*a**(15/2)*sqrt(c)*x**(9/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 16*I*a**(13/2)*sqrt(c)*x**(7/2)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 16*a**9*sqrt(c)*x**6*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 40*a**8*sqrt(c)*x**5*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 30*a**7*sqrt(c)*x**4*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 40*a**6*sqrt(c)*x**3*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 100*a**5*sqrt(c)*x**2*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) - 96*a**4*sqrt(c)*x*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)) + 30*a**3*sqrt(c)*sqrt(-a*x + 1)/(-105*I*a**(13/2)*x**(13/2) + 315*I*a**(11/2)*x**(11/2) - 315*I*a**(9/2)*x**(9/2) + 105*I*a**(7/2)*x**(7/2)), True))/a**4","C",0
519,1,777,0,25.991701," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(7/2),x)","- c^{3} \left(\begin{cases} \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{\sqrt{c} \operatorname{acosh}{\left(\sqrt{a} \sqrt{x} \right)}}{a} - \frac{\sqrt{c} \sqrt{x}}{\sqrt{a} \sqrt{a x - 1}} & \text{for}\: \left|{a x}\right| > 1 \\\frac{i \sqrt{c} \operatorname{asin}{\left(\sqrt{a} \sqrt{x} \right)}}{a} + \frac{i \sqrt{c} \sqrt{x} \sqrt{- a x + 1}}{\sqrt{a}} & \text{otherwise} \end{cases}\right) - \frac{2 c^{4} \operatorname{atan}{\left(\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right)}}{a \sqrt{- c}} - \frac{2 c^{3} \sqrt{c - \frac{c}{a x}}}{a} + \frac{c^{3} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{2 a \left(c - \frac{c}{a x}\right)^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{c^{3} \left(\begin{cases} \frac{4 i a^{\frac{11}{2}} \sqrt{c} x^{\frac{7}{2}}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 i a^{\frac{9}{2}} \sqrt{c} x^{\frac{5}{2}}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 i a^{5} \sqrt{c} x^{3} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{2 i a^{4} \sqrt{c} x^{2} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{8 i a^{3} \sqrt{c} x \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{6 i a^{2} \sqrt{c} \sqrt{a x - 1}}{- 15 i a^{\frac{7}{2}} x^{\frac{7}{2}} + 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} & \text{for}\: \left|{a x}\right| > 1 \\- \frac{4 i a^{\frac{11}{2}} \sqrt{c} x^{\frac{7}{2}}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{4 i a^{\frac{9}{2}} \sqrt{c} x^{\frac{5}{2}}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{4 a^{5} \sqrt{c} x^{3} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{2 a^{4} \sqrt{c} x^{2} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} + \frac{8 a^{3} \sqrt{c} x \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} - \frac{6 a^{2} \sqrt{c} \sqrt{- a x + 1}}{15 i a^{\frac{7}{2}} x^{\frac{7}{2}} - 15 i a^{\frac{5}{2}} x^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right)}{a^{3}}"," ",0,"-c**3*Piecewise((sqrt(a)*sqrt(c)*x**(3/2)/sqrt(a*x - 1) - sqrt(c)*acosh(sqrt(a)*sqrt(x))/a - sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(a*x - 1)), Abs(a*x) > 1), (I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)*sqrt(-a*x + 1)/sqrt(a), True)) - 2*c**4*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) - 2*c**3*sqrt(c - c/(a*x))/a + c**3*Piecewise((0, Eq(c, 0)), (2*a*(c - c/(a*x))**(3/2)/(3*c), True))/a**2 - c**3*Piecewise((4*I*a**(11/2)*sqrt(c)*x**(7/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**(9/2)*sqrt(c)*x**(5/2)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 4*I*a**5*sqrt(c)*x**3*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) + 2*I*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) + 8*I*a**3*sqrt(c)*x*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)) - 6*I*a**2*sqrt(c)*sqrt(a*x - 1)/(-15*I*a**(7/2)*x**(7/2) + 15*I*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*I*a**(11/2)*sqrt(c)*x**(7/2)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 4*I*a**(9/2)*sqrt(c)*x**(5/2)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 4*a**5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 2*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) + 8*a**3*sqrt(c)*x*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)) - 6*a**2*sqrt(c)*sqrt(-a*x + 1)/(15*I*a**(7/2)*x**(7/2) - 15*I*a**(5/2)*x**(5/2)), True))/a**3","C",0
520,1,143,0,6.734818," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(5/2),x)","- c^{2} \left(\begin{cases} \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{\sqrt{c} \operatorname{acosh}{\left(\sqrt{a} \sqrt{x} \right)}}{a} - \frac{\sqrt{c} \sqrt{x}}{\sqrt{a} \sqrt{a x - 1}} & \text{for}\: \left|{a x}\right| > 1 \\\frac{i \sqrt{c} \operatorname{asin}{\left(\sqrt{a} \sqrt{x} \right)}}{a} + \frac{i \sqrt{c} \sqrt{x} \sqrt{- a x + 1}}{\sqrt{a}} & \text{otherwise} \end{cases}\right) + \frac{c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{2 a \left(c - \frac{c}{a x}\right)^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"-c**2*Piecewise((sqrt(a)*sqrt(c)*x**(3/2)/sqrt(a*x - 1) - sqrt(c)*acosh(sqrt(a)*sqrt(x))/a - sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(a*x - 1)), Abs(a*x) > 1), (I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)*sqrt(-a*x + 1)/sqrt(a), True)) + c**2*Piecewise((0, Eq(c, 0)), (2*a*(c - c/(a*x))**(3/2)/(3*c), True))/a**2","C",0
521,1,163,0,30.513277," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(3/2),x)","- c \left(\begin{cases} \frac{\sqrt{a} \sqrt{c} x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{\sqrt{c} \operatorname{acosh}{\left(\sqrt{a} \sqrt{x} \right)}}{a} - \frac{\sqrt{c} \sqrt{x}}{\sqrt{a} \sqrt{a x - 1}} & \text{for}\: \left|{a x}\right| > 1 \\\frac{i \sqrt{c} \operatorname{asin}{\left(\sqrt{a} \sqrt{x} \right)}}{a} + \frac{i \sqrt{c} \sqrt{x} \sqrt{- a x + 1}}{\sqrt{a}} & \text{otherwise} \end{cases}\right) + \frac{2 c^{2} \operatorname{atan}{\left(\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right)}}{a \sqrt{- c}} + \frac{2 c \sqrt{c - \frac{c}{a x}}}{a}"," ",0,"-c*Piecewise((sqrt(a)*sqrt(c)*x**(3/2)/sqrt(a*x - 1) - sqrt(c)*acosh(sqrt(a)*sqrt(x))/a - sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(a*x - 1)), Abs(a*x) > 1), (I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)*sqrt(-a*x + 1)/sqrt(a), True)) + 2*c**2*atan(sqrt(c - c/(a*x))/sqrt(-c))/(a*sqrt(-c)) + 2*c*sqrt(c - c/(a*x))/a","C",0
522,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2),x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x - 1), x)","F",0
523,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**(1/2),x)","- \int \frac{a x}{a x \sqrt{c - \frac{c}{a x}} - \sqrt{c - \frac{c}{a x}}}\, dx - \int \frac{1}{a x \sqrt{c - \frac{c}{a x}} - \sqrt{c - \frac{c}{a x}}}\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(c - c/(a*x)) - sqrt(c - c/(a*x))), x) - Integral(1/(a*x*sqrt(c - c/(a*x)) - sqrt(c - c/(a*x))), x)","F",0
524,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**(3/2),x)","- \int \frac{a x}{a c x \sqrt{c - \frac{c}{a x}} - 2 c \sqrt{c - \frac{c}{a x}} + \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\, dx - \int \frac{1}{a c x \sqrt{c - \frac{c}{a x}} - 2 c \sqrt{c - \frac{c}{a x}} + \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\, dx"," ",0,"-Integral(a*x/(a*c*x*sqrt(c - c/(a*x)) - 2*c*sqrt(c - c/(a*x)) + c*sqrt(c - c/(a*x))/(a*x)), x) - Integral(1/(a*c*x*sqrt(c - c/(a*x)) - 2*c*sqrt(c - c/(a*x)) + c*sqrt(c - c/(a*x))/(a*x)), x)","F",0
525,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**(5/2),x)","- \int \frac{a x}{a c^{2} x \sqrt{c - \frac{c}{a x}} - 3 c^{2} \sqrt{c - \frac{c}{a x}} + \frac{3 c^{2} \sqrt{c - \frac{c}{a x}}}{a x} - \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}}}\, dx - \int \frac{1}{a c^{2} x \sqrt{c - \frac{c}{a x}} - 3 c^{2} \sqrt{c - \frac{c}{a x}} + \frac{3 c^{2} \sqrt{c - \frac{c}{a x}}}{a x} - \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}}}\, dx"," ",0,"-Integral(a*x/(a*c**2*x*sqrt(c - c/(a*x)) - 3*c**2*sqrt(c - c/(a*x)) + 3*c**2*sqrt(c - c/(a*x))/(a*x) - c**2*sqrt(c - c/(a*x))/(a**2*x**2)), x) - Integral(1/(a*c**2*x*sqrt(c - c/(a*x)) - 3*c**2*sqrt(c - c/(a*x)) + 3*c**2*sqrt(c - c/(a*x))/(a*x) - c**2*sqrt(c - c/(a*x))/(a**2*x**2)), x)","F",0
526,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a/x)**(7/2),x)","- \int \frac{a x}{a c^{3} x \sqrt{c - \frac{c}{a x}} - 4 c^{3} \sqrt{c - \frac{c}{a x}} + \frac{6 c^{3} \sqrt{c - \frac{c}{a x}}}{a x} - \frac{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} + \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}}}\, dx - \int \frac{1}{a c^{3} x \sqrt{c - \frac{c}{a x}} - 4 c^{3} \sqrt{c - \frac{c}{a x}} + \frac{6 c^{3} \sqrt{c - \frac{c}{a x}}}{a x} - \frac{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} + \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}}}\, dx"," ",0,"-Integral(a*x/(a*c**3*x*sqrt(c - c/(a*x)) - 4*c**3*sqrt(c - c/(a*x)) + 6*c**3*sqrt(c - c/(a*x))/(a*x) - 4*c**3*sqrt(c - c/(a*x))/(a**2*x**2) + c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x) - Integral(1/(a*c**3*x*sqrt(c - c/(a*x)) - 4*c**3*sqrt(c - c/(a*x)) + 6*c**3*sqrt(c - c/(a*x))/(a*x) - 4*c**3*sqrt(c - c/(a*x))/(a**2*x**2) + c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x)","F",0
527,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(9/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{9}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(9/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
528,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(7/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
529,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(5/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
530,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(3/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
531,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
532,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(1/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
533,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(3/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-c*(-1 + 1/(a*x)))**(3/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
534,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(5/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-c*(-1 + 1/(a*x)))**(5/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
535,-1,0,0,0.000000," ","integrate((c-c/a/x)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
536,0,0,0,0.000000," ","integrate((c-c/a/x)**(7/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(7/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
537,0,0,0,0.000000," ","integrate((c-c/a/x)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
538,0,0,0,0.000000," ","integrate((c-c/a/x)**(3/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(3/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
539,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
540,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)), x)","F",0
541,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(3/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x)))**(3/2)*(a*x + 1)), x)","F",0
542,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(5/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)), x)","F",0
543,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(7/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x)))**(7/2)*(a*x + 1)), x)","F",0
544,0,0,0,0.000000," ","integrate((c-c/a/x)**(9/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{5 c^{4} \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{10 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left(- \frac{10 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right)\, dx - \int \frac{5 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \left(- \frac{c^{4} \sqrt{c - \frac{c}{a x}}}{a^{5} x^{5} + a^{4} x^{4}}\right)\, dx - \int \frac{a c^{4} x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-5*c**4*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(10*c**4*sqrt(c - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(-10*c**4*sqrt(c - c/(a*x))/(a**3*x**3 + a**2*x**2), x) - Integral(5*c**4*sqrt(c - c/(a*x))/(a**4*x**4 + a**3*x**3), x) - Integral(-c**4*sqrt(c - c/(a*x))/(a**5*x**5 + a**4*x**4), x) - Integral(a*c**4*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
545,0,0,0,0.000000," ","integrate((c-c/a/x)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{6 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left(- \frac{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right)\, dx - \int \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \frac{a c^{3} x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-4*c**3*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(6*c**3*sqrt(c - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(-4*c**3*sqrt(c - c/(a*x))/(a**3*x**3 + a**2*x**2), x) - Integral(c**3*sqrt(c - c/(a*x))/(a**4*x**4 + a**3*x**3), x) - Integral(a*c**3*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
546,0,0,0,0.000000," ","integrate((c-c/a/x)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{3 c^{2} \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{3 c^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left(- \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right)\, dx - \int \frac{a c^{2} x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-3*c**2*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(3*c**2*sqrt(c - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(-c**2*sqrt(c - c/(a*x))/(a**3*x**3 + a**2*x**2), x) - Integral(a*c**2*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
547,0,0,0,0.000000," ","integrate((c-c/a/x)**(3/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{2 c \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{c \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \frac{a c x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-2*c*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(c*sqrt(c - c/(a*x))/(a**2*x**2 + a*x), x) - Integral(a*c*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
548,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
549,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(1/2),x)","- \int \frac{a x}{a x \sqrt{c - \frac{c}{a x}} + \sqrt{c - \frac{c}{a x}}}\, dx - \int \left(- \frac{1}{a x \sqrt{c - \frac{c}{a x}} + \sqrt{c - \frac{c}{a x}}}\right)\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(c - c/(a*x)) + sqrt(c - c/(a*x))), x) - Integral(-1/(a*x*sqrt(c - c/(a*x)) + sqrt(c - c/(a*x))), x)","F",0
550,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(3/2),x)","- \int \frac{a x}{a c x \sqrt{c - \frac{c}{a x}} - \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\, dx - \int \left(- \frac{1}{a c x \sqrt{c - \frac{c}{a x}} - \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\right)\, dx"," ",0,"-Integral(a*x/(a*c*x*sqrt(c - c/(a*x)) - c*sqrt(c - c/(a*x))/(a*x)), x) - Integral(-1/(a*c*x*sqrt(c - c/(a*x)) - c*sqrt(c - c/(a*x))/(a*x)), x)","F",0
551,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(5/2),x)","- \int \frac{a x}{a c^{2} x \sqrt{c - \frac{c}{a x}} - c^{2} \sqrt{c - \frac{c}{a x}} - \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a x} + \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}}}\, dx - \int \left(- \frac{1}{a c^{2} x \sqrt{c - \frac{c}{a x}} - c^{2} \sqrt{c - \frac{c}{a x}} - \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a x} + \frac{c^{2} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c**2*x*sqrt(c - c/(a*x)) - c**2*sqrt(c - c/(a*x)) - c**2*sqrt(c - c/(a*x))/(a*x) + c**2*sqrt(c - c/(a*x))/(a**2*x**2)), x) - Integral(-1/(a*c**2*x*sqrt(c - c/(a*x)) - c**2*sqrt(c - c/(a*x)) - c**2*sqrt(c - c/(a*x))/(a*x) + c**2*sqrt(c - c/(a*x))/(a**2*x**2)), x)","F",0
552,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(7/2),x)","- \int \frac{a x}{a c^{3} x \sqrt{c - \frac{c}{a x}} - 2 c^{3} \sqrt{c - \frac{c}{a x}} + \frac{2 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} - \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}}}\, dx - \int \left(- \frac{1}{a c^{3} x \sqrt{c - \frac{c}{a x}} - 2 c^{3} \sqrt{c - \frac{c}{a x}} + \frac{2 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} - \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c**3*x*sqrt(c - c/(a*x)) - 2*c**3*sqrt(c - c/(a*x)) + 2*c**3*sqrt(c - c/(a*x))/(a**2*x**2) - c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x) - Integral(-1/(a*c**3*x*sqrt(c - c/(a*x)) - 2*c**3*sqrt(c - c/(a*x)) + 2*c**3*sqrt(c - c/(a*x))/(a**2*x**2) - c**3*sqrt(c - c/(a*x))/(a**3*x**3)), x)","F",0
553,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(9/2),x)","- \int \frac{a x}{a c^{4} x \sqrt{c - \frac{c}{a x}} - 3 c^{4} \sqrt{c - \frac{c}{a x}} + \frac{2 c^{4} \sqrt{c - \frac{c}{a x}}}{a x} + \frac{2 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} - \frac{3 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}} + \frac{c^{4} \sqrt{c - \frac{c}{a x}}}{a^{4} x^{4}}}\, dx - \int \left(- \frac{1}{a c^{4} x \sqrt{c - \frac{c}{a x}} - 3 c^{4} \sqrt{c - \frac{c}{a x}} + \frac{2 c^{4} \sqrt{c - \frac{c}{a x}}}{a x} + \frac{2 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2}} - \frac{3 c^{4} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3}} + \frac{c^{4} \sqrt{c - \frac{c}{a x}}}{a^{4} x^{4}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c**4*x*sqrt(c - c/(a*x)) - 3*c**4*sqrt(c - c/(a*x)) + 2*c**4*sqrt(c - c/(a*x))/(a*x) + 2*c**4*sqrt(c - c/(a*x))/(a**2*x**2) - 3*c**4*sqrt(c - c/(a*x))/(a**3*x**3) + c**4*sqrt(c - c/(a*x))/(a**4*x**4)), x) - Integral(-1/(a*c**4*x*sqrt(c - c/(a*x)) - 3*c**4*sqrt(c - c/(a*x)) + 2*c**4*sqrt(c - c/(a*x))/(a*x) + 2*c**4*sqrt(c - c/(a*x))/(a**2*x**2) - 3*c**4*sqrt(c - c/(a*x))/(a**3*x**3) + c**4*sqrt(c - c/(a*x))/(a**4*x**4)), x)","F",0
554,-1,0,0,0.000000," ","integrate((c-c/a/x)**(9/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
555,-1,0,0,0.000000," ","integrate((c-c/a/x)**(7/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
556,0,0,0,0.000000," ","integrate((c-c/a/x)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(5/2)*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
557,0,0,0,0.000000," ","integrate((c-c/a/x)**(3/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**(3/2)*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
558,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
559,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(1/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3), x)","F",0
560,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x)))**(3/2)*(a*x + 1)**3), x)","F",0
561,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(5/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x)))**(5/2)*(a*x + 1)**3), x)","F",0
562,-1,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
563,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)/x**3,x)","\frac{a \left(\int \frac{3 a x}{- a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + a x^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c}"," ",0,"a*(Integral(3*a*x/(-a**3*x**5*sqrt(-a**2*x**2 + 1) + a**2*x**4*sqrt(-a**2*x**2 + 1) + a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**3*x**5*sqrt(-a**2*x**2 + 1) + a**2*x**4*sqrt(-a**2*x**2 + 1) + a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**3*x**5*sqrt(-a**2*x**2 + 1) + a**2*x**4*sqrt(-a**2*x**2 + 1) + a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**3*x**5*sqrt(-a**2*x**2 + 1) + a**2*x**4*sqrt(-a**2*x**2 + 1) + a*x**3*sqrt(-a**2*x**2 + 1) - x**2*sqrt(-a**2*x**2 + 1)), x))/c","F",0
564,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(c-c/a/x)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
565,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(c-c/a/x)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
566,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(c-c/a/x)**(1/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
567,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
568,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
569,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
570,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
571,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x^{4} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x**4*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
572,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a/x)**(1/2)/x**5,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x^{5} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)/(x**5*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
573,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(c-c/a/x)**(1/2),x)","- \int \frac{x^{3} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x^{4} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx"," ",0,"-Integral(x**3*sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x**4*sqrt(c - c/(a*x))/(a*x - 1), x)","F",0
574,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(c-c/a/x)**(1/2),x)","- \int \frac{x^{2} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x^{3} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx"," ",0,"-Integral(x**2*sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x**3*sqrt(c - c/(a*x))/(a*x - 1), x)","F",0
575,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(c-c/a/x)**(1/2),x)","- \int \frac{x \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x^{2} \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx"," ",0,"-Integral(x*sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x**2*sqrt(c - c/(a*x))/(a*x - 1), x)","F",0
576,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2),x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x - 1}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x - 1), x)","F",0
577,1,39,0,9.647237," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x,x)","\frac{2 c \operatorname{atan}{\left(\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - 2 \sqrt{c - \frac{c}{a x}}"," ",0,"2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sqrt(-c) - 2*sqrt(c - c/(a*x))","A",0
578,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**2,x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x^{3} - x^{2}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{3} - x^{2}}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x**3 - x**2), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**3 - x**2), x)","F",0
579,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**3,x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x^{4} - x^{3}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{4} - x^{3}}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x**4 - x**3), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**4 - x**3), x)","F",0
580,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**4,x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x^{5} - x^{4}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{5} - x^{4}}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x**5 - x**4), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**5 - x**4), x)","F",0
581,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**5,x)","- \int \frac{\sqrt{c - \frac{c}{a x}}}{a x^{6} - x^{5}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{6} - x^{5}}\, dx"," ",0,"-Integral(sqrt(c - c/(a*x))/(a*x**6 - x**5), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**6 - x**5), x)","F",0
582,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(c-c/a/x)**(1/2),x)","\int \frac{x^{3} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
583,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(c-c/a/x)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
584,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(c-c/a/x)**(1/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
585,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
586,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
587,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
588,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
589,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{4} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
590,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x**5,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{5} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x**5*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
591,0,0,0,0.000000," ","integrate(x**m*(c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
592,0,0,0,0.000000," ","integrate(x**2*(c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
593,0,0,0,0.000000," ","integrate(x*(c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
594,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
595,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(x*(a*x + 1)), x)","F",0
596,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)), x)","F",0
597,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{3} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)), x)","F",0
598,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{x^{4} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*sqrt(-(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)), x)","F",0
599,0,0,0,0.000000," ","integrate(x**3*(c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{3} \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{a x^{4} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-x**3*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x**4*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
600,0,0,0,0.000000," ","integrate(x**2*(c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{2} \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{a x^{3} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-x**2*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x**3*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
601,0,0,0,0.000000," ","integrate(x*(c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x \sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{a x^{2} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-x*sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x**2*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
602,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x + 1}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x + 1), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x + 1), x)","F",0
603,1,80,0,12.330547," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x,x)","\frac{2 c \operatorname{atan}{\left(\frac{\sqrt{c - \frac{c}{a x}}}{\sqrt{- c}} \right)}}{\sqrt{- c}} - \frac{4 \sqrt{2} c \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{c - \frac{c}{a x}}}{2 \sqrt{- c}} \right)}}{\sqrt{- c}} - 2 \sqrt{c - \frac{c}{a x}}"," ",0,"2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*c*atan(sqrt(2)*sqrt(c - c/(a*x))/(2*sqrt(-c)))/sqrt(-c) - 2*sqrt(c - c/(a*x))","A",0
604,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**2,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x^{3} + x^{2}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{3} + x^{2}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x**3 + x**2), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**3 + x**2), x)","F",0
605,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x^{4} + x^{3}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{4} + x^{3}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x**4 + x**3), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**4 + x**3), x)","F",0
606,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x^{5} + x^{4}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{5} + x^{4}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x**5 + x**4), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**5 + x**4), x)","F",0
607,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a x}}}{a x^{6} + x^{5}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a x}}}{a x^{6} + x^{5}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a*x))/(a*x**6 + x**5), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**6 + x**5), x)","F",0
608,0,0,0,0.000000," ","integrate(x**3*(c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{3} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
609,0,0,0,0.000000," ","integrate(x**2*(c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
610,0,0,0,0.000000," ","integrate(x*(c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
611,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
612,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x*(a*x + 1)**3), x)","F",0
613,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{2} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**2*(a*x + 1)**3), x)","F",0
614,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{3} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**3*(a*x + 1)**3), x)","F",0
615,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{4} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**4*(a*x + 1)**3), x)","F",0
616,0,0,0,0.000000," ","integrate((c-c/a/x)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{x^{5} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**5*(a*x + 1)**3), x)","F",0
617,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a/x)**p,x)","\int \left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*exp(n*atanh(a*x)), x)","F",0
618,0,0,0,0.000000," ","integrate((c-c/a/x)**p/exp(2*p*atanh(a*x)),x)","\int \left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} e^{- 2 p \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*exp(-2*p*atanh(a*x)), x)","F",0
619,0,0,0,0.000000," ","integrate(exp(2*p*atanh(a*x))*(c-c/a/x)**p,x)","\int \left(- c \left(-1 + \frac{1}{a x}\right)\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x)))**p*exp(2*p*atanh(a*x)), x)","F",0
620,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a/x)**2,x)","\frac{c^{2} \left(\int a^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2}}\, dx + \int \left(- \frac{2 a e^{n \operatorname{atanh}{\left(a x \right)}}}{x}\right)\, dx\right)}{a^{2}}"," ",0,"c**2*(Integral(a**2*exp(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x))/x**2, x) + Integral(-2*a*exp(n*atanh(a*x))/x, x))/a**2","F",0
621,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a/x),x)","\frac{c \left(\int a e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x}\right)\, dx\right)}{a}"," ",0,"c*(Integral(a*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x))/x, x))/a","F",0
622,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a/x),x)","\frac{a \int \frac{x e^{n \operatorname{atanh}{\left(a x \right)}}}{a x - 1}\, dx}{c}"," ",0,"a*Integral(x*exp(n*atanh(a*x))/(a*x - 1), x)/c","F",0
623,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a/x)**2,x)","\frac{a^{2} \int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}}"," ",0,"a**2*Integral(x**2*exp(n*atanh(a*x))/(a**2*x**2 - 2*a*x + 1), x)/c**2","F",0
624,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a/x)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
625,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a/x)**(1/2),x)","\int \sqrt{- c \left(-1 + \frac{1}{a x}\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x)))*exp(n*atanh(a*x)), x)","F",0
626,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a/x)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/sqrt(-c*(-1 + 1/(a*x))), x)","F",0
627,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a/x)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(-1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(-c*(-1 + 1/(a*x)))**(3/2), x)","F",0
628,1,1119,0,26.209893," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**4,x)","a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{4 c^{4} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{4 c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{6 c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{6 c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}} - \frac{4 c^{4} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{4 c^{4} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)}{a^{6}} + \frac{c^{4} \left(\begin{cases} - \frac{5 a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} + \frac{5 a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{5 a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{5 i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} - \frac{5 i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{5 i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{7}} + \frac{c^{4} \left(\begin{cases} - \frac{16 a^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{2}} - \frac{6 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{16 i a^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35} - \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{2}} - \frac{6 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right)}{a^{8}}"," ",0,"a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 4*c**4*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 4*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 6*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + 6*c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 4*c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - 4*c**4*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))/a**6 + c**4*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**7 + c**4*Piecewise((-16*a**7*sqrt(-1 + 1/(a**2*x**2))/35 - 8*a**5*sqrt(-1 + 1/(a**2*x**2))/(35*x**2) - 6*a**3*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (-16*I*a**7*sqrt(1 - 1/(a**2*x**2))/35 - 8*I*a**5*sqrt(1 - 1/(a**2*x**2))/(35*x**2) - 6*I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))/a**8","A",0
629,1,687,0,16.349025," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**3,x)","a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{3 c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{3 c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{3 c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{3 c^{3} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}} - \frac{c^{3} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{c^{3} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)}{a^{6}}"," ",0,"a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 3*c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 3*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 3*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + 3*c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))/a**6","A",0
630,1,354,0,9.860081," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**2,x)","a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{2 c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{2 c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 2*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 2*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4","A",0
631,1,144,0,5.602899," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2),x)","a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2","A",0
632,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2),x)","\frac{a^{2} \int \frac{x^{2}}{a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"a**2*Integral(x**2/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x)/c","F",0
633,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**2,x)","\frac{a^{4} \int \frac{x^{4}}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"a**4*Integral(x**4/(a**3*x**3*sqrt(-a**2*x**2 + 1) - a**2*x**2*sqrt(-a**2*x**2 + 1) - a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**2","F",0
634,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**3,x)","\frac{a^{6} \int \frac{x^{6}}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"a**6*Integral(x**6/(a**5*x**5*sqrt(-a**2*x**2 + 1) - a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) + 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x)/c**3","F",0
635,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**4,x)","\frac{a^{8} \int \frac{x^{8}}{a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 3 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"a**8*Integral(x**8/(a**7*x**7*sqrt(-a**2*x**2 + 1) - a**6*x**6*sqrt(-a**2*x**2 + 1) - 3*a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) + 3*a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) - a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**4","F",0
636,1,126,0,0.815847," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**5,x)","\frac{- a^{10} c^{5} x - 2 a^{9} c^{5} \log{\left(x \right)} - \frac{3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}}"," ",0,"(-a**10*c**5*x - 2*a**9*c**5*log(x) - (3780*a**8*c**5*x**8 + 5040*a**7*c**5*x**7 - 840*a**6*c**5*x**6 - 3780*a**5*c**5*x**5 - 504*a**4*c**5*x**4 + 1680*a**3*c**5*x**3 + 540*a**2*c**5*x**2 - 315*a*c**5*x - 140*c**5)/(1260*x**9))/a**10","A",0
637,1,90,0,0.488243," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**4,x)","\frac{- a^{8} c^{4} x - 2 a^{7} c^{4} \log{\left(x \right)} - \frac{420 a^{6} c^{4} x^{6} + 630 a^{5} c^{4} x^{5} - 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}}"," ",0,"(-a**8*c**4*x - 2*a**7*c**4*log(x) - (420*a**6*c**4*x**6 + 630*a**5*c**4*x**5 - 315*a**3*c**4*x**3 - 84*a**2*c**4*x**2 + 70*a*c**4*x + 30*c**4)/(210*x**7))/a**8","A",0
638,1,78,0,0.352187," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**3,x)","\frac{- a^{6} c^{3} x - 2 a^{5} c^{3} \log{\left(x \right)} - \frac{30 a^{4} c^{3} x^{4} + 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} - 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}}"," ",0,"(-a**6*c**3*x - 2*a**5*c**3*log(x) - (30*a**4*c**3*x**4 + 60*a**3*c**3*x**3 + 10*a**2*c**3*x**2 - 15*a*c**3*x - 6*c**3)/(30*x**5))/a**6","A",0
639,1,41,0,0.201987," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**2,x)","\frac{- a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left(x \right)} - \frac{3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}}"," ",0,"(-a**4*c**2*x - 2*a**3*c**2*log(x) - (3*a*c**2*x + c**2)/(3*x**3))/a**4","A",0
640,1,20,0,0.120694," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2),x)","\frac{- a^{2} c x - 2 a c \log{\left(x \right)} + \frac{c}{x}}{a^{2}}"," ",0,"(-a**2*c*x - 2*a*c*log(x) + c/x)/a**2","A",0
641,1,37,0,0.166342," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2),x)","- a^{2} \left(- \frac{1}{a^{4} c x - a^{3} c} + \frac{x}{a^{2} c} + \frac{2 \log{\left(a x - 1 \right)}}{a^{3} c}\right)"," ",0,"-a**2*(-1/(a**4*c*x - a**3*c) + x/(a**2*c) + 2*log(a*x - 1)/(a**3*c))","A",0
642,1,75,0,0.410702," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**2,x)","- a^{4} \left(\frac{- 7 a x + 6}{4 a^{7} c^{2} x^{2} - 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac{x}{a^{4} c^{2}} + \frac{\frac{17 \log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{5} c^{2}}\right)"," ",0,"-a**4*((-7*a*x + 6)/(4*a**7*c**2*x**2 - 8*a**6*c**2*x + 4*a**5*c**2) + x/(a**4*c**2) + (17*log(x - 1/a)/8 - log(x + 1/a)/8)/(a**5*c**2))","A",0
643,1,104,0,0.623780," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**3,x)","- a^{6} \left(\frac{- 15 a^{3} x^{3} + 12 a^{2} x^{2} + 13 a x - 11}{6 a^{11} c^{3} x^{4} - 12 a^{10} c^{3} x^{3} + 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac{x}{a^{6} c^{3}} + \frac{\frac{9 \log{\left(x - \frac{1}{a} \right)}}{4} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{7} c^{3}}\right)"," ",0,"-a**6*((-15*a**3*x**3 + 12*a**2*x**2 + 13*a*x - 11)/(6*a**11*c**3*x**4 - 12*a**10*c**3*x**3 + 12*a**8*c**3*x - 6*a**7*c**3) + x/(a**6*c**3) + (9*log(x - 1/a)/4 - log(x + 1/a)/4)/(a**7*c**3))","A",0
644,1,158,0,0.886640," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**4,x)","- a^{8} \left(\frac{- 627 a^{5} x^{5} + 486 a^{4} x^{4} + 1058 a^{3} x^{3} - 874 a^{2} x^{2} - 467 a x + 400}{192 a^{15} c^{4} x^{6} - 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} + 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} - 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{303 \log{\left(x - \frac{1}{a} \right)}}{128} - \frac{47 \log{\left(x + \frac{1}{a} \right)}}{128}}{a^{9} c^{4}}\right)"," ",0,"-a**8*((-627*a**5*x**5 + 486*a**4*x**4 + 1058*a**3*x**3 - 874*a**2*x**2 - 467*a*x + 400)/(192*a**15*c**4*x**6 - 384*a**14*c**4*x**5 - 192*a**13*c**4*x**4 + 768*a**12*c**4*x**3 - 192*a**11*c**4*x**2 - 384*a**10*c**4*x + 192*a**9*c**4) + x/(a**8*c**4) + (303*log(x - 1/a)/128 - 47*log(x + 1/a)/128)/(a**9*c**4))","A",0
645,1,935,0,30.595800," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**4,x)","- a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 c^{4} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + \frac{8 c^{4} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{6 c^{4} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} - \frac{6 c^{4} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}} - \frac{8 c^{4} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} + \frac{3 c^{4} \left(\begin{cases} - \frac{5 a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} + \frac{5 a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{5 a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{5 i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} - \frac{5 i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{5 i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{7}} + \frac{c^{4} \left(\begin{cases} - \frac{16 a^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{2}} - \frac{6 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{16 i a^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35} - \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{2}} - \frac{6 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right)}{a^{8}}"," ",0,"-a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**4*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 8*c**4*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 6*c**4*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 - 6*c**4*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 8*c**4*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 + 3*c**4*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**7 + c**4*Piecewise((-16*a**7*sqrt(-1 + 1/(a**2*x**2))/35 - 8*a**5*sqrt(-1 + 1/(a**2*x**2))/(35*x**2) - 6*a**3*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (-16*I*a**7*sqrt(1 - 1/(a**2*x**2))/35 - 8*I*a**5*sqrt(1 - 1/(a**2*x**2))/(35*x**2) - 6*I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))/a**8","A",0
646,1,687,0,19.486899," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**3,x)","- a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 c^{3} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{c^{3} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{5 c^{3} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{5 c^{3} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} - \frac{c^{3} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}} - \frac{3 c^{3} \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{c^{3} \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)}{a^{6}}"," ",0,"-a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**3*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c**3*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 5*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 5*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 - c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 3*c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))/a**6","A",0
647,1,357,0,16.833732," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**2,x)","- a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 c^{2} \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{2 c^{2} \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{2 c^{2} \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}} + \frac{3 c^{2} \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{2} \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"-a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**2*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 2*c**2*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 2*c**2*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 3*c**2*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + c**2*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4","A",0
648,1,150,0,12.327267," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2),x)","- a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) - 3 c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) - \frac{3 c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"-a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 3*c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2","A",0
649,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2),x)","\frac{a^{2} \left(\int \frac{x^{2}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{2 a x^{3}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{2} x^{4}}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c}"," ",0,"a**2*(Integral(x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(2*a*x**3/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**2*x**4/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c","F",0
650,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**2,x)","\frac{a^{4} \left(\int \frac{x^{4}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{- a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right)}{c^{2}}"," ",0,"a**4*(Integral(x**4/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(-a**4*x**4*sqrt(-a**2*x**2 + 1) + 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
651,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**3,x)","\frac{a^{6} \int \frac{x^{6}}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} - \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"a**6*Integral(x**6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x)/c**3","F",0
652,-1,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
653,1,112,0,0.662748," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**5,x)","\frac{a^{10} c^{5} x + 4 a^{9} c^{5} \log{\left(x \right)} + \frac{- 1890 a^{8} c^{5} x^{8} + 2520 a^{7} c^{5} x^{7} + 2940 a^{6} c^{5} x^{6} - 1764 a^{4} c^{5} x^{4} - 840 a^{3} c^{5} x^{3} + 270 a^{2} c^{5} x^{2} + 315 a c^{5} x + 70 c^{5}}{630 x^{9}}}{a^{10}}"," ",0,"(a**10*c**5*x + 4*a**9*c**5*log(x) + (-1890*a**8*c**5*x**8 + 2520*a**7*c**5*x**7 + 2940*a**6*c**5*x**6 - 1764*a**4*c**5*x**4 - 840*a**3*c**5*x**3 + 270*a**2*c**5*x**2 + 315*a*c**5*x + 70*c**5)/(630*x**9))/a**10","A",0
654,1,100,0,0.513084," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**4,x)","\frac{a^{8} c^{4} x + 4 a^{7} c^{4} \log{\left(x \right)} + \frac{- 420 a^{6} c^{4} x^{6} + 210 a^{5} c^{4} x^{5} + 350 a^{4} c^{4} x^{4} + 105 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x - 15 c^{4}}{105 x^{7}}}{a^{8}}"," ",0,"(a**8*c**4*x + 4*a**7*c**4*log(x) + (-420*a**6*c**4*x**6 + 210*a**5*c**4*x**5 + 350*a**4*c**4*x**4 + 105*a**3*c**4*x**3 - 84*a**2*c**4*x**2 - 70*a*c**4*x - 15*c**4)/(105*x**7))/a**8","A",0
655,1,65,0,0.319836," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**3,x)","\frac{a^{6} c^{3} x + 4 a^{5} c^{3} \log{\left(x \right)} + \frac{- 75 a^{4} c^{3} x^{4} + 25 a^{2} c^{3} x^{2} + 15 a c^{3} x + 3 c^{3}}{15 x^{5}}}{a^{6}}"," ",0,"(a**6*c**3*x + 4*a**5*c**3*log(x) + (-75*a**4*c**3*x**4 + 25*a**2*c**3*x**2 + 15*a*c**3*x + 3*c**3)/(15*x**5))/a**6","A",0
656,1,53,0,0.221358," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**2,x)","\frac{a^{4} c^{2} x + 4 a^{3} c^{2} \log{\left(x \right)} + \frac{- 18 a^{2} c^{2} x^{2} - 6 a c^{2} x - c^{2}}{3 x^{3}}}{a^{4}}"," ",0,"(a**4*c**2*x + 4*a**3*c**2*log(x) + (-18*a**2*c**2*x**2 - 6*a*c**2*x - c**2)/(3*x**3))/a**4","A",0
657,1,26,0,0.286285," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2),x)","c x + \frac{4 c \left(- \log{\left(x \right)} + 2 \log{\left(x - \frac{1}{a} \right)}\right)}{a} + \frac{c}{a^{2} x}"," ",0,"c*x + 4*c*(-log(x) + 2*log(x - 1/a))/a + c/(a**2*x)","A",0
658,1,41,0,0.244876," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a**2/x**2),x)","\frac{- 5 a x + 4}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac{x}{c} + \frac{4 \log{\left(a x - 1 \right)}}{a c}"," ",0,"(-5*a*x + 4)/(a**3*c*x**2 - 2*a**2*c*x + a*c) + x/c + 4*log(a*x - 1)/(a*c)","A",0
659,1,83,0,0.332628," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a**2/x**2)**2,x)","a^{4} \left(\frac{- 18 a^{2} x^{2} + 30 a x - 13}{3 a^{8} c^{2} x^{3} - 9 a^{7} c^{2} x^{2} + 9 a^{6} c^{2} x - 3 a^{5} c^{2}} + \frac{x}{a^{4} c^{2}} + \frac{4 \log{\left(a x - 1 \right)}}{a^{5} c^{2}}\right)"," ",0,"a**4*((-18*a**2*x**2 + 30*a*x - 13)/(3*a**8*c**2*x**3 - 9*a**7*c**2*x**2 + 9*a**6*c**2*x - 3*a**5*c**2) + x/(a**4*c**2) + 4*log(a*x - 1)/(a**5*c**2))","A",0
660,1,114,0,0.632777," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a**2/x**2)**3,x)","a^{6} \left(\frac{- 333 a^{3} x^{3} + 852 a^{2} x^{2} - 749 a x + 224}{48 a^{11} c^{3} x^{4} - 192 a^{10} c^{3} x^{3} + 288 a^{9} c^{3} x^{2} - 192 a^{8} c^{3} x + 48 a^{7} c^{3}} + \frac{x}{a^{6} c^{3}} + \frac{\frac{129 \log{\left(x - \frac{1}{a} \right)}}{32} - \frac{\log{\left(x + \frac{1}{a} \right)}}{32}}{a^{7} c^{3}}\right)"," ",0,"a**6*((-333*a**3*x**3 + 852*a**2*x**2 - 749*a*x + 224)/(48*a**11*c**3*x**4 - 192*a**10*c**3*x**3 + 288*a**9*c**3*x**2 - 192*a**8*c**3*x + 48*a**7*c**3) + x/(a**6*c**3) + (129*log(x - 1/a)/32 - log(x + 1/a)/32)/(a**7*c**3))","A",0
661,1,144,0,0.900081," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(c-c/a**2/x**2)**4,x)","a^{8} \left(\frac{- 3765 a^{5} x^{5} + 9300 a^{4} x^{4} - 4400 a^{3} x^{3} - 6820 a^{2} x^{2} + 8021 a x - 2384}{480 a^{15} c^{4} x^{6} - 1920 a^{14} c^{4} x^{5} + 2400 a^{13} c^{4} x^{4} - 2400 a^{11} c^{4} x^{2} + 1920 a^{10} c^{4} x - 480 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{261 \log{\left(x - \frac{1}{a} \right)}}{64} - \frac{5 \log{\left(x + \frac{1}{a} \right)}}{64}}{a^{9} c^{4}}\right)"," ",0,"a**8*((-3765*a**5*x**5 + 9300*a**4*x**4 - 4400*a**3*x**3 - 6820*a**2*x**2 + 8021*a*x - 2384)/(480*a**15*c**4*x**6 - 1920*a**14*c**4*x**5 + 2400*a**13*c**4*x**4 - 2400*a**11*c**4*x**2 + 1920*a**10*c**4*x - 480*a**9*c**4) + x/(a**8*c**4) + (261*log(x - 1/a)/64 - 5*log(x + 1/a)/64)/(a**9*c**4))","A",0
662,1,1110,0,16.184044," ","integrate((c-c/a**2/x**2)**4/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{4} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c^{4} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{3 c^{4} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{3 c^{4} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}} + \frac{3 c^{4} \left(\begin{cases} \frac{a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{3 c^{4} \left(\begin{cases} \frac{2 i a^{4} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right)}{a^{6}} - \frac{c^{4} \left(\begin{cases} \frac{a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{7}} + \frac{c^{4} \left(\begin{cases} \frac{8 a^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{8 i a^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right)}{a^{8}}"," ",0,"c**4*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**4*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - 3*c**4*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + 3*c**4*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + 3*c**4*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - 3*c**4*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True))/a**6 - c**4*Piecewise((a**6*acosh(1/(a*x))/16 - a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*asin(1/(a*x))/16 + I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**7 + c**4*Piecewise((8*a**7*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**5*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (8*I*a**7*sqrt(1 - 1/(a**2*x**2))/105 + 4*I*a**5*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))/a**8","C",0
663,1,692,0,10.021189," ","integrate((c-c/a**2/x**2)**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{3} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c^{3} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{2 c^{3} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{2 c^{3} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}} + \frac{c^{3} \left(\begin{cases} \frac{a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{c^{3} \left(\begin{cases} \frac{2 i a^{4} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right)}{a^{6}}"," ",0,"c**3*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**3*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - 2*c**3*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + 2*c**3*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + c**3*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True))/a**6","C",0
664,1,381,0,6.342052," ","integrate((c-c/a**2/x**2)**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c^{2} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c^{2} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{c^{2} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{2} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"c**2*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**2*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - c**2*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + c**2*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4","C",0
665,1,177,0,4.908056," ","integrate((c-c/a**2/x**2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\frac{c \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} - \frac{c \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"c*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2","C",0
666,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2),x)","\frac{a^{2} \int \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c}"," ",0,"a**2*Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + a**2*x**2 - a*x - 1), x)/c","F",0
667,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**2,x)","\frac{a^{4} \int \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx}{c^{2}}"," ",0,"a**4*Integral(x**4*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + a**4*x**4 - 2*a**3*x**3 - 2*a**2*x**2 + a*x + 1), x)/c**2","F",0
668,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**3,x)","\frac{a^{6} \int \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\, dx}{c^{3}}"," ",0,"a**6*Integral(x**6*sqrt(-a**2*x**2 + 1)/(a**7*x**7 + a**6*x**6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*a**3*x**3 + 3*a**2*x**2 - a*x - 1), x)/c**3","F",0
669,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**4,x)","\frac{a^{8} \int \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}}"," ",0,"a**8*Integral(x**8*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + a**8*x**8 - 4*a**7*x**7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x + 1), x)/c**4","F",0
670,1,88,0,0.460733," ","integrate((c-c/a**2/x**2)**4/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{- a^{8} c^{4} x + 2 a^{7} c^{4} \log{\left(x \right)} - \frac{420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}}"," ",0,"(-a**8*c**4*x + 2*a**7*c**4*log(x) - (420*a**6*c**4*x**6 - 630*a**5*c**4*x**5 + 315*a**3*c**4*x**3 - 84*a**2*c**4*x**2 - 70*a*c**4*x + 30*c**4)/(210*x**7))/a**8","A",0
671,1,76,0,0.332308," ","integrate((c-c/a**2/x**2)**3/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{- a^{6} c^{3} x + 2 a^{5} c^{3} \log{\left(x \right)} - \frac{30 a^{4} c^{3} x^{4} - 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} + 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}}"," ",0,"(-a**6*c**3*x + 2*a**5*c**3*log(x) - (30*a**4*c**3*x**4 - 60*a**3*c**3*x**3 + 10*a**2*c**3*x**2 + 15*a*c**3*x - 6*c**3)/(30*x**5))/a**6","A",0
672,1,39,0,0.189996," ","integrate((c-c/a**2/x**2)**2/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{- a^{4} c^{2} x + 2 a^{3} c^{2} \log{\left(x \right)} - \frac{- 3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}}"," ",0,"(-a**4*c**2*x + 2*a**3*c**2*log(x) - (-3*a*c**2*x + c**2)/(3*x**3))/a**4","A",0
673,1,20,0,0.113121," ","integrate((c-c/a**2/x**2)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{- a^{2} c x + 2 a c \log{\left(x \right)} + \frac{c}{x}}{a^{2}}"," ",0,"(-a**2*c*x + 2*a*c*log(x) + c/x)/a**2","A",0
674,1,37,0,0.152729," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2),x)","- a^{2} \left(- \frac{1}{a^{4} c x + a^{3} c} + \frac{x}{a^{2} c} - \frac{2 \log{\left(a x + 1 \right)}}{a^{3} c}\right)"," ",0,"-a**2*(-1/(a**4*c*x + a**3*c) + x/(a**2*c) - 2*log(a*x + 1)/(a**3*c))","A",0
675,1,76,0,0.391051," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**2,x)","- a^{4} \left(\frac{- 7 a x - 6}{4 a^{7} c^{2} x^{2} + 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac{x}{a^{4} c^{2}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{17 \log{\left(x + \frac{1}{a} \right)}}{8}}{a^{5} c^{2}}\right)"," ",0,"-a**4*((-7*a*x - 6)/(4*a**7*c**2*x**2 + 8*a**6*c**2*x + 4*a**5*c**2) + x/(a**4*c**2) + (log(x - 1/a)/8 - 17*log(x + 1/a)/8)/(a**5*c**2))","A",0
676,1,104,0,0.626140," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**3,x)","- a^{6} \left(\frac{- 15 a^{3} x^{3} - 12 a^{2} x^{2} + 13 a x + 11}{6 a^{11} c^{3} x^{4} + 12 a^{10} c^{3} x^{3} - 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac{x}{a^{6} c^{3}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{4} - \frac{9 \log{\left(x + \frac{1}{a} \right)}}{4}}{a^{7} c^{3}}\right)"," ",0,"-a**6*((-15*a**3*x**3 - 12*a**2*x**2 + 13*a*x + 11)/(6*a**11*c**3*x**4 + 12*a**10*c**3*x**3 - 12*a**8*c**3*x - 6*a**7*c**3) + x/(a**6*c**3) + (log(x - 1/a)/4 - 9*log(x + 1/a)/4)/(a**7*c**3))","A",0
677,1,158,0,0.884590," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**4,x)","- a^{8} \left(\frac{- 627 a^{5} x^{5} - 486 a^{4} x^{4} + 1058 a^{3} x^{3} + 874 a^{2} x^{2} - 467 a x - 400}{192 a^{15} c^{4} x^{6} + 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} - 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} + 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{47 \log{\left(x - \frac{1}{a} \right)}}{128} - \frac{303 \log{\left(x + \frac{1}{a} \right)}}{128}}{a^{9} c^{4}}\right)"," ",0,"-a**8*((-627*a**5*x**5 - 486*a**4*x**4 + 1058*a**3*x**3 + 874*a**2*x**2 - 467*a*x - 400)/(192*a**15*c**4*x**6 + 384*a**14*c**4*x**5 - 192*a**13*c**4*x**4 - 768*a**12*c**4*x**3 - 192*a**11*c**4*x**2 + 384*a**10*c**4*x + 192*a**9*c**4) + x/(a**8*c**4) + (47*log(x - 1/a)/128 - 303*log(x + 1/a)/128)/(a**9*c**4))","A",0
678,1,1110,0,18.713202," ","integrate((c-c/a**2/x**2)**4/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- \frac{c^{4} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{3 c^{4} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{c^{4} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} - \frac{5 c^{4} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}} + \frac{5 c^{4} \left(\begin{cases} \frac{a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} + \frac{c^{4} \left(\begin{cases} \frac{2 i a^{4} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right)}{a^{6}} - \frac{3 c^{4} \left(\begin{cases} \frac{a^{6} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{a^{5}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{6} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{i a^{5}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{7}} + \frac{c^{4} \left(\begin{cases} \frac{8 a^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{8 i a^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right)}{a^{8}}"," ",0,"-c**4*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a + 3*c**4*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - c**4*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 - 5*c**4*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + 5*c**4*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 + c**4*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True))/a**6 - 3*c**4*Piecewise((a**6*acosh(1/(a*x))/16 - a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*asin(1/(a*x))/16 + I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**7 + c**4*Piecewise((8*a**7*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**5*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (8*I*a**7*sqrt(1 - 1/(a**2*x**2))/105 + 4*I*a**5*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))/a**8","C",0
679,1,695,0,11.652451," ","integrate((c-c/a**2/x**2)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- \frac{c^{3} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{3 c^{3} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{2 c^{3} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} - \frac{2 c^{3} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}} + \frac{3 c^{3} \left(\begin{cases} \frac{a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{c^{3} \left(\begin{cases} \frac{2 i a^{4} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right)}{a^{6}}"," ",0,"-c**3*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a + 3*c**3*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - 2*c**3*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 - 2*c**3*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4 + 3*c**3*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**5 - c**3*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True))/a**6","C",0
680,1,384,0,7.629995," ","integrate((c-c/a**2/x**2)**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- \frac{c^{2} \left(\begin{cases} i \sqrt{a^{2} x^{2} - 1} - \log{\left(a x \right)} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} + i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{- a^{2} x^{2} + 1} + \frac{\log{\left(a^{2} x^{2} \right)}}{2} - \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)}{a} + \frac{3 c^{2} \left(\begin{cases} - \frac{i a^{2} x}{\sqrt{a^{2} x^{2} - 1}} + i a \operatorname{acosh}{\left(a x \right)} + \frac{i}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} x}{\sqrt{- a^{2} x^{2} + 1}} - a \operatorname{asin}{\left(a x \right)} - \frac{1}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{3 c^{2} \left(\begin{cases} \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{2} \left(\begin{cases} \frac{a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"-c**2*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a + 3*c**2*Piecewise((-I*a**2*x/sqrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - 3*c**2*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + c**2*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))/a**4","C",0
681,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\frac{c \left(\int \left(- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\right)\, dx + \int \frac{a x \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \frac{a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \left(- \frac{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\right)\, dx\right)}{a^{2}}"," ",0,"c*(Integral(-sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Integral(a*x*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Integral(a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x) + Integral(-a**3*x**3*sqrt(-a**2*x**2 + 1)/(a**2*x**4 + 2*a*x**3 + x**2), x))/a**2","F",0
682,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2),x)","\frac{a^{2} \left(\int \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + 3 a^{4} x^{4} + 2 a^{3} x^{3} - 2 a^{2} x^{2} - 3 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{5} x^{5} + 3 a^{4} x^{4} + 2 a^{3} x^{3} - 2 a^{2} x^{2} - 3 a x - 1}\right)\, dx\right)}{c}"," ",0,"a**2*(Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + 3*a**4*x**4 + 2*a**3*x**3 - 2*a**2*x**2 - 3*a*x - 1), x) + Integral(-a**2*x**4*sqrt(-a**2*x**2 + 1)/(a**5*x**5 + 3*a**4*x**4 + 2*a**3*x**3 - 2*a**2*x**2 - 3*a*x - 1), x))/c","F",0
683,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**2,x)","\frac{a^{4} \left(\int \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} + 3 a^{6} x^{6} + a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1}}{a^{7} x^{7} + 3 a^{6} x^{6} + a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + a^{2} x^{2} + 3 a x + 1}\right)\, dx\right)}{c^{2}}"," ",0,"a**4*(Integral(x**4*sqrt(-a**2*x**2 + 1)/(a**7*x**7 + 3*a**6*x**6 + a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + a**2*x**2 + 3*a*x + 1), x) + Integral(-a**2*x**6*sqrt(-a**2*x**2 + 1)/(a**7*x**7 + 3*a**6*x**6 + a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + a**2*x**2 + 3*a*x + 1), x))/c**2","F",0
684,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**3,x)","\frac{a^{6} \left(\int \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} + 3 a^{8} x^{8} - 8 a^{6} x^{6} - 6 a^{5} x^{5} + 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x - 1}\, dx + \int \left(- \frac{a^{2} x^{8} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} + 3 a^{8} x^{8} - 8 a^{6} x^{6} - 6 a^{5} x^{5} + 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x - 1}\right)\, dx\right)}{c^{3}}"," ",0,"a**6*(Integral(x**6*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + 3*a**8*x**8 - 8*a**6*x**6 - 6*a**5*x**5 + 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x - 1), x) + Integral(-a**2*x**8*sqrt(-a**2*x**2 + 1)/(a**9*x**9 + 3*a**8*x**8 - 8*a**6*x**6 - 6*a**5*x**5 + 6*a**4*x**4 + 8*a**3*x**3 - 3*a*x - 1), x))/c**3","F",0
685,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**4,x)","\frac{a^{8} \left(\int \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{a^{2} x^{10} \sqrt{- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\right)\, dx\right)}{c^{4}}"," ",0,"a**8*(Integral(x**8*sqrt(-a**2*x**2 + 1)/(a**11*x**11 + 3*a**10*x**10 - a**9*x**9 - 11*a**8*x**8 - 6*a**7*x**7 + 14*a**6*x**6 + 14*a**5*x**5 - 6*a**4*x**4 - 11*a**3*x**3 - a**2*x**2 + 3*a*x + 1), x) + Integral(-a**2*x**10*sqrt(-a**2*x**2 + 1)/(a**11*x**11 + 3*a**10*x**10 - a**9*x**9 - 11*a**8*x**8 - 6*a**7*x**7 + 14*a**6*x**6 + 14*a**5*x**5 - 6*a**4*x**4 - 11*a**3*x**3 - a**2*x**2 + 3*a*x + 1), x))/c**4","F",0
686,-1,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(9/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
687,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(7/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
688,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(5/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
689,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
690,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
691,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(1/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))), x)","F",0
692,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(3/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)), x)","F",0
693,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(5/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)), x)","F",0
694,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(7/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)), x)","F",0
695,-2,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(9/2),x)","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError","F(-2)",0
696,-2,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(7/2),x)","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError","F(-2)",0
697,-2,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(5/2),x)","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError","F(-2)",0
698,-2,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(3/2),x)","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError","F(-2)",0
699,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2),x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x - 1), x)","F",0
700,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(1/2),x)","- \int \frac{a x}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} - \sqrt{c - \frac{c}{a^{2} x^{2}}}}\, dx - \int \frac{1}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} - \sqrt{c - \frac{c}{a^{2} x^{2}}}}\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(c - c/(a**2*x**2)) - sqrt(c - c/(a**2*x**2))), x) - Integral(1/(a*x*sqrt(c - c/(a**2*x**2)) - sqrt(c - c/(a**2*x**2))), x)","F",0
701,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(3/2),x)","- \int \frac{a x}{a c x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx - \int \frac{1}{a c x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx"," ",0,"-Integral(a*x/(a*c*x*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2))/(a*x) + c*sqrt(c - c/(a**2*x**2))/(a**2*x**2)), x) - Integral(1/(a*c*x*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2))/(a*x) + c*sqrt(c - c/(a**2*x**2))/(a**2*x**2)), x)","F",0
702,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(5/2),x)","- \int \frac{a x}{a c^{2} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\, dx - \int \frac{1}{a c^{2} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\, dx"," ",0,"-Integral(a*x/(a*c**2*x*sqrt(c - c/(a**2*x**2)) - c**2*sqrt(c - c/(a**2*x**2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x) + 2*c**2*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x) - Integral(1/(a*c**2*x*sqrt(c - c/(a**2*x**2)) - c**2*sqrt(c - c/(a**2*x**2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x) + 2*c**2*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x)","F",0
703,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(7/2),x)","- \int \frac{a x}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx - \int \frac{1}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx"," ",0,"-Integral(a*x/(a*c**3*x*sqrt(c - c/(a**2*x**2)) - c**3*sqrt(c - c/(a**2*x**2)) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a*x) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + c**3*sqrt(c - c/(a**2*x**2))/(a**6*x**6)), x) - Integral(1/(a*c**3*x*sqrt(c - c/(a**2*x**2)) - c**3*sqrt(c - c/(a**2*x**2)) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a*x) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + c**3*sqrt(c - c/(a**2*x**2))/(a**6*x**6)), x)","F",0
704,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(9/2),x)","- \int \frac{a x}{a c^{4} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx - \int \frac{1}{a c^{4} x \sqrt{c - \frac{c}{a^{2} x^{2}}} - c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac{6 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac{4 c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac{c^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx"," ",0,"-Integral(a*x/(a*c**4*x*sqrt(c - c/(a**2*x**2)) - c**4*sqrt(c - c/(a**2*x**2)) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a*x) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 6*c**4*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 6*c**4*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**6*x**6) + c**4*sqrt(c - c/(a**2*x**2))/(a**7*x**7) - c**4*sqrt(c - c/(a**2*x**2))/(a**8*x**8)), x) - Integral(1/(a*c**4*x*sqrt(c - c/(a**2*x**2)) - c**4*sqrt(c - c/(a**2*x**2)) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a*x) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 6*c**4*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 6*c**4*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**6*x**6) + c**4*sqrt(c - c/(a**2*x**2))/(a**7*x**7) - c**4*sqrt(c - c/(a**2*x**2))/(a**8*x**8)), x)","F",0
705,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(9/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{9}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(9/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
706,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(7/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
707,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(5/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
708,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(3/2),x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
709,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
710,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(1/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))), x)","F",0
711,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(3/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)), x)","F",0
712,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(5/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)), x)","F",0
713,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(7/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)), x)","F",0
714,-1,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
715,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(7/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)/(a*x + 1), x)","F",0
716,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)/(a*x + 1), x)","F",0
717,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(3/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)/(a*x + 1), x)","F",0
718,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1), x)","F",0
719,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)), x)","F",0
720,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(3/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)*(a*x + 1)), x)","F",0
721,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(5/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)*(a*x + 1)), x)","F",0
722,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a**2/x**2)**(7/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)*(a*x + 1)), x)","F",0
723,1,1408,0,50.943872," ","integrate((c-c/a**2/x**2)**(9/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{4} \left(\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left(a x \right)}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} - \frac{i \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)}}{a} & \text{otherwise} \end{cases}\right) + \frac{2 c^{4} \left(\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a} + \frac{2 c^{4} \left(\begin{cases} \frac{i a \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{i \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{2 a^{2} x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{\sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{6 c^{4} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{a^{2} \left(c - \frac{c}{a^{2} x^{2}}\right)^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{6 c^{4} \left(\begin{cases} \frac{2 a^{3} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{a \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 a x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 i a^{3} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{i a \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 a x^{5}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{2 c^{4} \left(\begin{cases} \frac{i a^{5} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{i a^{4} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{i a^{2} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 i \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{6 a^{2} x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a^{5} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{a^{4} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{a^{2} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{6 a^{2} x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{6}} - \frac{2 c^{4} \left(\begin{cases} \frac{8 a^{5} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{105 x} + \frac{4 a^{3} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{105 x^{3}} + \frac{a \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{35 x^{5}} - \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{7 a x^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{8 i a^{5} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{105 x} + \frac{4 i a^{3} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{105 x^{3}} + \frac{i a \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{35 x^{5}} - \frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{7 a x^{7}} & \text{otherwise} \end{cases}\right)}{a^{7}} + \frac{c^{4} \left(\begin{cases} \frac{5 i a^{7} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{128} - \frac{5 i a^{6} \sqrt{c}}{128 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 i a^{4} \sqrt{c}}{384 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{i a^{2} \sqrt{c}}{192 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{7 i \sqrt{c}}{48 x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{8 a^{2} x^{9} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{5 a^{7} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{128} + \frac{5 a^{6} \sqrt{c}}{128 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 a^{4} \sqrt{c}}{384 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{a^{2} \sqrt{c}}{192 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{7 \sqrt{c}}{48 x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{8 a^{2} x^{9} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{8}}"," ",0,"-c**4*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**4*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + 2*c**4*Piecewise((I*a*sqrt(c)*acosh(1/(a*x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**2 - 6*c**4*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 + 6*c**4*Piecewise((2*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + a*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - sqrt(c)*sqrt(a**2*x**2 - 1)/(5*a*x**5), Abs(a**2*x**2) > 1), (2*I*a**3*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - I*sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*a*x**5), True))/a**5 - 2*c**4*Piecewise((I*a**5*sqrt(c)*acosh(1/(a*x))/16 - I*a**4*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + I*a**2*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*I*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(6*a**2*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**5*sqrt(c)*asin(1/(a*x))/16 + a**4*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - a**2*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(6*a**2*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**6 - 2*c**4*Piecewise((8*a**5*sqrt(c)*sqrt(a**2*x**2 - 1)/(105*x) + 4*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(105*x**3) + a*sqrt(c)*sqrt(a**2*x**2 - 1)/(35*x**5) - sqrt(c)*sqrt(a**2*x**2 - 1)/(7*a*x**7), Abs(a**2*x**2) > 1), (8*I*a**5*sqrt(c)*sqrt(-a**2*x**2 + 1)/(105*x) + 4*I*a**3*sqrt(c)*sqrt(-a**2*x**2 + 1)/(105*x**3) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(35*x**5) - I*sqrt(c)*sqrt(-a**2*x**2 + 1)/(7*a*x**7), True))/a**7 + c**4*Piecewise((5*I*a**7*sqrt(c)*acosh(1/(a*x))/128 - 5*I*a**6*sqrt(c)/(128*x*sqrt(-1 + 1/(a**2*x**2))) + 5*I*a**4*sqrt(c)/(384*x**3*sqrt(-1 + 1/(a**2*x**2))) + I*a**2*sqrt(c)/(192*x**5*sqrt(-1 + 1/(a**2*x**2))) + 7*I*sqrt(c)/(48*x**7*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(8*a**2*x**9*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-5*a**7*sqrt(c)*asin(1/(a*x))/128 + 5*a**6*sqrt(c)/(128*x*sqrt(1 - 1/(a**2*x**2))) - 5*a**4*sqrt(c)/(384*x**3*sqrt(1 - 1/(a**2*x**2))) - a**2*sqrt(c)/(192*x**5*sqrt(1 - 1/(a**2*x**2))) - 7*sqrt(c)/(48*x**7*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(8*a**2*x**9*sqrt(1 - 1/(a**2*x**2))), True))/a**8","C",0
724,1,1059,0,24.007422," ","integrate((c-c/a**2/x**2)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{3} \left(\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left(a x \right)}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} - \frac{i \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)}}{a} & \text{otherwise} \end{cases}\right) + \frac{2 c^{3} \left(\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a} + \frac{c^{3} \left(\begin{cases} \frac{i a \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{i \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{2 a^{2} x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{\sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{4 c^{3} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{a^{2} \left(c - \frac{c}{a^{2} x^{2}}\right)^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{3} \left(\begin{cases} \frac{i a^{3} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{i a^{2} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 i \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{4 a^{2} x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a^{3} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{a^{2} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{4 a^{2} x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{4}} + \frac{2 c^{3} \left(\begin{cases} \frac{2 a^{3} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{a \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 a x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 i a^{3} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{i a \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 a x^{5}} & \text{otherwise} \end{cases}\right)}{a^{5}} - \frac{c^{3} \left(\begin{cases} \frac{i a^{5} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{i a^{4} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{i a^{2} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 i \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{6 a^{2} x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a^{5} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{a^{4} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{a^{2} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{6 a^{2} x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{6}}"," ",0,"-c**3*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**3*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + c**3*Piecewise((I*a*sqrt(c)*acosh(1/(a*x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**2 - 4*c**3*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 + c**3*Piecewise((I*a**3*sqrt(c)*acosh(1/(a*x))/8 - I*a**2*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(4*a**2*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(4*a**2*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**4 + 2*c**3*Piecewise((2*a**3*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + a*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - sqrt(c)*sqrt(a**2*x**2 - 1)/(5*a*x**5), Abs(a**2*x**2) > 1), (2*I*a**3*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + I*a*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - I*sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*a*x**5), True))/a**5 - c**3*Piecewise((I*a**5*sqrt(c)*acosh(1/(a*x))/16 - I*a**4*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + I*a**2*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*I*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(6*a**2*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**5*sqrt(c)*asin(1/(a*x))/16 + a**4*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - a**2*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(6*a**2*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**6","C",0
725,1,500,0,13.612203," ","integrate((c-c/a**2/x**2)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- c^{2} \left(\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left(a x \right)}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} - \frac{i \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)}}{a} & \text{otherwise} \end{cases}\right) + \frac{2 c^{2} \left(\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a} - \frac{2 c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{a^{2} \left(c - \frac{c}{a^{2} x^{2}}\right)^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right)}{a^{3}} + \frac{c^{2} \left(\begin{cases} \frac{i a^{3} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{i a^{2} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 i \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{4 a^{2} x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a^{3} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{a^{2} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{\sqrt{c}}{4 a^{2} x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)}{a^{4}}"," ",0,"-c**2*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c**2*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a - 2*c**2*Piecewise((0, Eq(c, 0)), (a**2*(c - c/(a**2*x**2))**(3/2)/(3*c), True))/a**3 + c**2*Piecewise((I*a**3*sqrt(c)*acosh(1/(a*x))/8 - I*a**2*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*I*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(4*a**2*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a**3*sqrt(c)*asin(1/(a*x))/8 + a**2*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + sqrt(c)/(4*a**2*x**5*sqrt(1 - 1/(a**2*x**2))), True))/a**4","C",0
726,1,376,0,9.249037," ","integrate((c-c/a**2/x**2)**(3/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- c \left(\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left(a x \right)}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2 a} - \frac{i \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)}}{a} & \text{otherwise} \end{cases}\right) + \frac{2 c \left(\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)}{a} - \frac{c \left(\begin{cases} \frac{i a \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{i \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{2 a^{2} x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{a \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{\sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)}{a^{2}}"," ",0,"-c*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*asin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) + 2*c*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acosh(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqrt(c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a - c*Piecewise((I*a*sqrt(c)*acosh(1/(a*x))/2 + I*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**2","C",0
727,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x + 1), x)","F",0
728,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**(1/2),x)","- \int \frac{a x}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} + \sqrt{c - \frac{c}{a^{2} x^{2}}}}\, dx - \int \left(- \frac{1}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} + \sqrt{c - \frac{c}{a^{2} x^{2}}}}\right)\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(c - c/(a**2*x**2)) + sqrt(c - c/(a**2*x**2))), x) - Integral(-1/(a*x*sqrt(c - c/(a**2*x**2)) + sqrt(c - c/(a**2*x**2))), x)","F",0
729,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**(3/2),x)","- \int \frac{a x}{a c x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx - \int \left(- \frac{1}{a c x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{c \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c*x*sqrt(c - c/(a**2*x**2)) + c*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2))/(a*x) - c*sqrt(c - c/(a**2*x**2))/(a**2*x**2)), x) - Integral(-1/(a*c*x*sqrt(c - c/(a**2*x**2)) + c*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2))/(a*x) - c*sqrt(c - c/(a**2*x**2))/(a**2*x**2)), x)","F",0
730,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**(5/2),x)","- \int \frac{a x}{a c^{2} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\, dx - \int \left(- \frac{1}{a c^{2} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{2 c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{c^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c**2*x*sqrt(c - c/(a**2*x**2)) + c**2*sqrt(c - c/(a**2*x**2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x**2))/(a**3*x**3) + c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x) - Integral(-1/(a*c**2*x*sqrt(c - c/(a**2*x**2)) + c**2*sqrt(c - c/(a**2*x**2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x**2))/(a**3*x**3) + c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x)","F",0
731,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**(7/2),x)","- \int \frac{a x}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx - \int \left(- \frac{1}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\right)\, dx"," ",0,"-Integral(a*x/(a*c**3*x*sqrt(c - c/(a**2*x**2)) + c**3*sqrt(c - c/(a**2*x**2)) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a*x) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**3*x**3) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) - c**3*sqrt(c - c/(a**2*x**2))/(a**6*x**6)), x) - Integral(-1/(a*c**3*x*sqrt(c - c/(a**2*x**2)) + c**3*sqrt(c - c/(a**2*x**2)) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a*x) - 3*c**3*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**3*x**3) + 3*c**3*sqrt(c - c/(a**2*x**2))/(a**4*x**4) - c**3*sqrt(c - c/(a**2*x**2))/(a**5*x**5) - c**3*sqrt(c - c/(a**2*x**2))/(a**6*x**6)), x)","F",0
732,-1,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(9/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
733,-1,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(7/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
734,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)/(a*x + 1)**3, x)","F",0
735,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(3/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)/(a*x + 1)**3, x)","F",0
736,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1)**3, x)","F",0
737,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(1/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3), x)","F",0
738,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2)*(a*x + 1)**3), x)","F",0
739,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(5/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(5/2)*(a*x + 1)**3), x)","F",0
740,-1,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
741,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
742,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
743,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
744,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
745,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
746,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
747,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(c-c/a**2/x**2)**(1/2),x)","- \int \frac{x^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac{a x^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx"," ",0,"-Integral(x**3*sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x**4*sqrt(c - c/(a**2*x**2))/(a*x - 1), x)","F",0
748,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(c-c/a**2/x**2)**(1/2),x)","- \int \frac{x^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac{a x^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx"," ",0,"-Integral(x**2*sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x**3*sqrt(c - c/(a**2*x**2))/(a*x - 1), x)","F",0
749,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(c-c/a**2/x**2)**(1/2),x)","- \int \frac{x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac{a x^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx"," ",0,"-Integral(x*sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x**2*sqrt(c - c/(a**2*x**2))/(a*x - 1), x)","F",0
750,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2),x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x - 1}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x - 1), x)","F",0
751,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x,x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{2} - x}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{2} - x}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x**2 - x), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**2 - x), x)","F",0
752,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**2,x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} - x^{2}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} - x^{2}}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x**3 - x**2), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**3 - x**2), x)","F",0
753,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**3,x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{4} - x^{3}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{4} - x^{3}}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x**4 - x**3), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**4 - x**3), x)","F",0
754,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**4,x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x**5 - x**4), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**5 - x**4), x)","F",0
755,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**5,x)","- \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{6} - x^{5}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{6} - x^{5}}\, dx"," ",0,"-Integral(sqrt(c - c/(a**2*x**2))/(a*x**6 - x**5), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**6 - x**5), x)","F",0
756,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x^{3} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
757,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
758,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(c-c/a**2/x**2)**(1/2),x)","\int \frac{x \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
759,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2),x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
760,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
761,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
762,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
763,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{4} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
764,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**(1/2)/x**5,x)","\int \frac{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} \left(a x + 1\right)^{3}}{x^{5} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)**3/(x**5*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
765,0,0,0,0.000000," ","integrate(x**m*(c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1), x)","F",0
766,0,0,0,0.000000," ","integrate(x**2*(c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1), x)","F",0
767,0,0,0,0.000000," ","integrate(x*(c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1), x)","F",0
768,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1), x)","F",0
769,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x*(a*x + 1)), x)","F",0
770,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x**2*(a*x + 1)), x)","F",0
771,0,0,0,0.000000," ","integrate(x**3*(c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\right)\, dx - \int \frac{a x^{4} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\, dx"," ",0,"-Integral(-x**3*sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x**4*sqrt(c - c/(a**2*x**2))/(a*x + 1), x)","F",0
772,0,0,0,0.000000," ","integrate(x**2*(c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\right)\, dx - \int \frac{a x^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\, dx"," ",0,"-Integral(-x**2*sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x**3*sqrt(c - c/(a**2*x**2))/(a*x + 1), x)","F",0
773,0,0,0,0.000000," ","integrate(x*(c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\right)\, dx - \int \frac{a x^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\, dx"," ",0,"-Integral(-x*sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x**2*sqrt(c - c/(a**2*x**2))/(a*x + 1), x)","F",0
774,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x + 1}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x + 1), x)","F",0
775,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{2} + x}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{2} + x}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**2 + x), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**2 + x), x)","F",0
776,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**2,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} + x^{2}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} + x^{2}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**3 + x**2), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**3 + x**2), x)","F",0
777,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{4} + x^{3}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{4} + x^{3}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**4 + x**3), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**4 + x**3), x)","F",0
778,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{5} + x^{4}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{5} + x^{4}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**5 + x**4), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**5 + x**4), x)","F",0
779,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)","- \int \left(- \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\right)\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\, dx"," ",0,"-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**6 + x**5), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**6 + x**5), x)","F",0
780,0,0,0,0.000000," ","integrate(x**3*(c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1)**3, x)","F",0
781,0,0,0,0.000000," ","integrate(x**2*(c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1)**3, x)","F",0
782,0,0,0,0.000000," ","integrate(x*(c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1)**3, x)","F",0
783,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(a*x + 1)**3, x)","F",0
784,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x*(a*x + 1)**3), x)","F",0
785,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x^{2} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x**2*(a*x + 1)**3), x)","F",0
786,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x^{3} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x**3*(a*x + 1)**3), x)","F",0
787,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x^{4} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x**4*(a*x + 1)**3), x)","F",0
788,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}{x^{5} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))/(x**5*(a*x + 1)**3), x)","F",0
789,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**p/exp(2*p*atanh(a*x)),x)","\int \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p} e^{- 2 p \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*exp(-2*p*atanh(a*x)), x)","F",0
790,0,0,0,0.000000," ","integrate(exp(2*p*atanh(a*x))*(c-c/a**2/x**2)**p,x)","\int \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*exp(2*p*atanh(a*x)), x)","F",0
791,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a**2/x**2)**2,x)","\frac{c^{2} \left(\int a^{4} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{4}}\, dx + \int \left(- \frac{2 a^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2}}\right)\, dx\right)}{a^{4}}"," ",0,"c**2*(Integral(a**4*exp(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x))/x**4, x) + Integral(-2*a**2*exp(n*atanh(a*x))/x**2, x))/a**4","F",0
792,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a**2/x**2),x)","\frac{c \left(\int a^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2}}\right)\, dx\right)}{a^{2}}"," ",0,"c*(Integral(a**2*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x))/x**2, x))/a**2","F",0
793,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2),x)","\frac{a^{2} \int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"a**2*Integral(x**2*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
794,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**2,x)","\frac{a^{4} \int \frac{x^{4} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}}"," ",0,"a**4*Integral(x**4*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2","F",0
795,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a**2/x**2)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
796,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a**2/x**2)**(1/2),x)","\int \sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*exp(n*atanh(a*x)), x)","F",0
797,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x))), x)","F",0
798,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(3/2), x)","F",0
799,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(c-c/a**2/x**2)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
800,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(c-c/a**2/x**2)**p,x)","\int \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*exp(n*atanh(a*x)), x)","F",0
801,0,0,0,0.000000," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**p,x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p} \left(a x + 1\right)^{2}}{\left(a x - 1\right)^{2}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x + 1)**2/(a*x - 1)**2, x)","F",0
802,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a**2/x**2)**p,x)","\int \frac{\left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
803,1,697,0,11.345269," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**p,x)","- a \left(\begin{cases} \frac{0^{p} x}{a} - \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} + \frac{0^{p} \log{\left(-1 + \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{acoth}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{a a^{- 2 p} c^{p} p x^{3} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{0^{p} x}{a} - \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} + \frac{0^{p} \log{\left(1 - \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{atanh}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{a a^{- 2 p} c^{p} p x^{3} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{0^{p} \log{\left(a^{2} x^{2} - 1 \right)}}{2 a} - \frac{0^{p} \operatorname{acoth}{\left(a x \right)}}{a} - \frac{a a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} + \frac{a^{- 2 p} c^{p} p x x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{0^{p} \log{\left(- a^{2} x^{2} + 1 \right)}}{2 a} - \frac{0^{p} \operatorname{atanh}{\left(a x \right)}}{a} - \frac{a a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} + \frac{a^{- 2 p} c^{p} p x x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}"," ",0,"-a*Piecewise((0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) - a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) - a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), True)) - Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) - 0**p*acoth(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) + a**(-2*p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*gamma(p + 1/2)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) - 0**p*atanh(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) + a**(-2*p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*gamma(p + 1/2)*gamma(p + 1)), True))","C",0
804,1,178,0,14.932583," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**p,x)","\frac{a c^{p} x^{2} \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, 1 \\ 2, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(p + 1\right)} + \frac{c^{p} x \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} - \frac{1}{2}, 1, - p \\ \frac{1}{2}, \frac{1}{2} \end{matrix}\middle| {\frac{e^{2 i \pi}}{a^{2} x^{2}}} \right)}}{\sqrt{\pi} \Gamma\left(p + 1\right)} + \frac{c^{p} x \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, \frac{1}{2}, 1 \\ \frac{3}{2}, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{\sqrt{\pi} \Gamma\left(p + 1\right)} - \frac{c^{p} {G_{3, 3}^{2, 2}\left(\begin{matrix} -1, p & 1 \\-1, 0 & - \frac{1}{2} \end{matrix} \middle| {\frac{e^{i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 a \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"a*c**p*x**2*gamma(p + 1/2)*hyper((1/2, 1, 1), (2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(p + 1)) + c**p*x*gamma(p + 1/2)*hyper((-1/2, 1, -p), (1/2, 1/2), exp_polar(2*I*pi)/(a**2*x**2))/(sqrt(pi)*gamma(p + 1)) + c**p*x*gamma(p + 1/2)*hyper((1/2, 1/2, 1), (3/2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(sqrt(pi)*gamma(p + 1)) - c**p*meijerg(((-1, p), (1,)), ((-1, 0), (-1/2,)), exp_polar(I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a*gamma(-p)*gamma(p + 1))","C",0
805,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p/(a*x + 1), x)","F",0
806,1,695,0,11.768245," ","integrate((c-c/a**2/x**2)**p/(a*x+1)**2*(-a**2*x**2+1),x)","- a \left(\begin{cases} \frac{0^{p} x}{a} + \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \log{\left(-1 + \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{acoth}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{a a^{- 2 p} c^{p} p x^{3} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} + \frac{a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{0^{p} x}{a} + \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \log{\left(1 - \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{atanh}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{a a^{- 2 p} c^{p} p x^{3} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{3}{2} - p \\ \frac{5}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} + \frac{a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{0^{p} \log{\left(a^{2} x^{2} - 1 \right)}}{2 a} + \frac{0^{p} \operatorname{acoth}{\left(a x \right)}}{a} - \frac{a a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{- 2 p} c^{p} p x x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{0^{p} \log{\left(- a^{2} x^{2} + 1 \right)}}{2 a} + \frac{0^{p} \operatorname{atanh}{\left(a x \right)}}{a} - \frac{a a^{- 2 p} c^{p} p x^{2} x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, 1 - p \\ 2 - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{- 2 p} c^{p} p x x^{- 2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {a^{2} x^{2}} \right)}}{2 \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}"," ",0,"-a*Piecewise((0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) + a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 + a*a**(-2*p)*c**p*p*x**3*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*gamma(p - 1/2)*gamma(p + 1)) + a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)), True)) + Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) + 0**p*acoth(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) - a**(-2*p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*gamma(p + 1/2)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) + 0**p*atanh(a*x)/a - a*a**(-2*p)*c**p*p*x**2*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma(2 - p)*gamma(p + 1)) - a**(-2*p)*c**p*p*x*x**(-2*p)*exp(I*pi*p)*gamma(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*gamma(p + 1/2)*gamma(p + 1)), True))","C",0
807,0,0,0,0.000000," ","integrate((c-c/a**2/x**2)**p/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(-1 + \frac{1}{a x}\right) \left(1 + \frac{1}{a x}\right)\right)^{p}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p/(a*x + 1)**3, x)","F",0
808,-1,0,0,0.000000," ","integrate((1+x)**(3/2)/(-x**2+1)**(1/2)*x*sin(x),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
809,0,0,0,0.000000," ","integrate((1+x)**(3/2)/(-x**2+1)**(1/2)*sin(x),x)","\int \frac{\left(x + 1\right)^{\frac{3}{2}} \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral((x + 1)**(3/2)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)","F",0
810,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(1/2)*x*sin(x),x)","\int \frac{x \sqrt{1 - x} \left(x + 1\right) \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(1 - x)*(x + 1)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)","F",0
811,0,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(1/2)*sin(x),x)","\int \frac{\sqrt{1 - x} \left(x + 1\right) \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(sqrt(1 - x)*(x + 1)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)","F",0
812,-1,0,0,0.000000," ","integrate((1+x)**(5/2)/(-x**2+1)**(1/2)*x*sin(x),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
813,-1,0,0,0.000000," ","integrate((1+x)**(5/2)/(-x**2+1)**(1/2)*sin(x),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
814,-1,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(3/2)*x*sin(x),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
815,-1,0,0,0.000000," ","integrate((1+x)/(-x**2+1)**(1/2)*(1-x)**(3/2)*sin(x),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
816,0,0,0,0.000000," ","integrate((1+x)**(1/2)/(-x**2+1)**(1/2)*x*sin(x),x)","\int \frac{x \sqrt{x + 1} \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(x + 1)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)","F",0
817,0,0,0,0.000000," ","integrate((1+x)**(1/2)/(-x**2+1)**(1/2)*sin(x),x)","\int \frac{\sqrt{x + 1} \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)}}\, dx"," ",0,"Integral(sqrt(x + 1)*sin(x)/sqrt(-(x - 1)*(x + 1)), x)","F",0
818,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x**3,x)","\int \frac{x^{3} \left(a + b x + 1\right)}{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral(x**3*(a + b*x + 1)/sqrt(-(a + b*x - 1)*(a + b*x + 1)), x)","F",0
819,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x**2,x)","\int \frac{x^{2} \left(a + b x + 1\right)}{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral(x**2*(a + b*x + 1)/sqrt(-(a + b*x - 1)*(a + b*x + 1)), x)","F",0
820,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x,x)","\int \frac{x \left(a + b x + 1\right)}{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral(x*(a + b*x + 1)/sqrt(-(a + b*x - 1)*(a + b*x + 1)), x)","F",0
821,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2),x)","\int \frac{a + b x + 1}{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral((a + b*x + 1)/sqrt(-(a + b*x - 1)*(a + b*x + 1)), x)","F",0
822,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x,x)","\int \frac{a + b x + 1}{x \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral((a + b*x + 1)/(x*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)","F",0
823,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**2,x)","\int \frac{a + b x + 1}{x^{2} \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral((a + b*x + 1)/(x**2*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)","F",0
824,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**3,x)","\int \frac{a + b x + 1}{x^{3} \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral((a + b*x + 1)/(x**3*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)","F",0
825,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**4,x)","\int \frac{a + b x + 1}{x^{4} \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}\, dx"," ",0,"Integral((a + b*x + 1)/(x**4*sqrt(-(a + b*x - 1)*(a + b*x + 1))), x)","F",0
826,1,102,0,0.339952," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)*x**4,x)","- \frac{x^{5}}{5} - x^{3} \left(- \frac{2 a}{3 b^{2}} + \frac{2}{3 b^{2}}\right) - x^{2} \left(\frac{a^{2}}{b^{3}} - \frac{2 a}{b^{3}} + \frac{1}{b^{3}}\right) - x \left(- \frac{2 a^{3}}{b^{4}} + \frac{6 a^{2}}{b^{4}} - \frac{6 a}{b^{4}} + \frac{2}{b^{4}}\right) - \frac{x^{4}}{2 b} - \frac{2 \left(a - 1\right)^{4} \log{\left(a + b x - 1 \right)}}{b^{5}}"," ",0,"-x**5/5 - x**3*(-2*a/(3*b**2) + 2/(3*b**2)) - x**2*(a**2/b**3 - 2*a/b**3 + b**(-3)) - x*(-2*a**3/b**4 + 6*a**2/b**4 - 6*a/b**4 + 2/b**4) - x**4/(2*b) - 2*(a - 1)**4*log(a + b*x - 1)/b**5","A",0
827,1,66,0,0.258133," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)*x**3,x)","- \frac{x^{4}}{4} - x^{2} \left(- \frac{a}{b^{2}} + \frac{1}{b^{2}}\right) - x \left(\frac{2 a^{2}}{b^{3}} - \frac{4 a}{b^{3}} + \frac{2}{b^{3}}\right) - \frac{2 x^{3}}{3 b} + \frac{2 \left(a - 1\right)^{3} \log{\left(a + b x - 1 \right)}}{b^{4}}"," ",0,"-x**4/4 - x**2*(-a/b**2 + b**(-2)) - x*(2*a**2/b**3 - 4*a/b**3 + 2/b**3) - 2*x**3/(3*b) + 2*(a - 1)**3*log(a + b*x - 1)/b**4","A",0
828,1,42,0,0.206971," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)*x**2,x)","- \frac{x^{3}}{3} - x \left(- \frac{2 a}{b^{2}} + \frac{2}{b^{2}}\right) - \frac{x^{2}}{b} - \frac{2 \left(a - 1\right)^{2} \log{\left(a + b x - 1 \right)}}{b^{3}}"," ",0,"-x**3/3 - x*(-2*a/b**2 + 2/b**2) - x**2/b - 2*(a - 1)**2*log(a + b*x - 1)/b**3","A",0
829,1,26,0,0.155000," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)*x,x)","- \frac{x^{2}}{2} - \frac{2 x}{b} + \frac{2 \left(a - 1\right) \log{\left(a + b x - 1 \right)}}{b^{2}}"," ",0,"-x**2/2 - 2*x/b + 2*(a - 1)*log(a + b*x - 1)/b**2","A",0
830,1,14,0,0.115526," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2),x)","- x - \frac{2 \log{\left(a + b x - 1 \right)}}{b}"," ",0,"-x - 2*log(a + b*x - 1)/b","A",0
831,1,88,0,0.461171," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)/x,x)","- \frac{\left(a + 1\right) \log{\left(x + \frac{a^{2} - \frac{a^{2} \left(a + 1\right)}{a - 1} + \frac{2 a \left(a + 1\right)}{a - 1} - 1 - \frac{a + 1}{a - 1}}{a b + 3 b} \right)}}{a - 1} + \frac{2 \log{\left(x + \frac{a^{2} + \frac{2 a^{2}}{a - 1} - \frac{4 a}{a - 1} - 1 + \frac{2}{a - 1}}{a b + 3 b} \right)}}{a - 1}"," ",0,"-(a + 1)*log(x + (a**2 - a**2*(a + 1)/(a - 1) + 2*a*(a + 1)/(a - 1) - 1 - (a + 1)/(a - 1))/(a*b + 3*b))/(a - 1) + 2*log(x + (a**2 + 2*a**2/(a - 1) - 4*a/(a - 1) - 1 + 2/(a - 1))/(a*b + 3*b))/(a - 1)","B",0
832,1,144,0,0.352236," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)/x**2,x)","\frac{2 b \log{\left(x + \frac{- \frac{2 a^{3} b}{\left(a - 1\right)^{2}} + \frac{6 a^{2} b}{\left(a - 1\right)^{2}} + 2 a b - \frac{6 a b}{\left(a - 1\right)^{2}} - 2 b + \frac{2 b}{\left(a - 1\right)^{2}}}{4 b^{2}} \right)}}{\left(a - 1\right)^{2}} - \frac{2 b \log{\left(x + \frac{\frac{2 a^{3} b}{\left(a - 1\right)^{2}} - \frac{6 a^{2} b}{\left(a - 1\right)^{2}} + 2 a b + \frac{6 a b}{\left(a - 1\right)^{2}} - 2 b - \frac{2 b}{\left(a - 1\right)^{2}}}{4 b^{2}} \right)}}{\left(a - 1\right)^{2}} - \frac{- a - 1}{x \left(a - 1\right)}"," ",0,"2*b*log(x + (-2*a**3*b/(a - 1)**2 + 6*a**2*b/(a - 1)**2 + 2*a*b - 6*a*b/(a - 1)**2 - 2*b + 2*b/(a - 1)**2)/(4*b**2))/(a - 1)**2 - 2*b*log(x + (2*a**3*b/(a - 1)**2 - 6*a**2*b/(a - 1)**2 + 2*a*b + 6*a*b/(a - 1)**2 - 2*b - 2*b/(a - 1)**2)/(4*b**2))/(a - 1)**2 - (-a - 1)/(x*(a - 1))","B",0
833,1,209,0,0.447539," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)/x**3,x)","- \frac{2 b^{2} \log{\left(x + \frac{- \frac{2 a^{4} b^{2}}{\left(a - 1\right)^{3}} + \frac{8 a^{3} b^{2}}{\left(a - 1\right)^{3}} - \frac{12 a^{2} b^{2}}{\left(a - 1\right)^{3}} + 2 a b^{2} + \frac{8 a b^{2}}{\left(a - 1\right)^{3}} - 2 b^{2} - \frac{2 b^{2}}{\left(a - 1\right)^{3}}}{4 b^{3}} \right)}}{\left(a - 1\right)^{3}} + \frac{2 b^{2} \log{\left(x + \frac{\frac{2 a^{4} b^{2}}{\left(a - 1\right)^{3}} - \frac{8 a^{3} b^{2}}{\left(a - 1\right)^{3}} + \frac{12 a^{2} b^{2}}{\left(a - 1\right)^{3}} + 2 a b^{2} - \frac{8 a b^{2}}{\left(a - 1\right)^{3}} - 2 b^{2} + \frac{2 b^{2}}{\left(a - 1\right)^{3}}}{4 b^{3}} \right)}}{\left(a - 1\right)^{3}} - \frac{- a^{2} + 4 b x + 1}{x^{2} \left(2 a^{2} - 4 a + 2\right)}"," ",0,"-2*b**2*log(x + (-2*a**4*b**2/(a - 1)**3 + 8*a**3*b**2/(a - 1)**3 - 12*a**2*b**2/(a - 1)**3 + 2*a*b**2 + 8*a*b**2/(a - 1)**3 - 2*b**2 - 2*b**2/(a - 1)**3)/(4*b**3))/(a - 1)**3 + 2*b**2*log(x + (2*a**4*b**2/(a - 1)**3 - 8*a**3*b**2/(a - 1)**3 + 12*a**2*b**2/(a - 1)**3 + 2*a*b**2 - 8*a*b**2/(a - 1)**3 - 2*b**2 + 2*b**2/(a - 1)**3)/(4*b**3))/(a - 1)**3 - (-a**2 + 4*b*x + 1)/(x**2*(2*a**2 - 4*a + 2))","B",0
834,1,260,0,0.558554," ","integrate((b*x+a+1)**2/(1-(b*x+a)**2)/x**4,x)","\frac{2 b^{3} \log{\left(x + \frac{- \frac{2 a^{5} b^{3}}{\left(a - 1\right)^{4}} + \frac{10 a^{4} b^{3}}{\left(a - 1\right)^{4}} - \frac{20 a^{3} b^{3}}{\left(a - 1\right)^{4}} + \frac{20 a^{2} b^{3}}{\left(a - 1\right)^{4}} + 2 a b^{3} - \frac{10 a b^{3}}{\left(a - 1\right)^{4}} - 2 b^{3} + \frac{2 b^{3}}{\left(a - 1\right)^{4}}}{4 b^{4}} \right)}}{\left(a - 1\right)^{4}} - \frac{2 b^{3} \log{\left(x + \frac{\frac{2 a^{5} b^{3}}{\left(a - 1\right)^{4}} - \frac{10 a^{4} b^{3}}{\left(a - 1\right)^{4}} + \frac{20 a^{3} b^{3}}{\left(a - 1\right)^{4}} - \frac{20 a^{2} b^{3}}{\left(a - 1\right)^{4}} + 2 a b^{3} + \frac{10 a b^{3}}{\left(a - 1\right)^{4}} - 2 b^{3} - \frac{2 b^{3}}{\left(a - 1\right)^{4}}}{4 b^{4}} \right)}}{\left(a - 1\right)^{4}} - \frac{- a^{3} + a^{2} + a - 6 b^{2} x^{2} + x \left(3 a b - 3 b\right) - 1}{x^{3} \left(3 a^{3} - 9 a^{2} + 9 a - 3\right)}"," ",0,"2*b**3*log(x + (-2*a**5*b**3/(a - 1)**4 + 10*a**4*b**3/(a - 1)**4 - 20*a**3*b**3/(a - 1)**4 + 20*a**2*b**3/(a - 1)**4 + 2*a*b**3 - 10*a*b**3/(a - 1)**4 - 2*b**3 + 2*b**3/(a - 1)**4)/(4*b**4))/(a - 1)**4 - 2*b**3*log(x + (2*a**5*b**3/(a - 1)**4 - 10*a**4*b**3/(a - 1)**4 + 20*a**3*b**3/(a - 1)**4 - 20*a**2*b**3/(a - 1)**4 + 2*a*b**3 + 10*a*b**3/(a - 1)**4 - 2*b**3 - 2*b**3/(a - 1)**4)/(4*b**4))/(a - 1)**4 - (-a**3 + a**2 + a - 6*b**2*x**2 + x*(3*a*b - 3*b) - 1)/(x**3*(3*a**3 - 9*a**2 + 9*a - 3))","B",0
835,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)*x**3,x)","\int \frac{x^{3} \left(a + b x + 1\right)^{3}}{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*(a + b*x + 1)**3/(-(a + b*x - 1)*(a + b*x + 1))**(3/2), x)","F",0
836,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)*x**2,x)","\int \frac{x^{2} \left(a + b x + 1\right)^{3}}{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*(a + b*x + 1)**3/(-(a + b*x - 1)*(a + b*x + 1))**(3/2), x)","F",0
837,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)*x,x)","\int \frac{x \left(a + b x + 1\right)^{3}}{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*(a + b*x + 1)**3/(-(a + b*x - 1)*(a + b*x + 1))**(3/2), x)","F",0
838,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2),x)","\int \frac{\left(a + b x + 1\right)^{3}}{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a + b*x + 1)**3/(-(a + b*x - 1)*(a + b*x + 1))**(3/2), x)","F",0
839,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)/x,x)","\int \frac{\left(a + b x + 1\right)^{3}}{x \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a + b*x + 1)**3/(x*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)), x)","F",0
840,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)/x**2,x)","\int \frac{\left(a + b x + 1\right)^{3}}{x^{2} \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a + b*x + 1)**3/(x**2*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)), x)","F",0
841,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)/x**3,x)","\int \frac{\left(a + b x + 1\right)^{3}}{x^{3} \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a + b*x + 1)**3/(x**3*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)), x)","F",0
842,0,0,0,0.000000," ","integrate((b*x+a+1)**3/(1-(b*x+a)**2)**(3/2)/x**4,x)","\int \frac{\left(a + b x + 1\right)^{3}}{x^{4} \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a + b*x + 1)**3/(x**4*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)), x)","F",0
843,0,0,0,0.000000," ","integrate(x**3/(b*x+a+1)*(1-(b*x+a)**2)**(1/2),x)","\int \frac{x^{3} \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{a + b x + 1}\, dx"," ",0,"Integral(x**3*sqrt(-(a + b*x - 1)*(a + b*x + 1))/(a + b*x + 1), x)","F",0
844,0,0,0,0.000000," ","integrate(x**2/(b*x+a+1)*(1-(b*x+a)**2)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{a + b x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a + b*x - 1)*(a + b*x + 1))/(a + b*x + 1), x)","F",0
845,0,0,0,0.000000," ","integrate(x/(b*x+a+1)*(1-(b*x+a)**2)**(1/2),x)","\int \frac{x \sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{a + b x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a + b*x - 1)*(a + b*x + 1))/(a + b*x + 1), x)","F",0
846,0,0,0,0.000000," ","integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2),x)","\int \frac{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{a + b x + 1}\, dx"," ",0,"Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(a + b*x + 1), x)","F",0
847,0,0,0,0.000000," ","integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{x \left(a + b x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x*(a + b*x + 1)), x)","F",0
848,0,0,0,0.000000," ","integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{x^{2} \left(a + b x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x**2*(a + b*x + 1)), x)","F",0
849,0,0,0,0.000000," ","integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x**3,x)","\int \frac{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{x^{3} \left(a + b x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x**3*(a + b*x + 1)), x)","F",0
850,0,0,0,0.000000," ","integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x**4,x)","\int \frac{\sqrt{- \left(a + b x - 1\right) \left(a + b x + 1\right)}}{x^{4} \left(a + b x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x**4*(a + b*x + 1)), x)","F",0
851,1,102,0,0.283125," ","integrate(x**4/(b*x+a+1)**2*(1-(b*x+a)**2),x)","- \frac{x^{5}}{5} - x^{3} \left(\frac{2 a}{3 b^{2}} + \frac{2}{3 b^{2}}\right) - x^{2} \left(- \frac{a^{2}}{b^{3}} - \frac{2 a}{b^{3}} - \frac{1}{b^{3}}\right) - x \left(\frac{2 a^{3}}{b^{4}} + \frac{6 a^{2}}{b^{4}} + \frac{6 a}{b^{4}} + \frac{2}{b^{4}}\right) + \frac{x^{4}}{2 b} + \frac{2 \left(a + 1\right)^{4} \log{\left(a + b x + 1 \right)}}{b^{5}}"," ",0,"-x**5/5 - x**3*(2*a/(3*b**2) + 2/(3*b**2)) - x**2*(-a**2/b**3 - 2*a/b**3 - 1/b**3) - x*(2*a**3/b**4 + 6*a**2/b**4 + 6*a/b**4 + 2/b**4) + x**4/(2*b) + 2*(a + 1)**4*log(a + b*x + 1)/b**5","A",0
852,1,68,0,0.227676," ","integrate(x**3/(b*x+a+1)**2*(1-(b*x+a)**2),x)","- \frac{x^{4}}{4} - x^{2} \left(\frac{a}{b^{2}} + \frac{1}{b^{2}}\right) - x \left(- \frac{2 a^{2}}{b^{3}} - \frac{4 a}{b^{3}} - \frac{2}{b^{3}}\right) + \frac{2 x^{3}}{3 b} - \frac{2 \left(a + 1\right)^{3} \log{\left(a + b x + 1 \right)}}{b^{4}}"," ",0,"-x**4/4 - x**2*(a/b**2 + b**(-2)) - x*(-2*a**2/b**3 - 4*a/b**3 - 2/b**3) + 2*x**3/(3*b) - 2*(a + 1)**3*log(a + b*x + 1)/b**4","A",0
853,1,41,0,0.195573," ","integrate(x**2/(b*x+a+1)**2*(1-(b*x+a)**2),x)","- \frac{x^{3}}{3} - x \left(\frac{2 a}{b^{2}} + \frac{2}{b^{2}}\right) + \frac{x^{2}}{b} + \frac{2 \left(a + 1\right)^{2} \log{\left(a + b x + 1 \right)}}{b^{3}}"," ",0,"-x**3/3 - x*(2*a/b**2 + 2/b**2) + x**2/b + 2*(a + 1)**2*log(a + b*x + 1)/b**3","A",0
854,1,26,0,0.143747," ","integrate(x/(b*x+a+1)**2*(1-(b*x+a)**2),x)","- \frac{x^{2}}{2} + \frac{2 x}{b} - \frac{2 \left(a + 1\right) \log{\left(a + b x + 1 \right)}}{b^{2}}"," ",0,"-x**2/2 + 2*x/b - 2*(a + 1)*log(a + b*x + 1)/b**2","A",0
855,1,12,0,0.111545," ","integrate(1/(b*x+a+1)**2*(1-(b*x+a)**2),x)","- x + \frac{2 \log{\left(a + b x + 1 \right)}}{b}"," ",0,"-x + 2*log(a + b*x + 1)/b","A",0
856,1,90,0,0.456206," ","integrate(1/(b*x+a+1)**2*(1-(b*x+a)**2)/x,x)","- \frac{\left(a - 1\right) \log{\left(x + \frac{- \frac{a^{2} \left(a - 1\right)}{a + 1} + a^{2} - \frac{2 a \left(a - 1\right)}{a + 1} - \frac{a - 1}{a + 1} - 1}{a b - 3 b} \right)}}{a + 1} - \frac{2 \log{\left(x + \frac{a^{2} - \frac{2 a^{2}}{a + 1} - \frac{4 a}{a + 1} - 1 - \frac{2}{a + 1}}{a b - 3 b} \right)}}{a + 1}"," ",0,"-(a - 1)*log(x + (-a**2*(a - 1)/(a + 1) + a**2 - 2*a*(a - 1)/(a + 1) - (a - 1)/(a + 1) - 1)/(a*b - 3*b))/(a + 1) - 2*log(x + (a**2 - 2*a**2/(a + 1) - 4*a/(a + 1) - 1 - 2/(a + 1))/(a*b - 3*b))/(a + 1)","B",0
857,1,143,0,0.346365," ","integrate(1/(b*x+a+1)**2*(1-(b*x+a)**2)/x**2,x)","- \frac{2 b \log{\left(x + \frac{- \frac{2 a^{3} b}{\left(a + 1\right)^{2}} - \frac{6 a^{2} b}{\left(a + 1\right)^{2}} + 2 a b - \frac{6 a b}{\left(a + 1\right)^{2}} + 2 b - \frac{2 b}{\left(a + 1\right)^{2}}}{4 b^{2}} \right)}}{\left(a + 1\right)^{2}} + \frac{2 b \log{\left(x + \frac{\frac{2 a^{3} b}{\left(a + 1\right)^{2}} + \frac{6 a^{2} b}{\left(a + 1\right)^{2}} + 2 a b + \frac{6 a b}{\left(a + 1\right)^{2}} + 2 b + \frac{2 b}{\left(a + 1\right)^{2}}}{4 b^{2}} \right)}}{\left(a + 1\right)^{2}} - \frac{1 - a}{x \left(a + 1\right)}"," ",0,"-2*b*log(x + (-2*a**3*b/(a + 1)**2 - 6*a**2*b/(a + 1)**2 + 2*a*b - 6*a*b/(a + 1)**2 + 2*b - 2*b/(a + 1)**2)/(4*b**2))/(a + 1)**2 + 2*b*log(x + (2*a**3*b/(a + 1)**2 + 6*a**2*b/(a + 1)**2 + 2*a*b + 6*a*b/(a + 1)**2 + 2*b + 2*b/(a + 1)**2)/(4*b**2))/(a + 1)**2 - (1 - a)/(x*(a + 1))","B",0
858,1,209,0,0.445126," ","integrate(1/(b*x+a+1)**2*(1-(b*x+a)**2)/x**3,x)","\frac{2 b^{2} \log{\left(x + \frac{- \frac{2 a^{4} b^{2}}{\left(a + 1\right)^{3}} - \frac{8 a^{3} b^{2}}{\left(a + 1\right)^{3}} - \frac{12 a^{2} b^{2}}{\left(a + 1\right)^{3}} + 2 a b^{2} - \frac{8 a b^{2}}{\left(a + 1\right)^{3}} + 2 b^{2} - \frac{2 b^{2}}{\left(a + 1\right)^{3}}}{4 b^{3}} \right)}}{\left(a + 1\right)^{3}} - \frac{2 b^{2} \log{\left(x + \frac{\frac{2 a^{4} b^{2}}{\left(a + 1\right)^{3}} + \frac{8 a^{3} b^{2}}{\left(a + 1\right)^{3}} + \frac{12 a^{2} b^{2}}{\left(a + 1\right)^{3}} + 2 a b^{2} + \frac{8 a b^{2}}{\left(a + 1\right)^{3}} + 2 b^{2} + \frac{2 b^{2}}{\left(a + 1\right)^{3}}}{4 b^{3}} \right)}}{\left(a + 1\right)^{3}} - \frac{- a^{2} - 4 b x + 1}{x^{2} \left(2 a^{2} + 4 a + 2\right)}"," ",0,"2*b**2*log(x + (-2*a**4*b**2/(a + 1)**3 - 8*a**3*b**2/(a + 1)**3 - 12*a**2*b**2/(a + 1)**3 + 2*a*b**2 - 8*a*b**2/(a + 1)**3 + 2*b**2 - 2*b**2/(a + 1)**3)/(4*b**3))/(a + 1)**3 - 2*b**2*log(x + (2*a**4*b**2/(a + 1)**3 + 8*a**3*b**2/(a + 1)**3 + 12*a**2*b**2/(a + 1)**3 + 2*a*b**2 + 8*a*b**2/(a + 1)**3 + 2*b**2 + 2*b**2/(a + 1)**3)/(4*b**3))/(a + 1)**3 - (-a**2 - 4*b*x + 1)/(x**2*(2*a**2 + 4*a + 2))","B",0
859,1,262,0,0.534941," ","integrate(1/(b*x+a+1)**2*(1-(b*x+a)**2)/x**4,x)","- \frac{2 b^{3} \log{\left(x + \frac{- \frac{2 a^{5} b^{3}}{\left(a + 1\right)^{4}} - \frac{10 a^{4} b^{3}}{\left(a + 1\right)^{4}} - \frac{20 a^{3} b^{3}}{\left(a + 1\right)^{4}} - \frac{20 a^{2} b^{3}}{\left(a + 1\right)^{4}} + 2 a b^{3} - \frac{10 a b^{3}}{\left(a + 1\right)^{4}} + 2 b^{3} - \frac{2 b^{3}}{\left(a + 1\right)^{4}}}{4 b^{4}} \right)}}{\left(a + 1\right)^{4}} + \frac{2 b^{3} \log{\left(x + \frac{\frac{2 a^{5} b^{3}}{\left(a + 1\right)^{4}} + \frac{10 a^{4} b^{3}}{\left(a + 1\right)^{4}} + \frac{20 a^{3} b^{3}}{\left(a + 1\right)^{4}} + \frac{20 a^{2} b^{3}}{\left(a + 1\right)^{4}} + 2 a b^{3} + \frac{10 a b^{3}}{\left(a + 1\right)^{4}} + 2 b^{3} + \frac{2 b^{3}}{\left(a + 1\right)^{4}}}{4 b^{4}} \right)}}{\left(a + 1\right)^{4}} - \frac{- a^{3} - a^{2} + a + 6 b^{2} x^{2} + x \left(- 3 a b - 3 b\right) + 1}{x^{3} \left(3 a^{3} + 9 a^{2} + 9 a + 3\right)}"," ",0,"-2*b**3*log(x + (-2*a**5*b**3/(a + 1)**4 - 10*a**4*b**3/(a + 1)**4 - 20*a**3*b**3/(a + 1)**4 - 20*a**2*b**3/(a + 1)**4 + 2*a*b**3 - 10*a*b**3/(a + 1)**4 + 2*b**3 - 2*b**3/(a + 1)**4)/(4*b**4))/(a + 1)**4 + 2*b**3*log(x + (2*a**5*b**3/(a + 1)**4 + 10*a**4*b**3/(a + 1)**4 + 20*a**3*b**3/(a + 1)**4 + 20*a**2*b**3/(a + 1)**4 + 2*a*b**3 + 10*a*b**3/(a + 1)**4 + 2*b**3 + 2*b**3/(a + 1)**4)/(4*b**4))/(a + 1)**4 - (-a**3 - a**2 + a + 6*b**2*x**2 + x*(-3*a*b - 3*b) + 1)/(x**3*(3*a**3 + 9*a**2 + 9*a + 3))","B",0
860,0,0,0,0.000000," ","integrate(x**3/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2),x)","\int \frac{x^{3} \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{\left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(a + b*x + 1)**3, x)","F",0
861,0,0,0,0.000000," ","integrate(x**2/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2),x)","\int \frac{x^{2} \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{\left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(a + b*x + 1)**3, x)","F",0
862,0,0,0,0.000000," ","integrate(x/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2),x)","\int \frac{x \left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{\left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral(x*(-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(a + b*x + 1)**3, x)","F",0
863,0,0,0,0.000000," ","integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2),x)","\int \frac{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{\left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(a + b*x + 1)**3, x)","F",0
864,0,0,0,0.000000," ","integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x,x)","\int \frac{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{x \left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(x*(a + b*x + 1)**3), x)","F",0
865,0,0,0,0.000000," ","integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x**2,x)","\int \frac{\left(- \left(a + b x - 1\right) \left(a + b x + 1\right)\right)^{\frac{3}{2}}}{x^{2} \left(a + b x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a + b*x - 1)*(a + b*x + 1))**(3/2)/(x**2*(a + b*x + 1)**3), x)","F",0
866,-1,0,0,0.000000," ","integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
867,-1,0,0,0.000000," ","integrate(1/(b*x+a+1)**3*(1-(b*x+a)**2)**(3/2)/x**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
868,0,0,0,0.000000," ","integrate(1/(1-(b*x+1)**2)**(1/2),x)","\int \frac{1}{\sqrt{1 - \left(b x + 1\right)^{2}}}\, dx"," ",0,"Integral(1/sqrt(1 - (b*x + 1)**2), x)","F",0
869,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x**3/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{x^{3}}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(x**3/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
870,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x**2/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{x^{2}}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(x**2/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
871,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)*x/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{x}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(x/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
872,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{1}{a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(1/(a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
873,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{1}{a x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(1/(a*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
874,0,0,0,0.000000," ","integrate((b*x+a+1)/(1-(b*x+a)**2)**(1/2)/x**2/(-b**2*x**2-2*a*b*x-a**2+1),x)","- \int \frac{1}{a x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx"," ",0,"-Integral(1/(a*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)","F",0
875,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))*x**m,x)","\int x^{m} e^{n \operatorname{atanh}{\left(a + b x \right)}}\, dx"," ",0,"Integral(x**m*exp(n*atanh(a + b*x)), x)","F",0
876,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))*x**3,x)","\int x^{3} e^{n \operatorname{atanh}{\left(a + b x \right)}}\, dx"," ",0,"Integral(x**3*exp(n*atanh(a + b*x)), x)","F",0
877,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))*x**2,x)","\int x^{2} e^{n \operatorname{atanh}{\left(a + b x \right)}}\, dx"," ",0,"Integral(x**2*exp(n*atanh(a + b*x)), x)","F",0
878,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))*x,x)","\int x e^{n \operatorname{atanh}{\left(a + b x \right)}}\, dx"," ",0,"Integral(x*exp(n*atanh(a + b*x)), x)","F",0
879,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a)),x)","\int e^{n \operatorname{atanh}{\left(a + b x \right)}}\, dx"," ",0,"Integral(exp(n*atanh(a + b*x)), x)","F",0
880,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))/x,x)","\int \frac{e^{n \operatorname{atanh}{\left(a + b x \right)}}}{x}\, dx"," ",0,"Integral(exp(n*atanh(a + b*x))/x, x)","F",0
881,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))/x**2,x)","\int \frac{e^{n \operatorname{atanh}{\left(a + b x \right)}}}{x^{2}}\, dx"," ",0,"Integral(exp(n*atanh(a + b*x))/x**2, x)","F",0
882,0,0,0,0.000000," ","integrate(exp(n*atanh(b*x+a))/x**3,x)","\int \frac{e^{n \operatorname{atanh}{\left(a + b x \right)}}}{x^{3}}\, dx"," ",0,"Integral(exp(n*atanh(a + b*x))/x**3, x)","F",0
883,1,452,0,40.904778," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**4,x)","\begin{cases} \frac{- \frac{c^{4} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} + c^{4} \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 3 c^{4} \left(\begin{cases} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} + \frac{\operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 3 c^{4} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + 3 c^{4} \left(\begin{cases} - \frac{a^{3} x^{3} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{6} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{16} + \frac{\operatorname{asin}{\left(a x \right)}}{16} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + 3 c^{4} \left(\begin{cases} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{7}{2}}}{7} + \frac{2 \left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{4} \left(\begin{cases} - \frac{a^{3} x^{3} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{6} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{32} - \frac{a x \sqrt{- a^{2} x^{2} + 1} \left(- 16 a^{6} x^{6} + 24 a^{4} x^{4} - 10 a^{2} x^{2} + 1\right)}{128} + \frac{5 \operatorname{asin}{\left(a x \right)}}{128} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{4} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{9}{2}}}{9} - \frac{3 \left(- a^{2} x^{2} + 1\right)^{\frac{7}{2}}}{7} + \frac{3 \left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-c**4*(-a**2*x**2 + 1)**(3/2)/3 + c**4*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) - 3*c**4*Piecewise((-a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 + asin(a*x)/8, (a*x > -1) & (a*x < 1))) - 3*c**4*Piecewise(((-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))) + 3*c**4*Piecewise((-a**3*x**3*(-a**2*x**2 + 1)**(3/2)/6 - a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/16 + asin(a*x)/16, (a*x > -1) & (a*x < 1))) + 3*c**4*Piecewise((-(-a**2*x**2 + 1)**(7/2)/7 + 2*(-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))) - c**4*Piecewise((-a**3*x**3*(-a**2*x**2 + 1)**(3/2)/6 - a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/32 - a*x*sqrt(-a**2*x**2 + 1)*(-16*a**6*x**6 + 24*a**4*x**4 - 10*a**2*x**2 + 1)/128 + 5*asin(a*x)/128, (a*x > -1) & (a*x < 1))) - c**4*Piecewise(((-a**2*x**2 + 1)**(9/2)/9 - 3*(-a**2*x**2 + 1)**(7/2)/7 + 3*(-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))))/a, Ne(a, 0)), (c**4*x, True))","A",0
884,1,267,0,24.980908," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**3,x)","\begin{cases} \frac{- \frac{c^{3} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} + c^{3} \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 2 c^{3} \left(\begin{cases} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} + \frac{\operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - 2 c^{3} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{a^{3} x^{3} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{6} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{16} + \frac{\operatorname{asin}{\left(a x \right)}}{16} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) + c^{3} \left(\begin{cases} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{7}{2}}}{7} + \frac{2 \left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c^{3} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-c**3*(-a**2*x**2 + 1)**(3/2)/3 + c**3*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) - 2*c**3*Piecewise((-a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 + asin(a*x)/8, (a*x > -1) & (a*x < 1))) - 2*c**3*Piecewise(((-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))) + c**3*Piecewise((-a**3*x**3*(-a**2*x**2 + 1)**(3/2)/6 - a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/16 + asin(a*x)/16, (a*x > -1) & (a*x < 1))) + c**3*Piecewise((-(-a**2*x**2 + 1)**(7/2)/7 + 2*(-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))))/a, Ne(a, 0)), (c**3*x, True))","A",0
885,1,143,0,20.149334," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**2,x)","\begin{cases} \frac{- \frac{c^{2} \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} + c^{2} \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{2} \left(\begin{cases} - \frac{a x \left(- 2 a^{2} x^{2} + 1\right) \sqrt{- a^{2} x^{2} + 1}}{8} + \frac{\operatorname{asin}{\left(a x \right)}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right) - c^{2} \left(\begin{cases} \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{5}{2}}}{5} - \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c^{2} x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-c**2*(-a**2*x**2 + 1)**(3/2)/3 + c**2*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))) - c**2*Piecewise((-a*x*(-2*a**2*x**2 + 1)*sqrt(-a**2*x**2 + 1)/8 + asin(a*x)/8, (a*x > -1) & (a*x < 1))) - c**2*Piecewise(((-a**2*x**2 + 1)**(5/2)/5 - (-a**2*x**2 + 1)**(3/2)/3, (a*x > -1) & (a*x < 1))))/a, Ne(a, 0)), (c**2*x, True))","A",0
886,1,53,0,4.821707," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c),x)","\begin{cases} \frac{- \frac{c \left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3} + c \left(\begin{cases} \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right)}{a} & \text{for}\: a \neq 0 \\c x & \text{otherwise} \end{cases}"," ",0,"Piecewise(((-c*(-a**2*x**2 + 1)**(3/2)/3 + c*Piecewise((a*x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/2, (a*x > -1) & (a*x < 1))))/a, Ne(a, 0)), (c*x, True))","A",0
887,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{4}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x**4/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
888,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{3}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x**3/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
889,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{2}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x**2/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
890,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c),x)","\frac{\int \frac{x}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
891,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c),x)","\frac{\int \frac{a x}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(a*x/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
892,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c),x)","\frac{\int \frac{a}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(a/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c","F",0
893,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c),x)","\frac{\int \frac{a}{- a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(a/(-a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c","F",0
894,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c),x)","\frac{\int \frac{a}{- a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(a/(-a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c","F",0
895,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a**2*c*x**2+c),x)","\frac{\int \frac{a}{- a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(a/(-a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**6*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x))/c","F",0
896,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**6/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{6}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{7}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**6/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**7/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
897,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{6}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
898,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
899,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
900,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
901,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
902,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
903,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{a}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 2*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
904,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{a}{a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a/(a**4*x**5*sqrt(-a**2*x**2 + 1) - 2*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 2*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
905,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{a}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 2*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 2*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
906,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{a}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{8} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{6} \sqrt{- a^{2} x^{2} + 1} + x^{4} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(a/(a**4*x**7*sqrt(-a**2*x**2 + 1) - 2*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**8*sqrt(-a**2*x**2 + 1) - 2*a**2*x**6*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
907,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**7/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{7}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{8}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**8/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
908,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**6/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{6}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{7}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
909,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{5}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{6}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**5/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
910,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{4}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**4/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
911,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**4/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
912,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
913,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
914,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{a x}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(a*x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
915,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{a}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{7} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(a/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**7*sqrt(-a**2*x**2 + 1) + 3*a**4*x**5*sqrt(-a**2*x**2 + 1) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
916,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{a}{- a^{6} x^{7} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt{- a^{2} x^{2} + 1} + x \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{8} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(a/(-a**6*x**7*sqrt(-a**2*x**2 + 1) + 3*a**4*x**5*sqrt(-a**2*x**2 + 1) - 3*a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**8*sqrt(-a**2*x**2 + 1) + 3*a**4*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
917,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{a}{- a^{6} x^{8} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{9} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(a/(-a**6*x**8*sqrt(-a**2*x**2 + 1) + 3*a**4*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**9*sqrt(-a**2*x**2 + 1) + 3*a**4*x**7*sqrt(-a**2*x**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
918,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**4,x)","\frac{\int \frac{a x}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(a*x/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
919,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**5,x)","\frac{\int \frac{a x}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}}"," ",0,"(Integral(a*x/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**5","F",0
920,1,39,0,18.386407," ","integrate((a*x+1)/(-a**2*x**2+1)*x**4,x)","- \frac{x^{4}}{4 a} - \frac{x^{3}}{3 a^{2}} - \frac{x^{2}}{2 a^{3}} - \frac{x}{a^{4}} - \frac{\log{\left(a x - 1 \right)}}{a^{5}}"," ",0,"-x**4/(4*a) - x**3/(3*a**2) - x**2/(2*a**3) - x/a**4 - log(a*x - 1)/a**5","A",0
921,1,31,0,0.099816," ","integrate((a*x+1)/(-a**2*x**2+1)*x**3,x)","- \frac{x^{3}}{3 a} - \frac{x^{2}}{2 a^{2}} - \frac{x}{a^{3}} - \frac{\log{\left(a x - 1 \right)}}{a^{4}}"," ",0,"-x**3/(3*a) - x**2/(2*a**2) - x/a**3 - log(a*x - 1)/a**4","A",0
922,1,22,0,0.089229," ","integrate((a*x+1)/(-a**2*x**2+1)*x**2,x)","- \frac{x^{2}}{2 a} - \frac{x}{a^{2}} - \frac{\log{\left(a x - 1 \right)}}{a^{3}}"," ",0,"-x**2/(2*a) - x/a**2 - log(a*x - 1)/a**3","A",0
923,1,14,0,0.082028," ","integrate((a*x+1)/(-a**2*x**2+1)*x,x)","- \frac{x}{a} - \frac{\log{\left(a x - 1 \right)}}{a^{2}}"," ",0,"-x/a - log(a*x - 1)/a**2","A",0
924,1,8,0,0.058992," ","integrate((a*x+1)/(-a**2*x**2+1),x)","- \frac{\log{\left(a x - 1 \right)}}{a}"," ",0,"-log(a*x - 1)/a","A",0
925,1,8,0,0.109401," ","integrate((a*x+1)/(-a**2*x**2+1)/x,x)","\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}"," ",0,"log(x) - log(x - 1/a)","A",0
926,1,15,0,0.137554," ","integrate((a*x+1)/(-a**2*x**2+1)/x**2,x)","- a \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{1}{x}"," ",0,"-a*(-log(x) + log(x - 1/a)) - 1/x","A",0
927,1,26,0,0.170755," ","integrate((a*x+1)/(-a**2*x**2+1)/x**3,x)","- a^{2} \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{2 a x + 1}{2 x^{2}}"," ",0,"-a**2*(-log(x) + log(x - 1/a)) - (2*a*x + 1)/(2*x**2)","A",0
928,1,34,0,0.171912," ","integrate((a*x+1)/(-a**2*x**2+1)/x**4,x)","- a^{3} \left(- \log{\left(x \right)} + \log{\left(x - \frac{1}{a} \right)}\right) - \frac{6 a^{2} x^{2} + 3 a x + 2}{6 x^{3}}"," ",0,"-a**3*(-log(x) + log(x - 1/a)) - (6*a**2*x**2 + 3*a*x + 2)/(6*x**3)","A",0
929,1,48,0,0.256552," ","integrate((a*x+1)/(-a**2*x**2+1)**2*x**4,x)","- \frac{1}{2 a^{6} x - 2 a^{5}} + \frac{x^{2}}{2 a^{3}} + \frac{x}{a^{4}} + \frac{\frac{7 \log{\left(x - \frac{1}{a} \right)}}{4} + \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{5}}"," ",0,"-1/(2*a**6*x - 2*a**5) + x**2/(2*a**3) + x/a**4 + (7*log(x - 1/a)/4 + log(x + 1/a)/4)/a**5","A",0
930,1,39,0,0.252747," ","integrate((a*x+1)/(-a**2*x**2+1)**2*x**3,x)","- \frac{1}{2 a^{5} x - 2 a^{4}} + \frac{x}{a^{3}} + \frac{\frac{5 \log{\left(x - \frac{1}{a} \right)}}{4} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{4}}"," ",0,"-1/(2*a**5*x - 2*a**4) + x/a**3 + (5*log(x - 1/a)/4 - log(x + 1/a)/4)/a**4","A",0
931,1,34,0,0.210712," ","integrate((a*x+1)/(-a**2*x**2+1)**2*x**2,x)","- \frac{1}{2 a^{4} x - 2 a^{3}} + \frac{\frac{3 \log{\left(x - \frac{1}{a} \right)}}{4} + \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{3}}"," ",0,"-1/(2*a**4*x - 2*a**3) + (3*log(x - 1/a)/4 + log(x + 1/a)/4)/a**3","A",0
932,1,32,0,0.175147," ","integrate((a*x+1)/(-a**2*x**2+1)**2*x,x)","- \frac{1}{2 a^{3} x - 2 a^{2}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{4} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a^{2}}"," ",0,"-1/(2*a**3*x - 2*a**2) + (log(x - 1/a)/4 - log(x + 1/a)/4)/a**2","A",0
933,1,29,0,0.183888," ","integrate((a*x+1)/(-a**2*x**2+1)**2,x)","- \frac{1}{2 a^{2} x - 2 a} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{4} + \frac{\log{\left(x + \frac{1}{a} \right)}}{4}}{a}"," ",0,"-1/(2*a**2*x - 2*a) + (-log(x - 1/a)/4 + log(x + 1/a)/4)/a","A",0
934,1,29,0,0.257607," ","integrate((a*x+1)/(-a**2*x**2+1)**2/x,x)","\log{\left(x \right)} - \frac{3 \log{\left(x - \frac{1}{a} \right)}}{4} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4} - \frac{1}{2 a x - 2}"," ",0,"log(x) - 3*log(x - 1/a)/4 - log(x + 1/a)/4 - 1/(2*a*x - 2)","A",0
935,1,42,0,0.371428," ","integrate((a*x+1)/(-a**2*x**2+1)**2/x**2,x)","a \log{\left(x \right)} - \frac{5 a \log{\left(x - \frac{1}{a} \right)}}{4} + \frac{a \log{\left(x + \frac{1}{a} \right)}}{4} + \frac{- 3 a x + 2}{2 a x^{2} - 2 x}"," ",0,"a*log(x) - 5*a*log(x - 1/a)/4 + a*log(x + 1/a)/4 + (-3*a*x + 2)/(2*a*x**2 - 2*x)","A",0
936,1,58,0,0.404784," ","integrate((a*x+1)/(-a**2*x**2+1)**2/x**3,x)","2 a^{2} \log{\left(x \right)} - \frac{7 a^{2} \log{\left(x - \frac{1}{a} \right)}}{4} - \frac{a^{2} \log{\left(x + \frac{1}{a} \right)}}{4} + \frac{- 3 a^{2} x^{2} + a x + 1}{2 a x^{3} - 2 x^{2}}"," ",0,"2*a**2*log(x) - 7*a**2*log(x - 1/a)/4 - a**2*log(x + 1/a)/4 + (-3*a**2*x**2 + a*x + 1)/(2*a*x**3 - 2*x**2)","A",0
937,1,66,0,0.428448," ","integrate((a*x+1)/(-a**2*x**2+1)**2/x**4,x)","2 a^{3} \log{\left(x \right)} - \frac{9 a^{3} \log{\left(x - \frac{1}{a} \right)}}{4} + \frac{a^{3} \log{\left(x + \frac{1}{a} \right)}}{4} + \frac{- 15 a^{3} x^{3} + 9 a^{2} x^{2} + a x + 2}{6 a x^{4} - 6 x^{3}}"," ",0,"2*a**3*log(x) - 9*a**3*log(x - 1/a)/4 + a**3*log(x + 1/a)/4 + (-15*a**3*x**3 + 9*a**2*x**2 + a*x + 2)/(6*a*x**4 - 6*x**3)","A",0
938,1,82,0,0.435874," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**6,x)","- \frac{- 9 a^{2} x^{2} - 3 a x + 10}{8 a^{10} x^{3} - 8 a^{9} x^{2} - 8 a^{8} x + 8 a^{7}} - \frac{x^{2}}{2 a^{5}} - \frac{x}{a^{6}} - \frac{3 \left(\frac{13 \log{\left(x - \frac{1}{a} \right)}}{16} + \frac{3 \log{\left(x + \frac{1}{a} \right)}}{16}\right)}{a^{7}}"," ",0,"-(-9*a**2*x**2 - 3*a*x + 10)/(8*a**10*x**3 - 8*a**9*x**2 - 8*a**8*x + 8*a**7) - x**2/(2*a**5) - x/a**6 - 3*(13*log(x - 1/a)/16 + 3*log(x + 1/a)/16)/a**7","A",0
939,1,71,0,0.425837," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**5,x)","- \frac{- 9 a^{2} x^{2} + a x + 6}{8 a^{9} x^{3} - 8 a^{8} x^{2} - 8 a^{7} x + 8 a^{6}} - \frac{x}{a^{5}} - \frac{\frac{23 \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{7 \log{\left(x + \frac{1}{a} \right)}}{16}}{a^{6}}"," ",0,"-(-9*a**2*x**2 + a*x + 6)/(8*a**9*x**3 - 8*a**8*x**2 - 8*a**7*x + 8*a**6) - x/a**5 - (23*log(x - 1/a)/16 - 7*log(x + 1/a)/16)/a**6","A",0
940,1,68,0,0.378208," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**4,x)","- \frac{- 5 a^{2} x^{2} - 3 a x + 6}{8 a^{8} x^{3} - 8 a^{7} x^{2} - 8 a^{6} x + 8 a^{5}} - \frac{\frac{11 \log{\left(x - \frac{1}{a} \right)}}{16} + \frac{5 \log{\left(x + \frac{1}{a} \right)}}{16}}{a^{5}}"," ",0,"-(-5*a**2*x**2 - 3*a*x + 6)/(8*a**8*x**3 - 8*a**7*x**2 - 8*a**6*x + 8*a**5) - (11*log(x - 1/a)/16 + 5*log(x + 1/a)/16)/a**5","A",0
941,1,66,0,0.311286," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**3,x)","- \frac{- 5 a^{2} x^{2} + a x + 2}{8 a^{7} x^{3} - 8 a^{6} x^{2} - 8 a^{5} x + 8 a^{4}} - \frac{\frac{3 \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{3 \log{\left(x + \frac{1}{a} \right)}}{16}}{a^{4}}"," ",0,"-(-5*a**2*x**2 + a*x + 2)/(8*a**7*x**3 - 8*a**6*x**2 - 8*a**5*x + 8*a**4) - (3*log(x - 1/a)/16 - 3*log(x + 1/a)/16)/a**4","A",0
942,1,63,0,0.292456," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**2,x)","- \frac{- a^{2} x^{2} - 3 a x + 2}{8 a^{6} x^{3} - 8 a^{5} x^{2} - 8 a^{4} x + 8 a^{3}} - \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{16} + \frac{\log{\left(x + \frac{1}{a} \right)}}{16}}{a^{3}}"," ",0,"-(-a**2*x**2 - 3*a*x + 2)/(8*a**6*x**3 - 8*a**5*x**2 - 8*a**4*x + 8*a**3) - (-log(x - 1/a)/16 + log(x + 1/a)/16)/a**3","A",0
943,1,61,0,0.287946," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x,x)","- \frac{- a^{2} x^{2} + a x - 2}{8 a^{5} x^{3} - 8 a^{4} x^{2} - 8 a^{3} x + 8 a^{2}} - \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{16} + \frac{\log{\left(x + \frac{1}{a} \right)}}{16}}{a^{2}}"," ",0,"-(-a**2*x**2 + a*x - 2)/(8*a**5*x**3 - 8*a**4*x**2 - 8*a**3*x + 8*a**2) - (-log(x - 1/a)/16 + log(x + 1/a)/16)/a**2","A",0
944,1,65,0,0.304977," ","integrate((a*x+1)/(-a**2*x**2+1)**3,x)","- \frac{3 a^{2} x^{2} - 3 a x - 2}{8 a^{4} x^{3} - 8 a^{3} x^{2} - 8 a^{2} x + 8 a} - \frac{\frac{3 \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{3 \log{\left(x + \frac{1}{a} \right)}}{16}}{a}"," ",0,"-(3*a**2*x**2 - 3*a*x - 2)/(8*a**4*x**3 - 8*a**3*x**2 - 8*a**2*x + 8*a) - (3*log(x - 1/a)/16 - 3*log(x + 1/a)/16)/a","A",0
945,1,60,0,0.392669," ","integrate((a*x+1)/(-a**2*x**2+1)**3/x,x)","- \frac{3 a^{2} x^{2} + a x - 6}{8 a^{3} x^{3} - 8 a^{2} x^{2} - 8 a x + 8} + \log{\left(x \right)} - \frac{11 \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{5 \log{\left(x + \frac{1}{a} \right)}}{16}"," ",0,"-(3*a**2*x**2 + a*x - 6)/(8*a**3*x**3 - 8*a**2*x**2 - 8*a*x + 8) + log(x) - 11*log(x - 1/a)/16 - 5*log(x + 1/a)/16","A",0
946,1,78,0,0.515716," ","integrate((a*x+1)/(-a**2*x**2+1)**3/x**2,x)","a \log{\left(x \right)} - \frac{23 a \log{\left(x - \frac{1}{a} \right)}}{16} + \frac{7 a \log{\left(x + \frac{1}{a} \right)}}{16} - \frac{15 a^{3} x^{3} - 11 a^{2} x^{2} - 14 a x + 8}{8 a^{3} x^{4} - 8 a^{2} x^{3} - 8 a x^{2} + 8 x}"," ",0,"a*log(x) - 23*a*log(x - 1/a)/16 + 7*a*log(x + 1/a)/16 - (15*a**3*x**3 - 11*a**2*x**2 - 14*a*x + 8)/(8*a**3*x**4 - 8*a**2*x**3 - 8*a*x**2 + 8*x)","A",0
947,1,95,0,0.558045," ","integrate((a*x+1)/(-a**2*x**2+1)**3/x**3,x)","3 a^{2} \log{\left(x \right)} - \frac{39 a^{2} \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{9 a^{2} \log{\left(x + \frac{1}{a} \right)}}{16} - \frac{15 a^{4} x^{4} - 3 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 4}{8 a^{3} x^{5} - 8 a^{2} x^{4} - 8 a x^{3} + 8 x^{2}}"," ",0,"3*a**2*log(x) - 39*a**2*log(x - 1/a)/16 - 9*a**2*log(x + 1/a)/16 - (15*a**4*x**4 - 3*a**3*x**3 - 22*a**2*x**2 + 4*a*x + 4)/(8*a**3*x**5 - 8*a**2*x**4 - 8*a*x**3 + 8*x**2)","A",0
948,1,104,0,0.584790," ","integrate((a*x+1)/(-a**2*x**2+1)**3/x**4,x)","3 a^{3} \log{\left(x \right)} - \frac{59 a^{3} \log{\left(x - \frac{1}{a} \right)}}{16} + \frac{11 a^{3} \log{\left(x + \frac{1}{a} \right)}}{16} - \frac{105 a^{5} x^{5} - 69 a^{4} x^{4} - 106 a^{3} x^{3} + 52 a^{2} x^{2} + 4 a x + 8}{24 a^{3} x^{6} - 24 a^{2} x^{5} - 24 a x^{4} + 24 x^{3}}"," ",0,"3*a**3*log(x) - 59*a**3*log(x - 1/a)/16 + 11*a**3*log(x + 1/a)/16 - (105*a**5*x**5 - 69*a**4*x**4 - 106*a**3*x**3 + 52*a**2*x**2 + 4*a*x + 8)/(24*a**3*x**6 - 24*a**2*x**5 - 24*a*x**4 + 24*x**3)","A",0
949,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
950,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
951,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
952,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
953,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)","F",0
954,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
955,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(5/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(5/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
956,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**(7/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
957,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{4} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**4*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
958,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{3} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**3*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
959,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{2} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
960,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
961,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
962,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{a x + 1}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
963,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{a x + 1}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
964,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{a x + 1}{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
965,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{a x + 1}{x^{4} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)/(x**4*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
966,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{5} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**5*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
967,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{4} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**4*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
968,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{3} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
969,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{2} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
970,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
971,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
972,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{a x + 1}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
973,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{a x + 1}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
974,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{a x + 1}{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
975,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{a x + 1}{x^{4} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x**4*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
976,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**6/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{6} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**6*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
977,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{5} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**5*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
978,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{4} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**4*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
979,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{3} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**3*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
980,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{2} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**2*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
981,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
982,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
983,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{a x + 1}{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
984,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{a x + 1}{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
985,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{a x + 1}{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(x**3*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
986,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(7/2),x)","\int \frac{a x + 1}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}}}\, dx"," ",0,"Integral((a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(7/2)), x)","F",0
987,1,223,0,14.387516," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c)**2,x)","- \frac{a^{3} c^{2} x^{4} x^{m} \Gamma\left(\frac{m}{2} + 2\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + 3\right)} - \frac{a^{2} c^{2} x^{3} x^{m} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{5}{2}\right)} + \frac{a c^{2} x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{c^{2} x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"-a**3*c**2*x**4*x**m*gamma(m/2 + 2)*hyper((-1/2, m/2 + 2), (m/2 + 3,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3)) - a**2*c**2*x**3*x**m*gamma(m/2 + 3/2)*hyper((-1/2, m/2 + 3/2), (m/2 + 5/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 5/2)) + a*c**2*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 2)) + c**2*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3/2))","C",0
988,1,104,0,4.502795," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c),x)","\frac{a c x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{c x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"a*c*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 2)) + c*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3/2))","C",0
989,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{m}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x x^{m}}{- a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x**m/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x*x**m/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
990,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{m}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x x^{m}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(x**m/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x*x**m/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
991,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{m}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x x^{m}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(x**m/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x*x**m/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
992,1,1760,0,1.380779," ","integrate((a*x+1)*(-a**2*x**2+1)**2*x**m,x)","\begin{cases} a^{5} \log{\left(x \right)} - \frac{a^{4}}{x} + \frac{a^{3}}{x^{2}} + \frac{2 a^{2}}{3 x^{3}} - \frac{a}{4 x^{4}} - \frac{1}{5 x^{5}} & \text{for}\: m = -6 \\a^{5} x + a^{4} \log{\left(x \right)} + \frac{2 a^{3}}{x} + \frac{a^{2}}{x^{2}} - \frac{a}{3 x^{3}} - \frac{1}{4 x^{4}} & \text{for}\: m = -5 \\\frac{a^{5} x^{2}}{2} + a^{4} x - 2 a^{3} \log{\left(x \right)} + \frac{2 a^{2}}{x} - \frac{a}{2 x^{2}} - \frac{1}{3 x^{3}} & \text{for}\: m = -4 \\\frac{a^{5} x^{3}}{3} + \frac{a^{4} x^{2}}{2} - 2 a^{3} x - 2 a^{2} \log{\left(x \right)} - \frac{a}{x} - \frac{1}{2 x^{2}} & \text{for}\: m = -3 \\\frac{a^{5} x^{4}}{4} + \frac{a^{4} x^{3}}{3} - a^{3} x^{2} - 2 a^{2} x + a \log{\left(x \right)} - \frac{1}{x} & \text{for}\: m = -2 \\\frac{a^{5} x^{5}}{5} + \frac{a^{4} x^{4}}{4} - \frac{2 a^{3} x^{3}}{3} - a^{2} x^{2} + a x + \log{\left(x \right)} & \text{for}\: m = -1 \\\frac{a^{5} m^{5} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{15 a^{5} m^{4} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{85 a^{5} m^{3} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{225 a^{5} m^{2} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{274 a^{5} m x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{120 a^{5} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{a^{4} m^{5} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{16 a^{4} m^{4} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{95 a^{4} m^{3} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{260 a^{4} m^{2} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{324 a^{4} m x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{144 a^{4} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{2 a^{3} m^{5} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{34 a^{3} m^{4} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{214 a^{3} m^{3} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{614 a^{3} m^{2} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{792 a^{3} m x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{360 a^{3} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{2 a^{2} m^{5} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{36 a^{2} m^{4} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{242 a^{2} m^{3} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{744 a^{2} m^{2} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{1016 a^{2} m x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac{480 a^{2} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{a m^{5} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{19 a m^{4} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{137 a m^{3} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{461 a m^{2} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{702 a m x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{360 a x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{m^{5} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{20 m^{4} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{155 m^{3} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{580 m^{2} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{1044 m x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac{720 x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} & \text{otherwise} \end{cases}"," ",0,"Piecewise((a**5*log(x) - a**4/x + a**3/x**2 + 2*a**2/(3*x**3) - a/(4*x**4) - 1/(5*x**5), Eq(m, -6)), (a**5*x + a**4*log(x) + 2*a**3/x + a**2/x**2 - a/(3*x**3) - 1/(4*x**4), Eq(m, -5)), (a**5*x**2/2 + a**4*x - 2*a**3*log(x) + 2*a**2/x - a/(2*x**2) - 1/(3*x**3), Eq(m, -4)), (a**5*x**3/3 + a**4*x**2/2 - 2*a**3*x - 2*a**2*log(x) - a/x - 1/(2*x**2), Eq(m, -3)), (a**5*x**4/4 + a**4*x**3/3 - a**3*x**2 - 2*a**2*x + a*log(x) - 1/x, Eq(m, -2)), (a**5*x**5/5 + a**4*x**4/4 - 2*a**3*x**3/3 - a**2*x**2 + a*x + log(x), Eq(m, -1)), (a**5*m**5*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*a**5*m**4*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*a**5*m**3*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 225*a**5*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 274*a**5*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 120*a**5*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + a**4*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 16*a**4*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 95*a**4*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 260*a**4*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 324*a**4*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*a**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**3*m**5*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 34*a**3*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 214*a**3*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 614*a**3*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 792*a**3*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 360*a**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**2*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 36*a**2*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 242*a**2*m**3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 744*a**2*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1016*a**2*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 480*a**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + a*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 19*a*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 137*a*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 461*a*m**2*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 702*a*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*a*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*m**4*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*m**2*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1044*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720), True))","A",0
993,1,585,0,0.692448," ","integrate((a*x+1)*(-a**2*x**2+1)*x**m,x)","\begin{cases} - a^{3} \log{\left(x \right)} + \frac{a^{2}}{x} - \frac{a}{2 x^{2}} - \frac{1}{3 x^{3}} & \text{for}\: m = -4 \\- a^{3} x - a^{2} \log{\left(x \right)} - \frac{a}{x} - \frac{1}{2 x^{2}} & \text{for}\: m = -3 \\- \frac{a^{3} x^{2}}{2} - a^{2} x + a \log{\left(x \right)} - \frac{1}{x} & \text{for}\: m = -2 \\- \frac{a^{3} x^{3}}{3} - \frac{a^{2} x^{2}}{2} + a x + \log{\left(x \right)} & \text{for}\: m = -1 \\- \frac{a^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{6 a^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{11 a^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{6 a^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{a^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{7 a^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{14 a^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac{8 a^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{a m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{8 a m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{19 a m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{12 a x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases}"," ",0,"Piecewise((-a**3*log(x) + a**2/x - a/(2*x**2) - 1/(3*x**3), Eq(m, -4)), (-a**3*x - a**2*log(x) - a/x - 1/(2*x**2), Eq(m, -3)), (-a**3*x**2/2 - a**2*x + a*log(x) - 1/x, Eq(m, -2)), (-a**3*x**3/3 - a**2*x**2/2 + a*x + log(x), Eq(m, -1)), (-a**3*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 6*a**3*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 11*a**3*m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 6*a**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a**2*m**3*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 7*a**2*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 14*a**2*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + a*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*a*m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*a*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 12*a*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + m**3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*m**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))","A",0
994,1,82,0,0.253262," ","integrate((a*x+1)*x**m,x)","\begin{cases} a \log{\left(x \right)} - \frac{1}{x} & \text{for}\: m = -2 \\a x + \log{\left(x \right)} & \text{for}\: m = -1 \\\frac{a m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac{a x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac{m x x^{m}}{m^{2} + 3 m + 2} + \frac{2 x x^{m}}{m^{2} + 3 m + 2} & \text{otherwise} \end{cases}"," ",0,"Piecewise((a*log(x) - 1/x, Eq(m, -2)), (a*x + log(x), Eq(m, -1)), (a*m*x**2*x**m/(m**2 + 3*m + 2) + a*x**2*x**m/(m**2 + 3*m + 2) + m*x*x**m/(m**2 + 3*m + 2) + 2*x*x**m/(m**2 + 3*m + 2), True))","A",0
995,1,44,0,2.692724," ","integrate((a*x+1)/(-a**2*x**2+1)*x**m,x)","\frac{m x x^{m} \Phi\left(a x, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)} + \frac{x x^{m} \Phi\left(a x, 1, m + 1\right) \Gamma\left(m + 1\right)}{\Gamma\left(m + 2\right)}"," ",0,"m*x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma(m + 2) + x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma(m + 2)","B",0
996,1,673,0,29.316586," ","integrate((a*x+1)/(-a**2*x**2+1)**2*x**m,x)","- \frac{a^{2} m^{2} x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{a^{2} x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + a \left(- \frac{a^{2} m^{2} x^{4} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{2 a^{2} m x^{4} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{m^{2} x^{2} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{2 m x^{2} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{2 m x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{4 x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) - 8 \Gamma\left(\frac{m}{2} + 2\right)}\right) + \frac{m^{2} x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{2 m x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{2 x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{8 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 8 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"-a**2*m**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) + a**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) + a*(-a**2*m**2*x**4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) - 2*a**2*m*x**4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) + m**2*x**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) + 2*m*x**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) - 2*m*x**2*x**m*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) - 4*x**2*x**m*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2))) + m**2*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) - 2*m*x*x**m*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) - x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) - 2*x*x**m*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2))","C",0
997,1,2152,0,50.308232," ","integrate((a*x+1)/(-a**2*x**2+1)**3*x**m,x)","\frac{a^{4} m^{3} x^{5} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{3 a^{4} m^{2} x^{5} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{a^{4} m x^{5} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{3 a^{4} x^{5} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{2 a^{2} m^{3} x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{6 a^{2} m^{2} x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{2 a^{2} m^{2} x^{3} x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{2 a^{2} m x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{4 a^{2} m x^{3} x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{6 a^{2} x^{3} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{6 a^{2} x^{3} x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + a \left(\frac{a^{4} m^{3} x^{6} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{4 a^{4} m x^{6} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{2 a^{2} m^{3} x^{4} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{2 a^{2} m^{2} x^{4} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{8 a^{2} m x^{4} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{8 a^{2} x^{4} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{m^{3} x^{2} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{2 m^{2} x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} - \frac{4 m x^{2} x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + 1\right) \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{4 m x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)} + \frac{16 x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + 2\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + 2\right) + 32 \Gamma\left(\frac{m}{2} + 2\right)}\right) + \frac{m^{3} x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{3 m^{2} x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{2 m^{2} x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} - \frac{m x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{8 m x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{3 x x^{m} \Phi\left(a^{2} x^{2} e^{2 i \pi}, 1, \frac{m}{2} + \frac{1}{2}\right) \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)} + \frac{10 x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right)}{32 a^{4} x^{4} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) - 64 a^{2} x^{2} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) + 32 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"a**4*m**3*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 3*a**4*m**2*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - a**4*m*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 3*a**4*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 2*a**2*m**3*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 6*a**2*m**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 2*a**2*m**2*x**3*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 2*a**2*m*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 4*a**2*m*x**3*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 6*a**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 6*a**2*x**3*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + a*(a**4*m**3*x**6*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 4*a**4*m*x**6*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 2*a**2*m**3*x**4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 2*a**2*m**2*x**4*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 8*a**2*m*x**4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 8*a**2*x**4*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + m**3*x**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 2*m**2*x**2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 4*m*x**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 4*m*x**2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 16*x**2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2))) + m**3*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 3*m**2*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 2*m**2*x*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - m*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 8*m*x*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 3*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 10*x*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2))","C",0
998,-1,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
999,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{m} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*(-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
1000,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
1001,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{m} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**m*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
1002,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{m} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**m*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1003,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{m} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**m*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
1004,1,381,0,69.565765," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**m*(-a**2*c*x**2+c)**p,x)","- \frac{a a^{2 p} c^{p} x^{2} x^{m} x^{2 p} e^{i \pi p} \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(- \frac{m}{2} - p - 1\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, \frac{m}{2} + p + 1 \\ p + 1, \frac{m}{2} + p + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- \frac{m}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{2} x^{m} x^{2 p} e^{i \pi p} \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(- \frac{m}{2} - p - 1\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - \frac{m}{2} - p - 1 \\ \frac{1}{2}, - \frac{m}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- \frac{m}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x x^{m} x^{2 p} e^{i \pi p} \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(- \frac{m}{2} - p - \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, \frac{m}{2} + p + \frac{1}{2} \\ p + 1, \frac{m}{2} + p + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(p + 1\right) \Gamma\left(- \frac{m}{2} - p + \frac{1}{2}\right)} - \frac{a^{2 p} c^{p} x x^{m} x^{2 p} e^{i \pi p} \Gamma\left(p + \frac{1}{2}\right) \Gamma\left(- \frac{m}{2} - p - \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - \frac{m}{2} - p - \frac{1}{2} \\ \frac{1}{2}, - \frac{m}{2} - p + \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(p + 1\right) \Gamma\left(- \frac{m}{2} - p + \frac{1}{2}\right)}"," ",0,"-a*a**(2*p)*c**p*x**2*x**m*x**(2*p)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1)*hyper((1/2, 1, m/2 + p + 1), (p + 1, m/2 + p + 2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-m/2 - p)*gamma(p + 1)) - a*a**(2*p)*c**p*x**2*x**m*x**(2*p)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1)*hyper((1, -p, -m/2 - p - 1), (1/2, -m/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-m/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x*x**m*x**(2*p)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1/2)*hyper((1/2, 1, m/2 + p + 1/2), (p + 1, m/2 + p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(p + 1)*gamma(-m/2 - p + 1/2)) - a**(2*p)*c**p*x*x**m*x**(2*p)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1/2)*hyper((1, -p, -m/2 - p - 1/2), (1/2, -m/2 - p + 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(p + 1)*gamma(-m/2 - p + 1/2))","C",0
1005,1,258,0,19.096884," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a**2*x**2+1)**p,x)","- \frac{a a^{2 p} x^{5} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{5}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{5}{2} \\ p + 1, p + \frac{7}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x^{5} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{5}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{5}{2} \\ \frac{1}{2}, - p - \frac{3}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + 1\right)} - \frac{{G_{3, 3}^{2, 2}\left(\begin{matrix} - p - 1, 1 & -1 \\- p - \frac{3}{2}, - p - 1 & 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 \pi a^{4}} - \frac{{G_{3, 3}^{1, 3}\left(\begin{matrix} -1, - p - 2, 1 &  \\- p - 2 & - p - \frac{3}{2}, 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 a^{4} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x**5*x**(2*p)*exp(I*pi*p)*gamma(-p - 5/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 5/2), (p + 1, p + 7/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 3/2)*gamma(p + 1)) - a*a**(2*p)*x**5*x**(2*p)*exp(I*pi*p)*gamma(-p - 5/2)*gamma(p + 1/2)*hyper((1, -p, -p - 5/2), (1/2, -p - 3/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 3/2)*gamma(p + 1)) - meijerg(((-p - 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*pi*a**4) - meijerg(((-1, -p - 2, 1), ()), ((-p - 2,), (-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a**4*gamma(-p)*gamma(p + 1))","C",0
1006,1,255,0,13.508802," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a**2*x**2+1)**p,x)","- \frac{a^{2 p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{3}{2} \\ p + 1, p + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{3}{2} \\ \frac{1}{2}, - p - \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{{G_{3, 3}^{2, 2}\left(\begin{matrix} - p - 1, 1 & -1 \\- p - \frac{3}{2}, - p - 1 & 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 \pi a^{3}} - \frac{{G_{3, 3}^{1, 3}\left(\begin{matrix} -1, - p - 2, 1 &  \\- p - 2 & - p - \frac{3}{2}, 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 a^{3} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a**(2*p)*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a**(2*p)*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - meijerg(((-p - 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*pi*a**3) - meijerg(((-1, -p - 2, 1), ()), ((-p - 2,), (-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a**3*gamma(-p)*gamma(p + 1))","C",0
1007,1,301,0,12.100041," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a**2*x**2+1)**p,x)","- \frac{a a^{2 p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{3}{2} \\ p + 1, p + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{3}{2} \\ \frac{1}{2}, - p - \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a*a**(2*p)*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1))","C",0
1008,1,292,0,11.472679," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p,x)","- \frac{a a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{1}{2} \\ p + 1, p + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a*a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p)*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1))","C",0
1009,1,286,0,30.722519," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p/x,x)","- \frac{a a^{2 p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{1}{2} \\ p + 1, p + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p \\ p + 1, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p \\ \frac{1}{2}, 1 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a*a**(2*p)*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1/2, 1, p), (p + 1, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1))","C",0
1010,1,280,0,12.386891," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p/x**2,x)","- \frac{a a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p \\ p + 1, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p \\ \frac{1}{2}, 1 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - \frac{1}{2} \\ p + \frac{1}{2}, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, \frac{1}{2} - p \\ \frac{1}{2}, \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1/2, 1, p), (p + 1, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1/2), (p + 1/2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1, -p, 1/2 - p), (1/2, 3/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1))","C",0
1011,1,287,0,56.395235," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p/x**3,x)","- \frac{a a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - \frac{1}{2} \\ p + \frac{1}{2}, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, \frac{1}{2} - p \\ \frac{1}{2}, \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(1 - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - 1 \\ p, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x^{2} \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} x^{2 p} e^{i \pi p} \Gamma\left(1 - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, 1 - p \\ \frac{1}{2}, 2 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x^{2} \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1/2), (p + 1/2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1, -p, 1/2 - p), (1/2, 3/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1), (p, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x**2*gamma(2 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(1 - p)*gamma(p + 1/2)*hyper((1, -p, 1 - p), (1/2, 2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x**2*gamma(2 - p)*gamma(p + 1))","C",0
1012,1,272,0,54.205683," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3*(-a**2*c*x**2+c)**p,x)","- \frac{a a^{2 p} c^{p} x^{5} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{5}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{5}{2} \\ p + 1, p + \frac{7}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{5} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{5}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{5}{2} \\ \frac{1}{2}, - p - \frac{3}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + 1\right)} - \frac{c^{p} {G_{3, 3}^{2, 2}\left(\begin{matrix} - p - 1, 1 & -1 \\- p - \frac{3}{2}, - p - 1 & 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 \pi a^{4}} - \frac{c^{p} {G_{3, 3}^{1, 3}\left(\begin{matrix} -1, - p - 2, 1 &  \\- p - 2 & - p - \frac{3}{2}, 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 a^{4} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x**5*x**(2*p)*exp(I*pi*p)*gamma(-p - 5/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 5/2), (p + 1, p + 7/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 3/2)*gamma(p + 1)) - a*a**(2*p)*c**p*x**5*x**(2*p)*exp(I*pi*p)*gamma(-p - 5/2)*gamma(p + 1/2)*hyper((1, -p, -p - 5/2), (1/2, -p - 3/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 3/2)*gamma(p + 1)) - c**p*meijerg(((-p - 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*pi*a**4) - c**p*meijerg(((-1, -p - 2, 1), ()), ((-p - 2,), (-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a**4*gamma(-p)*gamma(p + 1))","C",0
1013,1,269,0,16.696982," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2*(-a**2*c*x**2+c)**p,x)","- \frac{a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{3}{2} \\ p + 1, p + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{3}{2} \\ \frac{1}{2}, - p - \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{c^{p} {G_{3, 3}^{2, 2}\left(\begin{matrix} - p - 1, 1 & -1 \\- p - \frac{3}{2}, - p - 1 & 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 \pi a^{3}} - \frac{c^{p} {G_{3, 3}^{1, 3}\left(\begin{matrix} -1, - p - 2, 1 &  \\- p - 2 & - p - \frac{3}{2}, 0 \end{matrix} \middle| {\frac{e^{- i \pi}}{a^{2} x^{2}}} \right)} \Gamma\left(p + \frac{1}{2}\right)}{2 a^{3} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a**(2*p)*c**p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a**(2*p)*c**p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - c**p*meijerg(((-p - 1, 1), (-1,)), ((-p - 3/2, -p - 1), (0,)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*pi*a**3) - c**p*meijerg(((-1, -p - 2, 1), ()), ((-p - 2,), (-p - 3/2, 0)), exp_polar(-I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a**3*gamma(-p)*gamma(p + 1))","C",0
1014,1,314,0,17.587226," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x*(-a**2*c*x**2+c)**p,x)","- \frac{a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{3}{2} \\ p + 1, p + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{3}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{3}{2} \\ \frac{1}{2}, - p - \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p + 5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a*a**(2*p)*c**p*x**3*x**(2*p)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p - 1/2)*gamma(p + 1)) - a**(2*p)*c**p*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1))","C",0
1015,1,306,0,23.174757," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**p,x)","- \frac{a a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ p + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma\left(- p - 1\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1, - p - 1 \\ \frac{1}{2} \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{1}{2} \\ p + 1, p + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a*a**(2*p)*c**p*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1))","C",0
1016,1,299,0,17.331623," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**p/x,x)","- \frac{a a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p + \frac{1}{2} \\ p + 1, p + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x x^{2 p} e^{i \pi p} \Gamma\left(- p - \frac{1}{2}\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p \\ p + 1, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p \\ \frac{1}{2}, 1 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a*a**(2*p)*c**p*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1/2, 1, p), (p + 1, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1))","C",0
1017,1,294,0,12.520290," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**p/x**2,x)","- \frac{a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p \\ p + 1, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(- p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, - p \\ \frac{1}{2}, 1 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} \Gamma\left(1 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - \frac{1}{2} \\ p + \frac{1}{2}, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, \frac{1}{2} - p \\ \frac{1}{2}, \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1/2, 1, p), (p + 1, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a*a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1/2), (p + 1/2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1, -p, 1/2 - p), (1/2, 3/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1))","C",0
1018,1,301,0,13.757773," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**p/x**3,x)","- \frac{a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - \frac{1}{2} \\ p + \frac{1}{2}, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, \frac{1}{2} - p \\ \frac{1}{2}, \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(1 - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} \frac{1}{2}, 1, p - 1 \\ p, p + 1 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \sqrt{\pi} x^{2} \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma\left(1 - p\right) \Gamma\left(p + \frac{1}{2}\right) {{}_{3}F_{2}\left(\begin{matrix} 1, - p, 1 - p \\ \frac{1}{2}, 2 - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 \sqrt{\pi} x^{2} \Gamma\left(2 - p\right) \Gamma\left(p + 1\right)}"," ",0,"-a*a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1/2), (p + 1/2, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a*a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1, -p, 1/2 - p), (1/2, 3/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x*gamma(3/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1 - p)*gamma(p + 1/2)*hyper((1/2, 1, p - 1), (p, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*x**2*gamma(2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p)*gamma(1 - p)*gamma(p + 1/2)*hyper((1, -p, 1 - p), (1/2, 2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*x**2*gamma(2 - p)*gamma(p + 1))","C",0
1019,1,24,0,0.072288," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4*(-a**2*c*x**2+c),x)","\frac{a^{2} c x^{7}}{7} + \frac{a c x^{6}}{3} + \frac{c x^{5}}{5}"," ",0,"a**2*c*x**7/7 + a*c*x**6/3 + c*x**5/5","A",0
1020,1,26,0,0.073637," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c),x)","\frac{a^{2} c x^{6}}{6} + \frac{2 a c x^{5}}{5} + \frac{c x^{4}}{4}"," ",0,"a**2*c*x**6/6 + 2*a*c*x**5/5 + c*x**4/4","A",0
1021,1,24,0,0.073857," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c),x)","\frac{a^{2} c x^{5}}{5} + \frac{a c x^{4}}{2} + \frac{c x^{3}}{3}"," ",0,"a**2*c*x**5/5 + a*c*x**4/2 + c*x**3/3","A",0
1022,1,26,0,0.070976," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c),x)","\frac{a^{2} c x^{4}}{4} + \frac{2 a c x^{3}}{3} + \frac{c x^{2}}{2}"," ",0,"a**2*c*x**4/4 + 2*a*c*x**3/3 + c*x**2/2","A",0
1023,1,19,0,0.067745," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c),x)","\frac{a^{2} c x^{3}}{3} + a c x^{2} + c x"," ",0,"a**2*c*x**3/3 + a*c*x**2 + c*x","A",0
1024,1,20,0,0.098905," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)/x,x)","\frac{a^{2} c x^{2}}{2} + 2 a c x + c \log{\left(x \right)}"," ",0,"a**2*c*x**2/2 + 2*a*c*x + c*log(x)","A",0
1025,1,17,0,0.115779," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)/x**2,x)","a^{2} c x + 2 a c \log{\left(x \right)} - \frac{c}{x}"," ",0,"a**2*c*x + 2*a*c*log(x) - c/x","A",0
1026,1,22,0,0.148922," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)/x**3,x)","a^{2} c \log{\left(x \right)} + \frac{- 4 a c x - c}{2 x^{2}}"," ",0,"a**2*c*log(x) + (-4*a*c*x - c)/(2*x**2)","A",0
1027,1,24,0,0.155089," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)/x**4,x)","\frac{- 3 a^{2} c x^{2} - 3 a c x - c}{3 x^{3}}"," ",0,"(-3*a**2*c*x**2 - 3*a*c*x - c)/(3*x**3)","A",0
1028,1,41,0,0.085380," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4*(-a**2*c*x**2+c)**2,x)","- \frac{a^{4} c^{2} x^{9}}{9} - \frac{a^{3} c^{2} x^{8}}{4} + \frac{a c^{2} x^{6}}{3} + \frac{c^{2} x^{5}}{5}"," ",0,"-a**4*c**2*x**9/9 - a**3*c**2*x**8/4 + a*c**2*x**6/3 + c**2*x**5/5","A",0
1029,1,44,0,0.085441," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**2,x)","- \frac{a^{4} c^{2} x^{8}}{8} - \frac{2 a^{3} c^{2} x^{7}}{7} + \frac{2 a c^{2} x^{5}}{5} + \frac{c^{2} x^{4}}{4}"," ",0,"-a**4*c**2*x**8/8 - 2*a**3*c**2*x**7/7 + 2*a*c**2*x**5/5 + c**2*x**4/4","A",0
1030,1,41,0,0.085927," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**2,x)","- \frac{a^{4} c^{2} x^{7}}{7} - \frac{a^{3} c^{2} x^{6}}{3} + \frac{a c^{2} x^{4}}{2} + \frac{c^{2} x^{3}}{3}"," ",0,"-a**4*c**2*x**7/7 - a**3*c**2*x**6/3 + a*c**2*x**4/2 + c**2*x**3/3","A",0
1031,1,44,0,0.084669," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**2,x)","- \frac{a^{4} c^{2} x^{6}}{6} - \frac{2 a^{3} c^{2} x^{5}}{5} + \frac{2 a c^{2} x^{3}}{3} + \frac{c^{2} x^{2}}{2}"," ",0,"-a**4*c**2*x**6/6 - 2*a**3*c**2*x**5/5 + 2*a*c**2*x**3/3 + c**2*x**2/2","A",0
1032,1,36,0,0.082552," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2,x)","- \frac{a^{4} c^{2} x^{5}}{5} - \frac{a^{3} c^{2} x^{4}}{2} + a c^{2} x^{2} + c^{2} x"," ",0,"-a**4*c**2*x**5/5 - a**3*c**2*x**4/2 + a*c**2*x**2 + c**2*x","A",0
1033,1,39,0,0.123733," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x,x)","- \frac{a^{4} c^{2} x^{4}}{4} - \frac{2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log{\left(x \right)}"," ",0,"-a**4*c**2*x**4/4 - 2*a**3*c**2*x**3/3 + 2*a*c**2*x + c**2*log(x)","A",0
1034,1,36,0,0.137172," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**2,x)","- \frac{a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log{\left(x \right)} - \frac{c^{2}}{x}"," ",0,"-a**4*c**2*x**3/3 - a**3*c**2*x**2 + 2*a*c**2*log(x) - c**2/x","A",0
1035,1,39,0,0.142187," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**3,x)","- \frac{a^{4} c^{2} x^{2}}{2} - 2 a^{3} c^{2} x - \frac{4 a c^{2} x + c^{2}}{2 x^{2}}"," ",0,"-a**4*c**2*x**2/2 - 2*a**3*c**2*x - (4*a*c**2*x + c**2)/(2*x**2)","B",0
1036,1,37,0,0.188926," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**4,x)","- a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left(x \right)} - \frac{3 a c^{2} x + c^{2}}{3 x^{3}}"," ",0,"-a**4*c**2*x - 2*a**3*c**2*log(x) - (3*a*c**2*x + c**2)/(3*x**3)","A",0
1037,1,41,0,0.222829," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**5,x)","- a^{4} c^{2} \log{\left(x \right)} - \frac{- 24 a^{3} c^{2} x^{3} + 8 a c^{2} x + 3 c^{2}}{12 x^{4}}"," ",0,"-a**4*c**2*log(x) - (-24*a**3*c**2*x**3 + 8*a*c**2*x + 3*c**2)/(12*x**4)","A",0
1038,1,42,0,0.248956," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**2/x**6,x)","- \frac{- 10 a^{4} c^{2} x^{4} - 10 a^{3} c^{2} x^{3} + 5 a c^{2} x + 2 c^{2}}{10 x^{5}}"," ",0,"-(-10*a**4*c**2*x**4 - 10*a**3*c**2*x**3 + 5*a*c**2*x + 2*c**2)/(10*x**5)","A",0
1039,1,76,0,0.099411," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4*(-a**2*c*x**2+c)**3,x)","\frac{a^{6} c^{3} x^{11}}{11} + \frac{a^{5} c^{3} x^{10}}{5} - \frac{a^{4} c^{3} x^{9}}{9} - \frac{a^{3} c^{3} x^{8}}{2} - \frac{a^{2} c^{3} x^{7}}{7} + \frac{a c^{3} x^{6}}{3} + \frac{c^{3} x^{5}}{5}"," ",0,"a**6*c**3*x**11/11 + a**5*c**3*x**10/5 - a**4*c**3*x**9/9 - a**3*c**3*x**8/2 - a**2*c**3*x**7/7 + a*c**3*x**6/3 + c**3*x**5/5","A",0
1040,1,82,0,0.096684," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**3,x)","\frac{a^{6} c^{3} x^{10}}{10} + \frac{2 a^{5} c^{3} x^{9}}{9} - \frac{a^{4} c^{3} x^{8}}{8} - \frac{4 a^{3} c^{3} x^{7}}{7} - \frac{a^{2} c^{3} x^{6}}{6} + \frac{2 a c^{3} x^{5}}{5} + \frac{c^{3} x^{4}}{4}"," ",0,"a**6*c**3*x**10/10 + 2*a**5*c**3*x**9/9 - a**4*c**3*x**8/8 - 4*a**3*c**3*x**7/7 - a**2*c**3*x**6/6 + 2*a*c**3*x**5/5 + c**3*x**4/4","A",0
1041,1,78,0,0.094434," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**3,x)","\frac{a^{6} c^{3} x^{9}}{9} + \frac{a^{5} c^{3} x^{8}}{4} - \frac{a^{4} c^{3} x^{7}}{7} - \frac{2 a^{3} c^{3} x^{6}}{3} - \frac{a^{2} c^{3} x^{5}}{5} + \frac{a c^{3} x^{4}}{2} + \frac{c^{3} x^{3}}{3}"," ",0,"a**6*c**3*x**9/9 + a**5*c**3*x**8/4 - a**4*c**3*x**7/7 - 2*a**3*c**3*x**6/3 - a**2*c**3*x**5/5 + a*c**3*x**4/2 + c**3*x**3/3","A",0
1042,1,82,0,0.095927," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**3,x)","\frac{a^{6} c^{3} x^{8}}{8} + \frac{2 a^{5} c^{3} x^{7}}{7} - \frac{a^{4} c^{3} x^{6}}{6} - \frac{4 a^{3} c^{3} x^{5}}{5} - \frac{a^{2} c^{3} x^{4}}{4} + \frac{2 a c^{3} x^{3}}{3} + \frac{c^{3} x^{2}}{2}"," ",0,"a**6*c**3*x**8/8 + 2*a**5*c**3*x**7/7 - a**4*c**3*x**6/6 - 4*a**3*c**3*x**5/5 - a**2*c**3*x**4/4 + 2*a*c**3*x**3/3 + c**3*x**2/2","A",0
1043,1,70,0,0.090647," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3,x)","\frac{a^{6} c^{3} x^{7}}{7} + \frac{a^{5} c^{3} x^{6}}{3} - \frac{a^{4} c^{3} x^{5}}{5} - a^{3} c^{3} x^{4} - \frac{a^{2} c^{3} x^{3}}{3} + a c^{3} x^{2} + c^{3} x"," ",0,"a**6*c**3*x**7/7 + a**5*c**3*x**6/3 - a**4*c**3*x**5/5 - a**3*c**3*x**4 - a**2*c**3*x**3/3 + a*c**3*x**2 + c**3*x","A",0
1044,1,76,0,0.152599," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3/x,x)","\frac{a^{6} c^{3} x^{6}}{6} + \frac{2 a^{5} c^{3} x^{5}}{5} - \frac{a^{4} c^{3} x^{4}}{4} - \frac{4 a^{3} c^{3} x^{3}}{3} - \frac{a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log{\left(x \right)}"," ",0,"a**6*c**3*x**6/6 + 2*a**5*c**3*x**5/5 - a**4*c**3*x**4/4 - 4*a**3*c**3*x**3/3 - a**2*c**3*x**2/2 + 2*a*c**3*x + c**3*log(x)","A",0
1045,1,70,0,0.167479," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3/x**2,x)","\frac{a^{6} c^{3} x^{5}}{5} + \frac{a^{5} c^{3} x^{4}}{2} - \frac{a^{4} c^{3} x^{3}}{3} - 2 a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 a c^{3} \log{\left(x \right)} - \frac{c^{3}}{x}"," ",0,"a**6*c**3*x**5/5 + a**5*c**3*x**4/2 - a**4*c**3*x**3/3 - 2*a**3*c**3*x**2 - a**2*c**3*x + 2*a*c**3*log(x) - c**3/x","A",0
1046,1,75,0,0.200185," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3/x**3,x)","\frac{a^{6} c^{3} x^{4}}{4} + \frac{2 a^{5} c^{3} x^{3}}{3} - \frac{a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log{\left(x \right)} + \frac{- 4 a c^{3} x - c^{3}}{2 x^{2}}"," ",0,"a**6*c**3*x**4/4 + 2*a**5*c**3*x**3/3 - a**4*c**3*x**2/2 - 4*a**3*c**3*x - a**2*c**3*log(x) + (-4*a*c**3*x - c**3)/(2*x**2)","A",0
1047,1,70,0,0.233028," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**3/x**4,x)","\frac{a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log{\left(x \right)} + \frac{3 a^{2} c^{3} x^{2} - 3 a c^{3} x - c^{3}}{3 x^{3}}"," ",0,"a**6*c**3*x**3/3 + a**5*c**3*x**2 - a**4*c**3*x - 4*a**3*c**3*log(x) + (3*a**2*c**3*x**2 - 3*a*c**3*x - c**3)/(3*x**3)","A",0
1048,1,87,0,0.100414," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**4,x)","- \frac{a^{8} c^{4} x^{9}}{9} - \frac{a^{7} c^{4} x^{8}}{4} + \frac{2 a^{6} c^{4} x^{7}}{7} + a^{5} c^{4} x^{6} - \frac{3 a^{3} c^{4} x^{4}}{2} - \frac{2 a^{2} c^{4} x^{3}}{3} + a c^{4} x^{2} + c^{4} x"," ",0,"-a**8*c**4*x**9/9 - a**7*c**4*x**8/4 + 2*a**6*c**4*x**7/7 + a**5*c**4*x**6 - 3*a**3*c**4*x**4/2 - 2*a**2*c**4*x**3/3 + a*c**4*x**2 + c**4*x","A",0
1049,1,53,0,0.193556," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4/(-a**2*c*x**2+c),x)","- \frac{1}{a^{6} c x - a^{5} c} + \frac{x^{3}}{3 a^{2} c} + \frac{x^{2}}{a^{3} c} + \frac{3 x}{a^{4} c} + \frac{4 \log{\left(a x - 1 \right)}}{a^{5} c}"," ",0,"-1/(a**6*c*x - a**5*c) + x**3/(3*a**2*c) + x**2/(a**3*c) + 3*x/(a**4*c) + 4*log(a*x - 1)/(a**5*c)","A",0
1050,1,44,0,0.174636," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c),x)","- \frac{1}{a^{5} c x - a^{4} c} + \frac{x^{2}}{2 a^{2} c} + \frac{2 x}{a^{3} c} + \frac{3 \log{\left(a x - 1 \right)}}{a^{4} c}"," ",0,"-1/(a**5*c*x - a**4*c) + x**2/(2*a**2*c) + 2*x/(a**3*c) + 3*log(a*x - 1)/(a**4*c)","A",0
1051,1,32,0,0.156837," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c),x)","- \frac{1}{a^{4} c x - a^{3} c} + \frac{x}{a^{2} c} + \frac{2 \log{\left(a x - 1 \right)}}{a^{3} c}"," ",0,"-1/(a**4*c*x - a**3*c) + x/(a**2*c) + 2*log(a*x - 1)/(a**3*c)","A",0
1052,1,24,0,0.131893," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c),x)","- \frac{1}{a^{3} c x - a^{2} c} + \frac{\log{\left(a x - 1 \right)}}{a^{2} c}"," ",0,"-1/(a**3*c*x - a**2*c) + log(a*x - 1)/(a**2*c)","A",0
1053,1,12,0,0.137523," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c),x)","- \frac{1}{a^{2} c x - a c}"," ",0,"-1/(a**2*c*x - a*c)","A",0
1054,1,19,0,0.219910," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c),x)","- \frac{1}{a c x - c} + \frac{\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}}{c}"," ",0,"-1/(a*c*x - c) + (log(x) - log(x - 1/a))/c","A",0
1055,1,31,0,0.238201," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c),x)","\frac{2 a \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right)}{c} + \frac{- 2 a x + 1}{a c x^{2} - c x}"," ",0,"2*a*(log(x) - log(x - 1/a))/c + (-2*a*x + 1)/(a*c*x**2 - c*x)","A",0
1056,1,46,0,0.269144," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3/(-a**2*c*x**2+c),x)","\frac{3 a^{2} \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right)}{c} + \frac{- 6 a^{2} x^{2} + 3 a x + 1}{2 a c x^{3} - 2 c x^{2}}"," ",0,"3*a**2*(log(x) - log(x - 1/a))/c + (-6*a**2*x**2 + 3*a*x + 1)/(2*a*c*x**3 - 2*c*x**2)","A",0
1057,1,54,0,0.290515," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**4/(-a**2*c*x**2+c),x)","\frac{4 a^{3} \left(\log{\left(x \right)} - \log{\left(x - \frac{1}{a} \right)}\right)}{c} + \frac{- 12 a^{3} x^{3} + 6 a^{2} x^{2} + 2 a x + 1}{3 a c x^{4} - 3 c x^{3}}"," ",0,"4*a**3*(log(x) - log(x - 1/a))/c + (-12*a**3*x**3 + 6*a**2*x**2 + 2*a*x + 1)/(3*a*c*x**4 - 3*c*x**3)","A",0
1058,1,71,0,0.459272," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4/(-a**2*c*x**2+c)**2,x)","- \frac{- 7 a x + 6}{4 a^{7} c^{2} x^{2} - 8 a^{6} c^{2} x + 4 a^{5} c^{2}} - \frac{x}{a^{4} c^{2}} - \frac{\frac{17 \log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{5} c^{2}}"," ",0,"-(-7*a*x + 6)/(4*a**7*c**2*x**2 - 8*a**6*c**2*x + 4*a**5*c**2) - x/(a**4*c**2) - (17*log(x - 1/a)/8 - log(x + 1/a)/8)/(a**5*c**2)","A",0
1059,1,63,0,0.346926," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c)**2,x)","- \frac{- 5 a x + 4}{4 a^{6} c^{2} x^{2} - 8 a^{5} c^{2} x + 4 a^{4} c^{2}} - \frac{\frac{7 \log{\left(x - \frac{1}{a} \right)}}{8} + \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{4} c^{2}}"," ",0,"-(-5*a*x + 4)/(4*a**6*c**2*x**2 - 8*a**5*c**2*x + 4*a**4*c**2) - (7*log(x - 1/a)/8 + log(x + 1/a)/8)/(a**4*c**2)","A",0
1060,1,61,0,0.295501," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**2,x)","- \frac{- 3 a x + 2}{4 a^{5} c^{2} x^{2} - 8 a^{4} c^{2} x + 4 a^{3} c^{2}} - \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{3} c^{2}}"," ",0,"-(-3*a*x + 2)/(4*a**5*c**2*x**2 - 8*a**4*c**2*x + 4*a**3*c**2) - (log(x - 1/a)/8 - log(x + 1/a)/8)/(a**3*c**2)","A",0
1061,1,53,0,0.275213," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c)**2,x)","\frac{x}{4 a^{3} c^{2} x^{2} - 8 a^{2} c^{2} x + 4 a c^{2}} - \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{8} + \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{2} c^{2}}"," ",0,"x/(4*a**3*c**2*x**2 - 8*a**2*c**2*x + 4*a*c**2) - (-log(x - 1/a)/8 + log(x + 1/a)/8)/(a**2*c**2)","A",0
1062,1,56,0,0.297397," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**2,x)","- \frac{a x - 2}{4 a^{3} c^{2} x^{2} - 8 a^{2} c^{2} x + 4 a c^{2}} - \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a c^{2}}"," ",0,"-(a*x - 2)/(4*a**3*c**2*x**2 - 8*a**2*c**2*x + 4*a*c**2) - (log(x - 1/a)/8 - log(x + 1/a)/8)/(a*c**2)","A",0
1063,1,58,0,0.483869," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c)**2,x)","- \frac{3 a x - 4}{4 a^{2} c^{2} x^{2} - 8 a c^{2} x + 4 c^{2}} - \frac{- \log{\left(x \right)} + \frac{7 \log{\left(x - \frac{1}{a} \right)}}{8} + \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{c^{2}}"," ",0,"-(3*a*x - 4)/(4*a**2*c**2*x**2 - 8*a*c**2*x + 4*c**2) - (-log(x) + 7*log(x - 1/a)/8 + log(x + 1/a)/8)/c**2","A",0
1064,1,76,0,0.578978," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c)**2,x)","- \frac{9 a^{2} x^{2} - 14 a x + 4}{4 a^{2} c^{2} x^{3} - 8 a c^{2} x^{2} + 4 c^{2} x} - \frac{- 2 a \log{\left(x \right)} + \frac{17 a \log{\left(x - \frac{1}{a} \right)}}{8} - \frac{a \log{\left(x + \frac{1}{a} \right)}}{8}}{c^{2}}"," ",0,"-(9*a**2*x**2 - 14*a*x + 4)/(4*a**2*c**2*x**3 - 8*a*c**2*x**2 + 4*c**2*x) - (-2*a*log(x) + 17*a*log(x - 1/a)/8 - a*log(x + 1/a)/8)/c**2","A",0
1065,1,92,0,0.612694," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3/(-a**2*c*x**2+c)**2,x)","- \frac{15 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 2}{4 a^{2} c^{2} x^{4} - 8 a c^{2} x^{3} + 4 c^{2} x^{2}} - \frac{- 4 a^{2} \log{\left(x \right)} + \frac{31 a^{2} \log{\left(x - \frac{1}{a} \right)}}{8} + \frac{a^{2} \log{\left(x + \frac{1}{a} \right)}}{8}}{c^{2}}"," ",0,"-(15*a**3*x**3 - 22*a**2*x**2 + 4*a*x + 2)/(4*a**2*c**2*x**4 - 8*a*c**2*x**3 + 4*c**2*x**2) - (-4*a**2*log(x) + 31*a**2*log(x - 1/a)/8 + a**2*log(x + 1/a)/8)/c**2","A",0
1066,1,100,0,0.651740," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**4/(-a**2*c*x**2+c)**2,x)","- \frac{75 a^{4} x^{4} - 114 a^{3} x^{3} + 28 a^{2} x^{2} + 4 a x + 4}{12 a^{2} c^{2} x^{5} - 24 a c^{2} x^{4} + 12 c^{2} x^{3}} - \frac{- 6 a^{3} \log{\left(x \right)} + \frac{49 a^{3} \log{\left(x - \frac{1}{a} \right)}}{8} - \frac{a^{3} \log{\left(x + \frac{1}{a} \right)}}{8}}{c^{2}}"," ",0,"-(75*a**4*x**4 - 114*a**3*x**3 + 28*a**2*x**2 + 4*a*x + 4)/(12*a**2*c**2*x**5 - 24*a*c**2*x**4 + 12*c**2*x**3) - (-6*a**3*log(x) + 49*a**3*log(x - 1/a)/8 - a**3*log(x + 1/a)/8)/c**2","A",0
1067,1,92,0,0.532704," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**5/(-a**2*c*x**2+c)**3,x)","\frac{- 33 a^{3} x^{3} + 18 a^{2} x^{2} + 37 a x - 26}{24 a^{10} c^{3} x^{4} - 48 a^{9} c^{3} x^{3} + 48 a^{7} c^{3} x - 24 a^{6} c^{3}} + \frac{\frac{13 \log{\left(x - \frac{1}{a} \right)}}{16} + \frac{3 \log{\left(x + \frac{1}{a} \right)}}{16}}{a^{6} c^{3}}"," ",0,"(-33*a**3*x**3 + 18*a**2*x**2 + 37*a*x - 26)/(24*a**10*c**3*x**4 - 48*a**9*c**3*x**3 + 48*a**7*c**3*x - 24*a**6*c**3) + (13*log(x - 1/a)/16 + 3*log(x + 1/a)/16)/(a**6*c**3)","A",0
1068,1,88,0,0.453583," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**4/(-a**2*c*x**2+c)**3,x)","\frac{- 9 a^{3} x^{3} + 6 a^{2} x^{2} + 5 a x - 4}{12 a^{9} c^{3} x^{4} - 24 a^{8} c^{3} x^{3} + 24 a^{6} c^{3} x - 12 a^{5} c^{3}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a^{5} c^{3}}"," ",0,"(-9*a**3*x**3 + 6*a**2*x**2 + 5*a*x - 4)/(12*a**9*c**3*x**4 - 24*a**8*c**3*x**3 + 24*a**6*c**3*x - 12*a**5*c**3) + (log(x - 1/a)/8 - log(x + 1/a)/8)/(a**5*c**3)","A",0
1069,1,88,0,0.429770," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c)**3,x)","\frac{- 3 a^{3} x^{3} - 6 a^{2} x^{2} + 7 a x - 2}{24 a^{8} c^{3} x^{4} - 48 a^{7} c^{3} x^{3} + 48 a^{5} c^{3} x - 24 a^{4} c^{3}} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{16} + \frac{\log{\left(x + \frac{1}{a} \right)}}{16}}{a^{4} c^{3}}"," ",0,"(-3*a**3*x**3 - 6*a**2*x**2 + 7*a*x - 2)/(24*a**8*c**3*x**4 - 48*a**7*c**3*x**3 + 48*a**5*c**3*x - 24*a**4*c**3) + (-log(x - 1/a)/16 + log(x + 1/a)/16)/(a**4*c**3)","A",0
1070,1,48,0,0.362243," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**3,x)","\frac{- 2 a x + 1}{6 a^{7} c^{3} x^{4} - 12 a^{6} c^{3} x^{3} + 12 a^{4} c^{3} x - 6 a^{3} c^{3}}"," ",0,"(-2*a*x + 1)/(6*a**7*c**3*x**4 - 12*a**6*c**3*x**3 + 12*a**4*c**3*x - 6*a**3*c**3)","A",0
1071,1,87,0,0.448663," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c)**3,x)","\frac{3 a^{3} x^{3} - 6 a^{2} x^{2} + a x - 2}{24 a^{6} c^{3} x^{4} - 48 a^{5} c^{3} x^{3} + 48 a^{3} c^{3} x - 24 a^{2} c^{3}} + \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{16} - \frac{\log{\left(x + \frac{1}{a} \right)}}{16}}{a^{2} c^{3}}"," ",0,"(3*a**3*x**3 - 6*a**2*x**2 + a*x - 2)/(24*a**6*c**3*x**4 - 48*a**5*c**3*x**3 + 48*a**3*c**3*x - 24*a**2*c**3) + (log(x - 1/a)/16 - log(x + 1/a)/16)/(a**2*c**3)","A",0
1072,1,83,0,0.432739," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**3,x)","\frac{- 3 a^{3} x^{3} + 6 a^{2} x^{2} - a x - 4}{12 a^{5} c^{3} x^{4} - 24 a^{4} c^{3} x^{3} + 24 a^{2} c^{3} x - 12 a c^{3}} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{8} + \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a c^{3}}"," ",0,"(-3*a**3*x**3 + 6*a**2*x**2 - a*x - 4)/(12*a**5*c**3*x**4 - 24*a**4*c**3*x**3 + 24*a**2*c**3*x - 12*a*c**3) + (-log(x - 1/a)/8 + log(x + 1/a)/8)/(a*c**3)","A",0
1073,1,87,0,0.697508," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c)**3,x)","\frac{- 15 a^{3} x^{3} + 18 a^{2} x^{2} + 19 a x - 26}{24 a^{4} c^{3} x^{4} - 48 a^{3} c^{3} x^{3} + 48 a c^{3} x - 24 c^{3}} + \frac{\log{\left(x \right)} - \frac{13 \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{3 \log{\left(x + \frac{1}{a} \right)}}{16}}{c^{3}}"," ",0,"(-15*a**3*x**3 + 18*a**2*x**2 + 19*a*x - 26)/(24*a**4*c**3*x**4 - 48*a**3*c**3*x**3 + 48*a*c**3*x - 24*c**3) + (log(x) - 13*log(x - 1/a)/16 - 3*log(x + 1/a)/16)/c**3","A",0
1074,1,104,0,0.744097," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c)**3,x)","\frac{- 15 a^{4} x^{4} + 24 a^{3} x^{3} + 7 a^{2} x^{2} - 23 a x + 6}{6 a^{4} c^{3} x^{5} - 12 a^{3} c^{3} x^{4} + 12 a c^{3} x^{2} - 6 c^{3} x} + \frac{2 a \log{\left(x \right)} - \frac{9 a \log{\left(x - \frac{1}{a} \right)}}{4} + \frac{a \log{\left(x + \frac{1}{a} \right)}}{4}}{c^{3}}"," ",0,"(-15*a**4*x**4 + 24*a**3*x**3 + 7*a**2*x**2 - 23*a*x + 6)/(6*a**4*c**3*x**5 - 12*a**3*c**3*x**4 + 12*a*c**3*x**2 - 6*c**3*x) + (2*a*log(x) - 9*a*log(x - 1/a)/4 + a*log(x + 1/a)/4)/c**3","A",0
1075,1,121,0,0.881844," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3/(-a**2*c*x**2+c)**3,x)","\frac{- 105 a^{5} x^{5} + 150 a^{4} x^{4} + 85 a^{3} x^{3} - 170 a^{2} x^{2} + 24 a x + 12}{24 a^{4} c^{3} x^{6} - 48 a^{3} c^{3} x^{5} + 48 a c^{3} x^{3} - 24 c^{3} x^{2}} + \frac{5 a^{2} \log{\left(x \right)} - \frac{75 a^{2} \log{\left(x - \frac{1}{a} \right)}}{16} - \frac{5 a^{2} \log{\left(x + \frac{1}{a} \right)}}{16}}{c^{3}}"," ",0,"(-105*a**5*x**5 + 150*a**4*x**4 + 85*a**3*x**3 - 170*a**2*x**2 + 24*a*x + 12)/(24*a**4*c**3*x**6 - 48*a**3*c**3*x**5 + 48*a*c**3*x**3 - 24*c**3*x**2) + (5*a**2*log(x) - 75*a**2*log(x - 1/a)/16 - 5*a**2*log(x + 1/a)/16)/c**3","A",0
1076,1,143,0,0.608178," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**4,x)","- \frac{15 a^{5} x^{5} - 30 a^{4} x^{4} - 10 a^{3} x^{3} + 50 a^{2} x^{2} - 17 a x - 16}{64 a^{7} c^{4} x^{6} - 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} + 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} - 128 a^{2} c^{4} x + 64 a c^{4}} - \frac{\frac{15 \log{\left(x - \frac{1}{a} \right)}}{128} - \frac{15 \log{\left(x + \frac{1}{a} \right)}}{128}}{a c^{4}}"," ",0,"-(15*a**5*x**5 - 30*a**4*x**4 - 10*a**3*x**3 + 50*a**2*x**2 - 17*a*x - 16)/(64*a**7*c**4*x**6 - 128*a**6*c**4*x**5 - 64*a**5*c**4*x**4 + 256*a**4*c**4*x**3 - 64*a**3*c**4*x**2 - 128*a**2*c**4*x + 64*a*c**4) - (15*log(x - 1/a)/128 - 15*log(x + 1/a)/128)/(a*c**4)","A",0
1077,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{3} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x^{4} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx"," ",0,"-Integral(x**3*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x**4*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x)","F",0
1078,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x^{3} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx"," ",0,"-Integral(x**2*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x**3*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x)","F",0
1079,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx"," ",0,"-Integral(x*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x**2*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x)","F",0
1080,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x)","F",0
1081,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x,x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{2} - x}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{2} - x}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**2 - x), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**2 - x), x)","F",0
1082,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x**2,x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{3} - x^{2}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{3} - x^{2}}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**3 - x**2), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**3 - x**2), x)","F",0
1083,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x**3,x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{4} - x^{3}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{4} - x^{3}}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**4 - x**3), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**4 - x**3), x)","F",0
1084,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x**4,x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{5} - x^{4}}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**5 - x**4), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**5 - x**4), x)","F",0
1085,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(1/2)/x**5,x)","- \int \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx"," ",0,"-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**6 - x**5), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**6 - x**5), x)","F",0
1086,1,420,0,18.314113," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**(3/2),x)","a^{2} c \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) + 2*a*c*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + c*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True))","A",0
1087,1,515,0,29.497133," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**(3/2),x)","a^{2} c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 2*a*c*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + c*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True))","C",0
1088,1,306,0,16.119483," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**(3/2),x)","a^{2} c \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + 2*a*c*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True))","A",0
1089,1,340,0,18.343545," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2),x)","a^{2} c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + 2*a*c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1090,1,267,0,10.398722," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x,x)","a^{2} c \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + 2*a*c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) + c*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True))","C",0
1091,1,350,0,19.256808," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**2,x)","a^{2} c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) + 2*a*c*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True)) + c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True))","C",0
1092,1,366,0,8.223884," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**3,x)","a^{2} c \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True)) + 2*a*c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True)) + c*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))","C",0
1093,1,359,0,18.177235," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**4,x)","a^{2} c \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True)) + 2*a*c*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))","C",0
1094,1,447,0,14.700497," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**5,x)","a^{2} c \left(\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + 2*a*c*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True)) + c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
1095,1,484,0,24.934403," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**6,x)","a^{2} c \left(\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{2 i a^{4} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True)) + 2*a*c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(c)*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*x**5), True))","C",0
1096,1,636,0,15.988361," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**7,x)","a^{2} c \left(\begin{cases} \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{2 i a^{4} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{a^{6} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{a^{5} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{6} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{i a^{5} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + 2*a*c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(c)*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*x**5), True)) + c*Piecewise((a**6*sqrt(c)*acosh(1/(a*x))/16 - a**5*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*a*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*sqrt(c)*asin(1/(a*x))/16 + I*a**5*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a*sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))","C",0
1097,1,660,0,39.567981," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**8,x)","a^{2} c \left(\begin{cases} \frac{2 i a^{4} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{2 a^{4} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right) + 2 a c \left(\begin{cases} \frac{a^{6} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{16} - \frac{a^{5} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{6} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{16} + \frac{i a^{5} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{8 a^{7} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 a^{5} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{8 i a^{7} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 i a^{5} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(c)*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + a**2*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*x**5), True)) + 2*a*c*Piecewise((a**6*sqrt(c)*acosh(1/(a*x))/16 - a**5*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3*sqrt(c)/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) + 5*a*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*sqrt(c)*asin(1/(a*x))/16 + I*a**5*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a*sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True)) + c*Piecewise((8*a**7*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**5*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (8*I*a**7*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/105 + 4*I*a**5*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))","C",0
1098,1,763,0,84.269216," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**(5/2),x)","- a^{4} c^{2} \left(\begin{cases} \frac{x^{8} \sqrt{- a^{2} c x^{2} + c}}{9} - \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{63 a^{2}} - \frac{2 x^{4} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} c x^{2} + c}}{315 a^{6}} - \frac{16 \sqrt{- a^{2} c x^{2} + c}}{315 a^{8}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{8}}{8} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i \sqrt{c} x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i \sqrt{c} x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 \sqrt{c} x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 \sqrt{c} x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((x**8*sqrt(-a**2*c*x**2 + c)/9 - x**6*sqrt(-a**2*c*x**2 + c)/(63*a**2) - 2*x**4*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*x**2*sqrt(-a**2*c*x**2 + c)/(315*a**6) - 16*sqrt(-a**2*c*x**2 + c)/(315*a**8), Ne(a, 0)), (sqrt(c)*x**8/8, True)) - 2*a**3*c**2*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*asin(a*x)/(128*a**7), True)) + 2*a*c**2*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True))","A",0
1099,1,687,0,21.275867," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2*(-a**2*c*x**2+c)**(5/2),x)","- a^{4} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i \sqrt{c} x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i \sqrt{c} x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 \sqrt{c} x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 \sqrt{c} x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*asin(a*x)/(128*a**7), True)) - 2*a**3*c**2*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) + 2*a*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + c**2*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True))","C",0
1100,1,586,0,70.643202," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x*(-a**2*c*x**2+c)**(5/2),x)","- a^{4} c^{2} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) - 2*a**3*c**2*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 2*a*c**2*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True))","A",0
1101,1,478,0,15.690790," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2),x)","- a^{4} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) - 2*a**3*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + 2*a*c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1102,1,508,0,33.540097," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x,x)","- a^{4} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) - 2*a**3*c**2*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + 2*a*c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) + c**2*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True))","C",0
1103,1,483,0,69.465805," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**2,x)","- a^{4} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a**3*c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + 2*a*c**2*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True)) + c**2*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True))","C",0
1104,1,401,0,118.626824," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**3,x)","- a^{4} c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) - 2*a**3*c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) + 2*a*c**2*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True)) + c**2*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))","C",0
1105,1,478,0,26.912254," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**4,x)","- a^{4} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True)) - 2*a**3*c**2*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True)) + 2*a*c**2*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c**2*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))","C",0
1106,1,575,0,65.524711," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**5,x)","- a^{4} c^{2} \left(\begin{cases} i \sqrt{c} \sqrt{a^{2} x^{2} - 1} - \sqrt{c} \log{\left(a x \right)} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} + i \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\sqrt{c} \sqrt{- a^{2} x^{2} + 1} + \frac{\sqrt{c} \log{\left(a^{2} x^{2} \right)}}{2} - \sqrt{c} \log{\left(\sqrt{- a^{2} x^{2} + 1} + 1 \right)} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{2} \left(\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left(a x \right)} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left(a x \right)} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right) + 2 a c^{2} \left(\begin{cases} \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{3 x^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{a^{4} \sqrt{c} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} - \frac{a^{3} \sqrt{c}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{i a^{4} \sqrt{c} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} + \frac{i a^{3} \sqrt{c}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{3 i a \sqrt{c}}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2*x**2 + 1) + 1), True)) - 2*a**3*c**2*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), True)) + 2*a*c**2*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True)) + c**2*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
1107,1,1091,0,27.967958," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(7/2),x)","a^{6} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i \sqrt{c} x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i \sqrt{c} x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 \sqrt{c} x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 \sqrt{c} x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) + 2 a^{5} c^{3} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right) - a^{4} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) - 4 a^{3} c^{3} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) - a^{2} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 2 a c^{3} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**6*c**3*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*asin(a*x)/(128*a**7), True)) + 2*a**5*c**3*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) - a**4*c**3*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) - 4*a**3*c**3*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) - a**2*c**3*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) + 2*a*c**3*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c**3*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1108,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{3}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{4}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**3/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**4/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)","F",0
1109,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{2}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{3}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**2/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**3/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)","F",0
1110,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{2}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**2/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)","F",0
1111,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)","F",0
1112,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x^{2} \sqrt{- a^{2} c x^{2} + c} - x \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x^{2} \sqrt{- a^{2} c x^{2} + c} - x \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a*x**2*sqrt(-a**2*c*x**2 + c) - x*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a*x**2*sqrt(-a**2*c*x**2 + c) - x*sqrt(-a**2*c*x**2 + c)), x)","F",0
1113,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x^{3} \sqrt{- a^{2} c x^{2} + c} - x^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x^{3} \sqrt{- a^{2} c x^{2} + c} - x^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a*x**3*sqrt(-a**2*c*x**2 + c) - x**2*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a*x**3*sqrt(-a**2*c*x**2 + c) - x**2*sqrt(-a**2*c*x**2 + c)), x)","F",0
1114,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x^{4} \sqrt{- a^{2} c x^{2} + c} - x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x^{4} \sqrt{- a^{2} c x^{2} + c} - x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a*x**4*sqrt(-a**2*c*x**2 + c) - x**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a*x**4*sqrt(-a**2*c*x**2 + c) - x**3*sqrt(-a**2*c*x**2 + c)), x)","F",0
1115,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**4/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x^{5} \sqrt{- a^{2} c x^{2} + c} - x^{4} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a x^{5} \sqrt{- a^{2} c x^{2} + c} - x^{4} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a*x**5*sqrt(-a**2*c*x**2 + c) - x**4*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a*x**5*sqrt(-a**2*c*x**2 + c) - x**4*sqrt(-a**2*c*x**2 + c)), x)","F",0
1116,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**3/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{x^{3}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{4}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**3/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**4/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1117,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{x^{2}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{3}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**2/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**3/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1118,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{x}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x^{2}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x**2/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1119,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{a x}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1120,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{a x}{- a^{3} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a c x^{2} \sqrt{- a^{2} c x^{2} + c} - c x \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a c x^{2} \sqrt{- a^{2} c x^{2} + c} - c x \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*sqrt(-a**2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*c*x**2 + c) - c*x*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*sqrt(-a**2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*c*x**2 + c) - c*x*sqrt(-a**2*c*x**2 + c)), x)","F",0
1121,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**2/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{a x}{- a^{3} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a c x^{3} \sqrt{- a^{2} c x^{2} + c} - c x^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt{- a^{2} c x^{2} + c} + a c x^{3} \sqrt{- a^{2} c x^{2} + c} - c x^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(-a**3*c*x**5*sqrt(-a**2*c*x**2 + c) + a**2*c*x**4*sqrt(-a**2*c*x**2 + c) + a*c*x**3*sqrt(-a**2*c*x**2 + c) - c*x**2*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**5*sqrt(-a**2*c*x**2 + c) + a**2*c*x**4*sqrt(-a**2*c*x**2 + c) + a*c*x**3*sqrt(-a**2*c*x**2 + c) - c*x**2*sqrt(-a**2*c*x**2 + c)), x)","F",0
1122,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/x**3/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{a x}{- a^{3} c x^{6} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a c x^{4} \sqrt{- a^{2} c x^{2} + c} - c x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{3} c x^{6} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{5} \sqrt{- a^{2} c x^{2} + c} + a c x^{4} \sqrt{- a^{2} c x^{2} + c} - c x^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(-a**3*c*x**6*sqrt(-a**2*c*x**2 + c) + a**2*c*x**5*sqrt(-a**2*c*x**2 + c) + a*c*x**4*sqrt(-a**2*c*x**2 + c) - c*x**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**6*sqrt(-a**2*c*x**2 + c) + a**2*c*x**5*sqrt(-a**2*c*x**2 + c) + a*c*x**4*sqrt(-a**2*c*x**2 + c) - c*x**3*sqrt(-a**2*c*x**2 + c)), x)","F",0
1123,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**(5/2),x)","- \int \frac{a x}{a^{5} c^{2} x^{5} \sqrt{- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt{- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt{- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{2} x \sqrt{- a^{2} c x^{2} + c} - c^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{a^{5} c^{2} x^{5} \sqrt{- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt{- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt{- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{2} x \sqrt{- a^{2} c x^{2} + c} - c^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) - a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) + 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) - c**2*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) - a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) + 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) - c**2*sqrt(-a**2*c*x**2 + c)), x)","F",0
1124,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**(7/2),x)","- \int \frac{a x}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} - c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{1}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} + a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} - 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} + 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} - c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) + a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) - 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) + 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) + a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) - 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) + 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) - c**3*sqrt(-a**2*c*x**2 + c)), x)","F",0
1125,1,3009,0,3.472007," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**3,x)","\begin{cases} a^{6} c^{3} \log{\left(x \right)} - \frac{2 a^{5} c^{3}}{x} + \frac{a^{4} c^{3}}{2 x^{2}} + \frac{4 a^{3} c^{3}}{3 x^{3}} + \frac{a^{2} c^{3}}{4 x^{4}} - \frac{2 a c^{3}}{5 x^{5}} - \frac{c^{3}}{6 x^{6}} & \text{for}\: m = -7 \\a^{6} c^{3} x + 2 a^{5} c^{3} \log{\left(x \right)} + \frac{a^{4} c^{3}}{x} + \frac{2 a^{3} c^{3}}{x^{2}} + \frac{a^{2} c^{3}}{3 x^{3}} - \frac{a c^{3}}{2 x^{4}} - \frac{c^{3}}{5 x^{5}} & \text{for}\: m = -6 \\\frac{a^{6} c^{3} x^{2}}{2} + 2 a^{5} c^{3} x - a^{4} c^{3} \log{\left(x \right)} + \frac{4 a^{3} c^{3}}{x} + \frac{a^{2} c^{3}}{2 x^{2}} - \frac{2 a c^{3}}{3 x^{3}} - \frac{c^{3}}{4 x^{4}} & \text{for}\: m = -5 \\\frac{a^{6} c^{3} x^{3}}{3} + a^{5} c^{3} x^{2} - a^{4} c^{3} x - 4 a^{3} c^{3} \log{\left(x \right)} + \frac{a^{2} c^{3}}{x} - \frac{a c^{3}}{x^{2}} - \frac{c^{3}}{3 x^{3}} & \text{for}\: m = -4 \\\frac{a^{6} c^{3} x^{4}}{4} + \frac{2 a^{5} c^{3} x^{3}}{3} - \frac{a^{4} c^{3} x^{2}}{2} - 4 a^{3} c^{3} x - a^{2} c^{3} \log{\left(x \right)} - \frac{2 a c^{3}}{x} - \frac{c^{3}}{2 x^{2}} & \text{for}\: m = -3 \\\frac{a^{6} c^{3} x^{5}}{5} + \frac{a^{5} c^{3} x^{4}}{2} - \frac{a^{4} c^{3} x^{3}}{3} - 2 a^{3} c^{3} x^{2} - a^{2} c^{3} x + 2 a c^{3} \log{\left(x \right)} - \frac{c^{3}}{x} & \text{for}\: m = -2 \\\frac{a^{6} c^{3} x^{6}}{6} + \frac{2 a^{5} c^{3} x^{5}}{5} - \frac{a^{4} c^{3} x^{4}}{4} - \frac{4 a^{3} c^{3} x^{3}}{3} - \frac{a^{2} c^{3} x^{2}}{2} + 2 a c^{3} x + c^{3} \log{\left(x \right)} & \text{for}\: m = -1 \\\frac{a^{6} c^{3} m^{6} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{21 a^{6} c^{3} m^{5} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{175 a^{6} c^{3} m^{4} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{735 a^{6} c^{3} m^{3} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{1624 a^{6} c^{3} m^{2} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{1764 a^{6} c^{3} m x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{720 a^{6} c^{3} x^{7} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{2 a^{5} c^{3} m^{6} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{44 a^{5} c^{3} m^{5} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{380 a^{5} c^{3} m^{4} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{1640 a^{5} c^{3} m^{3} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{3698 a^{5} c^{3} m^{2} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{4076 a^{5} c^{3} m x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{1680 a^{5} c^{3} x^{6} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{a^{4} c^{3} m^{6} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{23 a^{4} c^{3} m^{5} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{207 a^{4} c^{3} m^{4} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{925 a^{4} c^{3} m^{3} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{2144 a^{4} c^{3} m^{2} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{2412 a^{4} c^{3} m x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{1008 a^{4} c^{3} x^{5} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{4 a^{3} c^{3} m^{6} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{96 a^{3} c^{3} m^{5} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{904 a^{3} c^{3} m^{4} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{4224 a^{3} c^{3} m^{3} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{10180 a^{3} c^{3} m^{2} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{11808 a^{3} c^{3} m x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{5040 a^{3} c^{3} x^{4} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{a^{2} c^{3} m^{6} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{25 a^{2} c^{3} m^{5} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{247 a^{2} c^{3} m^{4} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{1219 a^{2} c^{3} m^{3} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{3112 a^{2} c^{3} m^{2} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{3796 a^{2} c^{3} m x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} - \frac{1680 a^{2} c^{3} x^{3} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{2 a c^{3} m^{6} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{52 a c^{3} m^{5} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{540 a c^{3} m^{4} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{2840 a c^{3} m^{3} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{7858 a c^{3} m^{2} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{10548 a c^{3} m x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{5040 a c^{3} x^{2} x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{c^{3} m^{6} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{27 c^{3} m^{5} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{295 c^{3} m^{4} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{1665 c^{3} m^{3} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{5104 c^{3} m^{2} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{8028 c^{3} m x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} + \frac{5040 c^{3} x x^{m}}{m^{7} + 28 m^{6} + 322 m^{5} + 1960 m^{4} + 6769 m^{3} + 13132 m^{2} + 13068 m + 5040} & \text{otherwise} \end{cases}"," ",0,"Piecewise((a**6*c**3*log(x) - 2*a**5*c**3/x + a**4*c**3/(2*x**2) + 4*a**3*c**3/(3*x**3) + a**2*c**3/(4*x**4) - 2*a*c**3/(5*x**5) - c**3/(6*x**6), Eq(m, -7)), (a**6*c**3*x + 2*a**5*c**3*log(x) + a**4*c**3/x + 2*a**3*c**3/x**2 + a**2*c**3/(3*x**3) - a*c**3/(2*x**4) - c**3/(5*x**5), Eq(m, -6)), (a**6*c**3*x**2/2 + 2*a**5*c**3*x - a**4*c**3*log(x) + 4*a**3*c**3/x + a**2*c**3/(2*x**2) - 2*a*c**3/(3*x**3) - c**3/(4*x**4), Eq(m, -5)), (a**6*c**3*x**3/3 + a**5*c**3*x**2 - a**4*c**3*x - 4*a**3*c**3*log(x) + a**2*c**3/x - a*c**3/x**2 - c**3/(3*x**3), Eq(m, -4)), (a**6*c**3*x**4/4 + 2*a**5*c**3*x**3/3 - a**4*c**3*x**2/2 - 4*a**3*c**3*x - a**2*c**3*log(x) - 2*a*c**3/x - c**3/(2*x**2), Eq(m, -3)), (a**6*c**3*x**5/5 + a**5*c**3*x**4/2 - a**4*c**3*x**3/3 - 2*a**3*c**3*x**2 - a**2*c**3*x + 2*a*c**3*log(x) - c**3/x, Eq(m, -2)), (a**6*c**3*x**6/6 + 2*a**5*c**3*x**5/5 - a**4*c**3*x**4/4 - 4*a**3*c**3*x**3/3 - a**2*c**3*x**2/2 + 2*a*c**3*x + c**3*log(x), Eq(m, -1)), (a**6*c**3*m**6*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 21*a**6*c**3*m**5*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 175*a**6*c**3*m**4*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 735*a**6*c**3*m**3*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1624*a**6*c**3*m**2*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1764*a**6*c**3*m*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 720*a**6*c**3*x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2*a**5*c**3*m**6*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 44*a**5*c**3*m**5*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 380*a**5*c**3*m**4*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1640*a**5*c**3*m**3*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 3698*a**5*c**3*m**2*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 4076*a**5*c**3*m*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1680*a**5*c**3*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - a**4*c**3*m**6*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 23*a**4*c**3*m**5*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 207*a**4*c**3*m**4*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 925*a**4*c**3*m**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 2144*a**4*c**3*m**2*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 2412*a**4*c**3*m*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 1008*a**4*c**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 4*a**3*c**3*m**6*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 96*a**3*c**3*m**5*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 904*a**3*c**3*m**4*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 4224*a**3*c**3*m**3*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 10180*a**3*c**3*m**2*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 11808*a**3*c**3*m*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 5040*a**3*c**3*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - a**2*c**3*m**6*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 25*a**2*c**3*m**5*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 247*a**2*c**3*m**4*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 1219*a**2*c**3*m**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 3112*a**2*c**3*m**2*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 3796*a**2*c**3*m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 1680*a**2*c**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2*a*c**3*m**6*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 52*a*c**3*m**5*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 540*a*c**3*m**4*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2840*a*c**3*m**3*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 7858*a*c**3*m**2*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 10548*a*c**3*m*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 5040*a*c**3*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + c**3*m**6*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 27*c**3*m**5*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 295*c**3*m**4*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1665*c**3*m**3*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 5104*c**3*m**2*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 8028*c**3*m*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 5040*c**3*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040), True))","A",0
1126,1,706,0,1.850736," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**2,x)","\begin{cases} - a^{4} c^{2} \log{\left(x \right)} + \frac{2 a^{3} c^{2}}{x} - \frac{2 a c^{2}}{3 x^{3}} - \frac{c^{2}}{4 x^{4}} & \text{for}\: m = -5 \\- a^{4} c^{2} x - 2 a^{3} c^{2} \log{\left(x \right)} - \frac{a c^{2}}{x^{2}} - \frac{c^{2}}{3 x^{3}} & \text{for}\: m = -4 \\- \frac{a^{4} c^{2} x^{3}}{3} - a^{3} c^{2} x^{2} + 2 a c^{2} \log{\left(x \right)} - \frac{c^{2}}{x} & \text{for}\: m = -2 \\- \frac{a^{4} c^{2} x^{4}}{4} - \frac{2 a^{3} c^{2} x^{3}}{3} + 2 a c^{2} x + c^{2} \log{\left(x \right)} & \text{for}\: m = -1 \\- \frac{a^{4} c^{2} m^{3} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{7 a^{4} c^{2} m^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{14 a^{4} c^{2} m x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{8 a^{4} c^{2} x^{5} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{2 a^{3} c^{2} m^{3} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{16 a^{3} c^{2} m^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{34 a^{3} c^{2} m x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac{20 a^{3} c^{2} x^{4} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{2 a c^{2} m^{3} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{20 a c^{2} m^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{58 a c^{2} m x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{40 a c^{2} x^{2} x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{c^{2} m^{3} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{11 c^{2} m^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{38 c^{2} m x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac{40 c^{2} x x^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text{otherwise} \end{cases}"," ",0,"Piecewise((-a**4*c**2*log(x) + 2*a**3*c**2/x - 2*a*c**2/(3*x**3) - c**2/(4*x**4), Eq(m, -5)), (-a**4*c**2*x - 2*a**3*c**2*log(x) - a*c**2/x**2 - c**2/(3*x**3), Eq(m, -4)), (-a**4*c**2*x**3/3 - a**3*c**2*x**2 + 2*a*c**2*log(x) - c**2/x, Eq(m, -2)), (-a**4*c**2*x**4/4 - 2*a**3*c**2*x**3/3 + 2*a*c**2*x + c**2*log(x), Eq(m, -1)), (-a**4*c**2*m**3*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 7*a**4*c**2*m**2*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 14*a**4*c**2*m*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 8*a**4*c**2*x**5*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 2*a**3*c**2*m**3*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 16*a**3*c**2*m**2*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 34*a**3*c**2*m*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 20*a**3*c**2*x**4*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 2*a*c**2*m**3*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 20*a*c**2*m**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 58*a*c**2*m*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*a*c**2*x**2*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + c**2*m**3*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 11*c**2*m**2*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 38*c**2*m*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*c**2*x*x**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40), True))","A",0
1127,1,299,0,1.104181," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c),x)","\begin{cases} a^{2} c \log{\left(x \right)} - \frac{2 a c}{x} - \frac{c}{2 x^{2}} & \text{for}\: m = -3 \\a^{2} c x + 2 a c \log{\left(x \right)} - \frac{c}{x} & \text{for}\: m = -2 \\\frac{a^{2} c x^{2}}{2} + 2 a c x + c \log{\left(x \right)} & \text{for}\: m = -1 \\\frac{a^{2} c m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 a^{2} c m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 a^{2} c x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 a c m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{8 a c m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a c x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{c m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 c m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 c x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases}"," ",0,"Piecewise((a**2*c*log(x) - 2*a*c/x - c/(2*x**2), Eq(m, -3)), (a**2*c*x + 2*a*c*log(x) - c/x, Eq(m, -2)), (a**2*c*x**2/2 + 2*a*c*x + c*log(x), Eq(m, -1)), (a**2*c*m**2*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 3*a**2*c*m*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*a**2*c*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*a*c*m**2*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 8*a*c*m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 6*a*c*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + c*m**2*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 5*c*m*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 6*c*x*x**m/(m**3 + 6*m**2 + 11*m + 6), True))","A",0
1128,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{m}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c}"," ",0,"Integral(x**m/(a**2*x**2 - 2*a*x + 1), x)/c","F",0
1129,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m/(-a**2*c*x**2+c)**2,x)","- \frac{\int \frac{x^{m}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{2}}"," ",0,"-Integral(x**m/(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1), x)/c**2","F",0
1130,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{x^{m}}{a^{6} x^{6} - 2 a^{5} x^{5} - a^{4} x^{4} + 4 a^{3} x^{3} - a^{2} x^{2} - 2 a x + 1}\, dx}{c^{3}}"," ",0,"Integral(x**m/(a**6*x**6 - 2*a**5*x**5 - a**4*x**4 + 4*a**3*x**3 - a**2*x**2 - 2*a*x + 1), x)/c**3","F",0
1131,1,226,0,23.158077," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**(5/2),x)","- \frac{a^{4} c^{\frac{5}{2}} x^{5} x^{m} \Gamma\left(\frac{m}{2} + \frac{5}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{5}{2} \\ \frac{m}{2} + \frac{7}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{7}{2}\right)} - \frac{a^{3} c^{\frac{5}{2}} x^{4} x^{m} \Gamma\left(\frac{m}{2} + 2\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{\Gamma\left(\frac{m}{2} + 3\right)} + \frac{a c^{\frac{5}{2}} x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{\Gamma\left(\frac{m}{2} + 2\right)} + \frac{c^{\frac{5}{2}} x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"-a**4*c**(5/2)*x**5*x**m*gamma(m/2 + 5/2)*hyper((-1/2, m/2 + 5/2), (m/2 + 7/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 7/2)) - a**3*c**(5/2)*x**4*x**m*gamma(m/2 + 2)*hyper((-1/2, m/2 + 2), (m/2 + 3,), a**2*x**2*exp_polar(2*I*pi))/gamma(m/2 + 3) + a*c**(5/2)*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/gamma(m/2 + 2) + c**(5/2)*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3/2))","C",0
1132,1,172,0,10.261580," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**(3/2),x)","\frac{a^{2} c^{\frac{3}{2}} x^{3} x^{m} \Gamma\left(\frac{m}{2} + \frac{3}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{5}{2}\right)} + \frac{a c^{\frac{3}{2}} x^{2} x^{m} \Gamma\left(\frac{m}{2} + 1\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{\Gamma\left(\frac{m}{2} + 2\right)} + \frac{c^{\frac{3}{2}} x x^{m} \Gamma\left(\frac{m}{2} + \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(\frac{m}{2} + \frac{3}{2}\right)}"," ",0,"a**2*c**(3/2)*x**3*x**m*gamma(m/2 + 3/2)*hyper((-1/2, m/2 + 3/2), (m/2 + 5/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 5/2)) + a*c**(3/2)*x**2*x**m*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/gamma(m/2 + 2) + c**(3/2)*x*x**m*gamma(m/2 + 1/2)*hyper((-1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3/2))","C",0
1133,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{m} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x x^{m} \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx"," ",0,"-Integral(x**m*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x*x**m*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x)","F",0
1134,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{x^{m}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x x^{m}}{a x \sqrt{- a^{2} c x^{2} + c} - \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**m/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x*x**m/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)","F",0
1135,0,0,0,0.000000," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**m/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{x^{m}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \frac{a x x^{m}}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} - c \sqrt{- a^{2} c x^{2} + c}}\, dx"," ",0,"-Integral(x**m/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x) - Integral(a*x*x**m/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) + a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) - c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1136,1,653,0,12.137700," ","integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**p,x)","- a \left(\begin{cases} \frac{0^{p} x}{a} - \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} + \frac{0^{p} \log{\left(-1 + \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{acoth}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(- p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, - p - \frac{1}{2} \\ \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{0^{p} x}{a} - \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} + \frac{0^{p} \log{\left(1 - \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{atanh}{\left(\frac{1}{a x} \right)}}{a^{2}} + \frac{c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(- p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, - p - \frac{1}{2} \\ \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{0^{p} \log{\left(a^{2} x^{2} - 1 \right)}}{2 a} - \frac{0^{p} \operatorname{acoth}{\left(a x \right)}}{a} + \frac{a c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(\frac{1}{2} - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a^{2} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{0^{p} \log{\left(- a^{2} x^{2} + 1 \right)}}{2 a} - \frac{0^{p} \operatorname{atanh}{\left(a x \right)}}{a} + \frac{a c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(\frac{1}{2} - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a^{2} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}"," ",0,"-a*Piecewise((0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 + c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*a*gamma(1/2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 + c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*a*gamma(1/2 - p)*gamma(p + 1)), True)) - Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) - 0**p*acoth(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) - 0**p*atanh(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), True))","C",0
1137,1,483,0,21.363668," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a**2*c*x**2+c),x)","a^{3} c \left(\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{16 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{16 a^{7}} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-I*x**7/(6*sqrt(a**2*x**2 - 1)) - I*x**5/(24*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(48*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(16*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(16*a**7), Abs(a**2*x**2) > 1), (x**7/(6*sqrt(-a**2*x**2 + 1)) + x**5/(24*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(48*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(16*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(16*a**7), True)) + 3*a**2*c*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 3*a*c*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + c*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True))","A",0
1138,1,371,0,18.454406," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(-a**2*c*x**2+c),x)","a^{3} c \left(\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-x**4*sqrt(-a**2*x**2 + 1)/(5*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(15*a**4) - 8*sqrt(-a**2*x**2 + 1)/(15*a**6), Ne(a, 0)), (x**6/6, True)) + 3*a**2*c*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + 3*a*c*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True))","C",0
1139,1,326,0,16.328359," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(-a**2*c*x**2+c),x)","a^{3} c \left(\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left(a x \right)}}{8 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left(a x \right)}}{8 a^{5}} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-I*x**5/(4*sqrt(a**2*x**2 - 1)) - I*x**3/(8*a**2*sqrt(a**2*x**2 - 1)) + 3*I*x/(8*a**4*sqrt(a**2*x**2 - 1)) - 3*I*acosh(a*x)/(8*a**5), Abs(a**2*x**2) > 1), (x**5/(4*sqrt(-a**2*x**2 + 1)) + x**3/(8*a**2*sqrt(-a**2*x**2 + 1)) - 3*x/(8*a**4*sqrt(-a**2*x**2 + 1)) + 3*asin(a*x)/(8*a**5), True)) + 3*a**2*c*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + 3*a*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True))","A",0
1140,1,218,0,14.313659," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c),x)","a^{3} c \left(\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right)"," ",0,"a**3*c*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a, 0)), (x**4/4, True)) + 3*a**2*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 3*a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0))","A",0
1141,1,197,0,13.684170," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x,x)","a^{3} c \left(\begin{cases} - \frac{i x \sqrt{a^{2} x^{2} - 1}}{2 a^{2}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x^{3}}{2 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{2 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a^{3}} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-I*x*sqrt(a**2*x**2 - 1)/(2*a**2) - I*acosh(a*x)/(2*a**3), Abs(a**2*x**2) > 1), (x**3/(2*sqrt(-a**2*x**2 + 1)) - x/(2*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(2*a**3), True)) + 3*a**2*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + 3*a*c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))","C",0
1142,1,150,0,13.977473," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**2,x)","a^{3} c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + 3 a c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + 3*a**2*c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 3*a*c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))","C",0
1143,1,223,0,10.010726," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**3,x)","a^{3} c \left(\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left(x \sqrt{a^{2}} \right)} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left(x \sqrt{- a^{2}} \right)} & \text{for}\: a^{2} < 0 \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((sqrt(a**(-2))*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 3*a**2*c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 3*a*c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + c*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True))","C",0
1144,1,267,0,28.083619," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**4,x)","a^{3} c \left(\begin{cases} - \operatorname{acosh}{\left(\frac{1}{a x} \right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\i \operatorname{asin}{\left(\frac{1}{a x} \right)} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + 3*a**2*c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 3*a*c*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + c*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True))","C",0
1145,1,411,0,12.003037," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**5,x)","a^{3} c \left(\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True)) + 3*a**2*c*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + 3*a*c*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + c*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))","C",0
1146,1,518,0,49.285172," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)/x**6,x)","a^{3} c \left(\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + 3 a^{2} c \left(\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right) + 3 a c \left(\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left(\frac{1}{a x} \right)}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left(\frac{1}{a x} \right)}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} - \frac{8 a^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 a^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\- \frac{8 i a^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15} - \frac{4 i a^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{15 x^{2}} - \frac{i a \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{5 x^{4}} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*x**3*sqrt(1 - 1/(a**2*x**2))), True)) + 3*a**2*c*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3), True)) + 3*a*c*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + c*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))","C",0
1147,1,340,0,20.165348," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**2,x)","a^{3} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 3 a^{2} c^{2} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 3 a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 3*a**2*c**2*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) + 3*a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**2*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1148,1,632,0,25.474987," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**3,x)","- a^{5} c^{3} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{3} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) - 2 a^{3} c^{3} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{3} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 3 a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 3*a**4*c**3*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) - 2*a**3*c**3*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**2*c**3*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) + 3*a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1149,1,996,0,46.239834," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**4,x)","a^{7} c^{4} \left(\begin{cases} \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{9} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{63 a^{2}} - \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{315 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{315 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) + 3 a^{6} c^{4} \left(\begin{cases} \frac{i a^{2} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) + a^{5} c^{4} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 5 a^{4} c^{4} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) - 5 a^{3} c^{4} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + a^{2} c^{4} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) + 3 a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**7*c**4*Piecewise((x**8*sqrt(-a**2*x**2 + 1)/9 - x**6*sqrt(-a**2*x**2 + 1)/(63*a**2) - 2*x**4*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(315*a**6) - 16*sqrt(-a**2*x**2 + 1)/(315*a**8), Ne(a, 0)), (x**8/8, True)) + 3*a**6*c**4*Piecewise((I*a**2*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*x**7/(48*sqrt(a**2*x**2 - 1)) - I*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*x**7/(48*sqrt(-a**2*x**2 + 1)) + x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(128*a**7), True)) + a**5*c**4*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 5*a**4*c**4*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) - 5*a**3*c**4*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + a**2*c**4*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) + 3*a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**4*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1150,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2/(-a**2*c*x**2+c),x)","\frac{\int \frac{x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
1151,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x/(-a**2*c*x**2+c),x)","\frac{\int \frac{x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
1152,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c),x)","\frac{\int \frac{3 a x}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"(Integral(3*a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c","F",0
1153,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{3 a x}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"(Integral(3*a*x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2","F",0
1154,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{3 a x}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"(Integral(3*a*x/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3","F",0
1155,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**4,x)","\frac{\int \frac{3 a x}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{3 a^{2} x^{2}}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a^{3} x^{3}}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{10} x^{10} \sqrt{- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"(Integral(3*a*x/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4","F",0
1156,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{3} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1157,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{2} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1158,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1159,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1160,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1161,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1162,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x**3,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1163,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x**4,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{x^{4} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x**4*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1164,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(1/2)/x**5,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{x^{5} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(x**5*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1165,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1166,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(5/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(5/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1167,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(7/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1168,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(9/2),x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{9}{2}} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(9/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1169,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
1170,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1171,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
1172,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(7/2),x)","\int \frac{\left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}}}\, dx"," ",0,"Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(7/2)), x)","F",0
1173,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**m*(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{m} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**m*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1174,-1,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**m*(-a**2*c*x**2+c)**p,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1175,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(-a**2*c*x**2+c)**p,x)","\int \frac{x^{3} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*(-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1176,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2*(-a**2*c*x**2+c)**p,x)","\int \frac{x^{2} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*(-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1177,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x*(-a**2*c*x**2+c)**p,x)","\int \frac{x \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*(-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1178,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**p,x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1179,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**p/x,x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1180,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**p/x**2,x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1181,0,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**p/x**3,x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{3}}{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1182,1,109,0,0.120951," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**5,x)","- \frac{a^{10} c^{5} x^{11}}{11} - \frac{2 a^{9} c^{5} x^{10}}{5} - \frac{a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac{14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x"," ",0,"-a**10*c**5*x**11/11 - 2*a**9*c**5*x**10/5 - a**8*c**5*x**9/3 + a**7*c**5*x**8 + 2*a**6*c**5*x**7 - 14*a**4*c**5*x**5/5 - 2*a**3*c**5*x**4 + a**2*c**5*x**3 + 2*a*c**5*x**2 + c**5*x","B",0
1183,1,100,0,0.111770," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**4,x)","\frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{2} + \frac{4 a^{6} c^{4} x^{7}}{7} - \frac{2 a^{5} c^{4} x^{6}}{3} - 2 a^{4} c^{4} x^{5} - a^{3} c^{4} x^{4} + \frac{4 a^{2} c^{4} x^{3}}{3} + 2 a c^{4} x^{2} + c^{4} x"," ",0,"a**8*c**4*x**9/9 + a**7*c**4*x**8/2 + 4*a**6*c**4*x**7/7 - 2*a**5*c**4*x**6/3 - 2*a**4*c**4*x**5 - a**3*c**4*x**4 + 4*a**2*c**4*x**3/3 + 2*a*c**4*x**2 + c**4*x","B",0
1184,1,63,0,0.114203," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**3,x)","- \frac{a^{6} c^{3} x^{7}}{7} - \frac{2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac{5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x"," ",0,"-a**6*c**3*x**7/7 - 2*a**5*c**3*x**6/3 - a**4*c**3*x**5 + 5*a**2*c**3*x**3/3 + 2*a*c**3*x**2 + c**3*x","B",0
1185,1,48,0,0.114108," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**2,x)","\frac{a^{4} c^{2} x^{5}}{5} + a^{3} c^{2} x^{4} + 2 a^{2} c^{2} x^{3} + 2 a c^{2} x^{2} + c^{2} x"," ",0,"a**4*c**2*x**5/5 + a**3*c**2*x**4 + 2*a**2*c**2*x**3 + 2*a*c**2*x**2 + c**2*x","B",0
1186,1,36,0,0.207926," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c),x)","- \frac{a^{2} c x^{3}}{3} - 2 a c x^{2} - 7 c x - \frac{8 c \log{\left(a x - 1 \right)}}{a}"," ",0,"-a**2*c*x**3/3 - 2*a*c*x**2 - 7*c*x - 8*c*log(a*x - 1)/a","A",0
1187,1,17,0,0.235560," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a**2*c*x**2+c),x)","\frac{x}{a^{2} c x^{2} - 2 a c x + c}"," ",0,"x/(a**2*c*x**2 - 2*a*c*x + c)","B",0
1188,1,42,0,0.286306," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**2,x)","- \frac{1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}}"," ",0,"-1/(3*a**4*c**2*x**3 - 9*a**3*c**2*x**2 + 9*a**2*c**2*x - 3*a*c**2)","B",0
1189,1,99,0,0.599823," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**3,x)","- \frac{3 a^{3} x^{3} - 12 a^{2} x^{2} + 19 a x - 16}{48 a^{5} c^{3} x^{4} - 192 a^{4} c^{3} x^{3} + 288 a^{3} c^{3} x^{2} - 192 a^{2} c^{3} x + 48 a c^{3}} - \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{32} - \frac{\log{\left(x + \frac{1}{a} \right)}}{32}}{a c^{3}}"," ",0,"-(3*a**3*x**3 - 12*a**2*x**2 + 19*a*x - 16)/(48*a**5*c**3*x**4 - 192*a**4*c**3*x**3 + 288*a**3*c**3*x**2 - 192*a**2*c**3*x + 48*a*c**3) - (log(x - 1/a)/32 - log(x + 1/a)/32)/(a*c**3)","A",0
1190,1,129,0,0.606951," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**4,x)","\frac{- 15 a^{5} x^{5} + 60 a^{4} x^{4} - 80 a^{3} x^{3} + 20 a^{2} x^{2} + 47 a x - 48}{160 a^{7} c^{4} x^{6} - 640 a^{6} c^{4} x^{5} + 800 a^{5} c^{4} x^{4} - 800 a^{3} c^{4} x^{2} + 640 a^{2} c^{4} x - 160 a c^{4}} + \frac{- \frac{3 \log{\left(x - \frac{1}{a} \right)}}{64} + \frac{3 \log{\left(x + \frac{1}{a} \right)}}{64}}{a c^{4}}"," ",0,"(-15*a**5*x**5 + 60*a**4*x**4 - 80*a**3*x**3 + 20*a**2*x**2 + 47*a*x - 48)/(160*a**7*c**4*x**6 - 640*a**6*c**4*x**5 + 800*a**5*c**4*x**4 - 800*a**3*c**4*x**2 + 640*a**2*c**4*x - 160*a*c**4) + (-3*log(x - 1/a)/64 + 3*log(x + 1/a)/64)/(a*c**4)","A",0
1191,0,0,0,0.000000," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**p,x)","\int \frac{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} \left(a x + 1\right)^{2}}{\left(a x - 1\right)^{2}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**2/(a*x - 1)**2, x)","F",0
1192,1,996,0,20.604133," ","integrate((-a**2*c*x**2+c)**4/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","a^{7} c^{4} \left(\begin{cases} \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{9} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{63 a^{2}} - \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{315 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{315 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) - a^{6} c^{4} \left(\begin{cases} \frac{i a^{2} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) - 3 a^{5} c^{4} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + 3 a^{4} c^{4} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 3 a^{3} c^{4} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - 3 a^{2} c^{4} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**7*c**4*Piecewise((x**8*sqrt(-a**2*x**2 + 1)/9 - x**6*sqrt(-a**2*x**2 + 1)/(63*a**2) - 2*x**4*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(315*a**6) - 16*sqrt(-a**2*x**2 + 1)/(315*a**8), Ne(a, 0)), (x**8/8, True)) - a**6*c**4*Piecewise((I*a**2*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*x**7/(48*sqrt(a**2*x**2 - 1)) - I*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*x**7/(48*sqrt(-a**2*x**2 + 1)) + x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(128*a**7), True)) - 3*a**5*c**4*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) + 3*a**4*c**4*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) + 3*a**3*c**4*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) - 3*a**2*c**4*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**4*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1193,1,629,0,11.909166," ","integrate((-a**2*c*x**2+c)**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","- a^{5} c^{3} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) + a^{4} c^{3} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - 2 a^{2} c^{3} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) + a**4*c**3*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) + 2*a**3*c**3*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) - 2*a**2*c**3*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1194,1,337,0,6.892497," ","integrate((-a**2*c*x**2+c)**2/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","a^{3} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) - a^{2} c^{2} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**3*c**2*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) - a**2*c**2*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**2*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1195,1,109,0,4.437120," ","integrate((-a**2*c*x**2+c)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","- a c \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a*c*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1196,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c),x)","\frac{\int \frac{1}{a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c}"," ",0,"Integral(1/(a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c","F",0
1197,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{1}{- a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"Integral(1/(-a**3*x**3*sqrt(-a**2*x**2 + 1) - a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**2","F",0
1198,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{1}{a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"Integral(1/(a**5*x**5*sqrt(-a**2*x**2 + 1) + a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**3","F",0
1199,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**4,x)","\frac{\int \frac{1}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"Integral(1/(-a**7*x**7*sqrt(-a**2*x**2 + 1) - a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 3*a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**4","F",0
1200,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**5,x)","\frac{\int \frac{1}{a^{9} x^{9} \sqrt{- a^{2} x^{2} + 1} + a^{8} x^{8} \sqrt{- a^{2} x^{2} + 1} - 4 a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{5}}"," ",0,"Integral(1/(a**9*x**9*sqrt(-a**2*x**2 + 1) + a**8*x**8*sqrt(-a**2*x**2 + 1) - 4*a**7*x**7*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 6*a**5*x**5*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**5","F",0
1201,0,0,0,0.000000," ","integrate(x**m*(-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
1202,0,0,0,0.000000," ","integrate(x**2*(-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
1203,0,0,0,0.000000," ","integrate(x*(-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
1204,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1), x)","F",0
1205,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(x*(a*x + 1)), x)","F",0
1206,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)), x)","F",0
1207,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(3/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1), x)","F",0
1208,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(5/2)/(a*x + 1), x)","F",0
1209,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(7/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(7/2)/(a*x + 1), x)","F",0
1210,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{9}{2}}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(9/2)/(a*x + 1), x)","F",0
1211,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)), x)","F",0
1212,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)), x)","F",0
1213,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1)*(a*x + 1))**(5/2)*(a*x + 1)), x)","F",0
1214,0,0,0,0.000000," ","integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(7/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)), x)","F",0
1215,0,0,0,0.000000," ","integrate(x**m*(-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{m} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x**m*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1216,0,0,0,0.000000," ","integrate(x**3*(-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x**3*sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1217,0,0,0,0.000000," ","integrate(x**2*(-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1218,0,0,0,0.000000," ","integrate(x*(-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1219,0,0,0,0.000000," ","integrate((-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1220,0,0,0,0.000000," ","integrate((-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(x*(a*x + 1)), x)","F",0
1221,0,0,0,0.000000," ","integrate((-a**2*x**2+1)**p/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-(a*x - 1)*(a*x + 1))**p/(x**2*(a*x + 1)), x)","F",0
1222,0,0,0,0.000000," ","integrate(x**3*(-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{3} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x**3*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1223,0,0,0,0.000000," ","integrate(x**2*(-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1224,0,0,0,0.000000," ","integrate(x*(-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{x \sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(x*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1225,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{a x + 1}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1), x)","F",0
1226,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2)/x,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{x \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(x*(a*x + 1)), x)","F",0
1227,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**p/(a*x+1)*(-a**2*x**2+1)**(1/2)/x**2,x)","\int \frac{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{x^{2} \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**p/(x**2*(a*x + 1)), x)","F",0
1228,1,87,0,0.092191," ","integrate((-a**2*c*x**2+c)**4/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{a^{8} c^{4} x^{9}}{9} + \frac{a^{7} c^{4} x^{8}}{4} + \frac{2 a^{6} c^{4} x^{7}}{7} - a^{5} c^{4} x^{6} + \frac{3 a^{3} c^{4} x^{4}}{2} - \frac{2 a^{2} c^{4} x^{3}}{3} - a c^{4} x^{2} + c^{4} x"," ",0,"-a**8*c**4*x**9/9 + a**7*c**4*x**8/4 + 2*a**6*c**4*x**7/7 - a**5*c**4*x**6 + 3*a**3*c**4*x**4/2 - 2*a**2*c**4*x**3/3 - a*c**4*x**2 + c**4*x","A",0
1229,1,70,0,0.085337," ","integrate((-a**2*c*x**2+c)**3/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{a^{6} c^{3} x^{7}}{7} - \frac{a^{5} c^{3} x^{6}}{3} - \frac{a^{4} c^{3} x^{5}}{5} + a^{3} c^{3} x^{4} - \frac{a^{2} c^{3} x^{3}}{3} - a c^{3} x^{2} + c^{3} x"," ",0,"a**6*c**3*x**7/7 - a**5*c**3*x**6/3 - a**4*c**3*x**5/5 + a**3*c**3*x**4 - a**2*c**3*x**3/3 - a*c**3*x**2 + c**3*x","A",0
1230,1,36,0,0.076706," ","integrate((-a**2*c*x**2+c)**2/(a*x+1)**2*(-a**2*x**2+1),x)","- \frac{a^{4} c^{2} x^{5}}{5} + \frac{a^{3} c^{2} x^{4}}{2} - a c^{2} x^{2} + c^{2} x"," ",0,"-a**4*c**2*x**5/5 + a**3*c**2*x**4/2 - a*c**2*x**2 + c**2*x","A",0
1231,1,19,0,0.067484," ","integrate((-a**2*c*x**2+c)/(a*x+1)**2*(-a**2*x**2+1),x)","\frac{a^{2} c x^{3}}{3} - a c x^{2} + c x"," ",0,"a**2*c*x**3/3 - a*c*x**2 + c*x","A",0
1232,1,12,0,0.122889," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c),x)","- \frac{1}{a^{2} c x + a c}"," ",0,"-1/(a**2*c*x + a*c)","A",0
1233,1,56,0,0.281602," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**2,x)","- \frac{a x + 2}{4 a^{3} c^{2} x^{2} + 8 a^{2} c^{2} x + 4 a c^{2}} - \frac{\frac{\log{\left(x - \frac{1}{a} \right)}}{8} - \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a c^{2}}"," ",0,"-(a*x + 2)/(4*a**3*c**2*x**2 + 8*a**2*c**2*x + 4*a*c**2) - (log(x - 1/a)/8 - log(x + 1/a)/8)/(a*c**2)","A",0
1234,1,83,0,0.412963," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**3,x)","\frac{- 3 a^{3} x^{3} - 6 a^{2} x^{2} - a x + 4}{12 a^{5} c^{3} x^{4} + 24 a^{4} c^{3} x^{3} - 24 a^{2} c^{3} x - 12 a c^{3}} + \frac{- \frac{\log{\left(x - \frac{1}{a} \right)}}{8} + \frac{\log{\left(x + \frac{1}{a} \right)}}{8}}{a c^{3}}"," ",0,"(-3*a**3*x**3 - 6*a**2*x**2 - a*x + 4)/(12*a**5*c**3*x**4 + 24*a**4*c**3*x**3 - 24*a**2*c**3*x - 12*a*c**3) + (-log(x - 1/a)/8 + log(x + 1/a)/8)/(a*c**3)","A",0
1235,1,143,0,0.577808," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**4,x)","- \frac{15 a^{5} x^{5} + 30 a^{4} x^{4} - 10 a^{3} x^{3} - 50 a^{2} x^{2} - 17 a x + 16}{64 a^{7} c^{4} x^{6} + 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} - 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} + 128 a^{2} c^{4} x + 64 a c^{4}} - \frac{\frac{15 \log{\left(x - \frac{1}{a} \right)}}{128} - \frac{15 \log{\left(x + \frac{1}{a} \right)}}{128}}{a c^{4}}"," ",0,"-(15*a**5*x**5 + 30*a**4*x**4 - 10*a**3*x**3 - 50*a**2*x**2 - 17*a*x + 16)/(64*a**7*c**4*x**6 + 128*a**6*c**4*x**5 - 64*a**5*c**4*x**4 - 256*a**4*c**4*x**3 - 64*a**3*c**4*x**2 + 128*a**2*c**4*x + 64*a*c**4) - (15*log(x - 1/a)/128 - 15*log(x + 1/a)/128)/(a*c**4)","A",0
1236,0,0,0,0.000000," ","integrate(x**3*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{3} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\right)\, dx - \int \frac{a x^{4} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx"," ",0,"-Integral(-x**3*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x**4*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x)","F",0
1237,0,0,0,0.000000," ","integrate(x**2*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\right)\, dx - \int \frac{a x^{3} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx"," ",0,"-Integral(-x**2*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x**3*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x)","F",0
1238,0,0,0,0.000000," ","integrate(x*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\right)\, dx - \int \frac{a x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx"," ",0,"-Integral(-x*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x**2*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x)","F",0
1239,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x + 1}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x)","F",0
1240,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x,x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{2} + x}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{2} + x}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**2 + x), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**2 + x), x)","F",0
1241,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**2,x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{3} + x^{2}}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{3} + x^{2}}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**3 + x**2), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**3 + x**2), x)","F",0
1242,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**3,x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{4} + x^{3}}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**4 + x**3), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**4 + x**3), x)","F",0
1243,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**5 + x**4), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**5 + x**4), x)","F",0
1244,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**5,x)","- \int \left(- \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\right)\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\, dx"," ",0,"-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**6 + x**5), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**6 + x**5), x)","F",0
1245,1,340,0,6.784210," ","integrate((-a**2*c*x**2+c)**(3/2)/(a*x+1)**2*(-a**2*x**2+1),x)","a^{2} c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 2 a c \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**2*c*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a*c*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1246,1,478,0,10.273818," ","integrate((-a**2*c*x**2+c)**(5/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- a^{4} c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) - 2 a c^{2} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**4*c**2*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 2*a**3*c**2*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) - 2*a*c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1247,1,1091,0,20.108524," ","integrate((-a**2*c*x**2+c)**(7/2)/(a*x+1)**2*(-a**2*x**2+1),x)","a^{6} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i \sqrt{c} x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i \sqrt{c} x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 \sqrt{c} x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 \sqrt{c} x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) - 2 a^{5} c^{3} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} c x^{2} + c}}{7} - \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} c x^{2} + c}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} c x^{2} + c}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right) - a^{4} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \sqrt{c} x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \sqrt{c} x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 4 a^{3} c^{3} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} c x^{2} + c}}{5} - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} c x^{2} + c}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right) - a^{2} c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \sqrt{c} x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i \sqrt{c} x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} \sqrt{c} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \sqrt{c} x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{\sqrt{c} x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 2 a c^{3} \left(\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} c x^{2} + c\right)^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**6*c**3*Piecewise((I*a**2*sqrt(c)*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*asin(a*x)/(128*a**7), True)) - 2*a**5*c**3*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) - a**4*c**3*Piecewise((I*a**2*sqrt(c)*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*sqrt(c)*x**5/(24*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**5/(24*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(16*a**4*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(16*a**5), True)) + 4*a**3*c**3*Piecewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) - a**2*c**3*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2 + 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a*c**3*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), True)) + c**3*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a), True))","C",0
1248,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(1/2),x)","- \int \frac{a x}{a x \sqrt{- a^{2} c x^{2} + c} + \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \left(- \frac{1}{a x \sqrt{- a^{2} c x^{2} + c} + \sqrt{- a^{2} c x^{2} + c}}\right)\, dx"," ",0,"-Integral(a*x/(a*x*sqrt(-a**2*c*x**2 + c) + sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(a*x*sqrt(-a**2*c*x**2 + c) + sqrt(-a**2*c*x**2 + c)), x)","F",0
1249,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(3/2),x)","- \int \frac{a x}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} + c \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \left(- \frac{1}{- a^{3} c x^{3} \sqrt{- a^{2} c x^{2} + c} - a^{2} c x^{2} \sqrt{- a^{2} c x^{2} + c} + a c x \sqrt{- a^{2} c x^{2} + c} + c \sqrt{- a^{2} c x^{2} + c}}\right)\, dx"," ",0,"-Integral(a*x/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) - a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) + c*sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(-a**3*c*x**3*sqrt(-a**2*c*x**2 + c) - a**2*c*x**2*sqrt(-a**2*c*x**2 + c) + a*c*x*sqrt(-a**2*c*x**2 + c) + c*sqrt(-a**2*c*x**2 + c)), x)","F",0
1250,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(5/2),x)","- \int \frac{a x}{a^{5} c^{2} x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt{- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt{- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{2} x \sqrt{- a^{2} c x^{2} + c} + c^{2} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \left(- \frac{1}{a^{5} c^{2} x^{5} \sqrt{- a^{2} c x^{2} + c} + a^{4} c^{2} x^{4} \sqrt{- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt{- a^{2} c x^{2} + c} - 2 a^{2} c^{2} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{2} x \sqrt{- a^{2} c x^{2} + c} + c^{2} \sqrt{- a^{2} c x^{2} + c}}\right)\, dx"," ",0,"-Integral(a*x/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) + a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) - 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) + c**2*sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(a**5*c**2*x**5*sqrt(-a**2*c*x**2 + c) + a**4*c**2*x**4*sqrt(-a**2*c*x**2 + c) - 2*a**3*c**2*x**3*sqrt(-a**2*c*x**2 + c) - 2*a**2*c**2*x**2*sqrt(-a**2*c*x**2 + c) + a*c**2*x*sqrt(-a**2*c*x**2 + c) + c**2*sqrt(-a**2*c*x**2 + c)), x)","F",0
1251,0,0,0,0.000000," ","integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(7/2),x)","- \int \frac{a x}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} + c^{3} \sqrt{- a^{2} c x^{2} + c}}\, dx - \int \left(- \frac{1}{- a^{7} c^{3} x^{7} \sqrt{- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt{- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt{- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt{- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt{- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt{- a^{2} c x^{2} + c} + a c^{3} x \sqrt{- a^{2} c x^{2} + c} + c^{3} \sqrt{- a^{2} c x^{2} + c}}\right)\, dx"," ",0,"-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) - a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) + 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) - 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) + c**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) - a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) + 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) - 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) + c**3*sqrt(-a**2*c*x**2 + c)), x)","F",0
1252,0,0,0,0.000000," ","integrate(x**m*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)","- \int \left(- \frac{x^{m} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\right)\, dx - \int \frac{a x x^{m} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx"," ",0,"-Integral(-x**m*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x) - Integral(a*x*x**m*sqrt(-a**2*c*x**2 + c)/(a*x + 1), x)","F",0
1253,1,651,0,10.900927," ","integrate((-a**2*c*x**2+c)**p/(a*x+1)**2*(-a**2*x**2+1),x)","- a \left(\begin{cases} \frac{0^{p} x}{a} + \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \log{\left(-1 + \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{acoth}{\left(\frac{1}{a x} \right)}}{a^{2}} - \frac{c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(- p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, - p - \frac{1}{2} \\ \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \frac{1}{\left|{a^{2} x^{2}}\right|} > 1 \\\frac{0^{p} x}{a} + \frac{0^{p} \log{\left(\frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \log{\left(1 - \frac{1}{a^{2} x^{2}} \right)}}{2 a^{2}} - \frac{0^{p} \operatorname{atanh}{\left(\frac{1}{a x} \right)}}{a^{2}} - \frac{c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} - \frac{a^{2 p} c^{p} p x x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(- p - \frac{1}{2}\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, - p - \frac{1}{2} \\ \frac{1}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a \Gamma\left(\frac{1}{2} - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{0^{p} \log{\left(a^{2} x^{2} - 1 \right)}}{2 a} + \frac{0^{p} \operatorname{acoth}{\left(a x \right)}}{a} + \frac{a c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} + \frac{a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(\frac{1}{2} - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a^{2} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{0^{p} \log{\left(- a^{2} x^{2} + 1 \right)}}{2 a} + \frac{0^{p} \operatorname{atanh}{\left(a x \right)}}{a} + \frac{a c^{p} x^{2} \Gamma\left(p\right) \Gamma\left(1 - p\right) {{}_{3}F_{2}\left(\begin{matrix} 2, 1, 1 - p \\ 2, 2 \end{matrix}\middle| {a^{2} x^{2} e^{2 i \pi}} \right)}}{2 \Gamma\left(- p\right) \Gamma\left(p + 1\right)} + \frac{a^{2 p} c^{p} p x^{2 p} e^{i \pi p} \Gamma\left(p\right) \Gamma\left(\frac{1}{2} - p\right) {{}_{2}F_{1}\left(\begin{matrix} 1 - p, \frac{1}{2} - p \\ \frac{3}{2} - p \end{matrix}\middle| {\frac{1}{a^{2} x^{2}}} \right)}}{2 a^{2} x \Gamma\left(\frac{3}{2} - p\right) \Gamma\left(p + 1\right)} & \text{otherwise} \end{cases}"," ",0,"-a*Piecewise((0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 - c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*a*gamma(1/2 - p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 - c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*p*x*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*a*gamma(1/2 - p)*gamma(p + 1)), True)) + Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) + 0**p*acoth(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) + a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) + 0**p*atanh(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) + a**(2*p)*c**p*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*a**2*x*gamma(3/2 - p)*gamma(p + 1)), True))","C",0
1254,1,996,0,20.376967," ","integrate((-a**2*c*x**2+c)**4/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- a^{7} c^{4} \left(\begin{cases} \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{9} - \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{63 a^{2}} - \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1}}{315 a^{6}} - \frac{16 \sqrt{- a^{2} x^{2} + 1}}{315 a^{8}} & \text{for}\: a \neq 0 \\\frac{x^{8}}{8} & \text{otherwise} \end{cases}\right) + 3 a^{6} c^{4} \left(\begin{cases} \frac{i a^{2} x^{9}}{8 \sqrt{a^{2} x^{2} - 1}} - \frac{7 i x^{7}}{48 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{192 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{384 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{128 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left(a x \right)}}{128 a^{7}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{9}}{8 \sqrt{- a^{2} x^{2} + 1}} + \frac{7 x^{7}}{48 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{192 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{384 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{128 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left(a x \right)}}{128 a^{7}} & \text{otherwise} \end{cases}\right) - a^{5} c^{4} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 5 a^{4} c^{4} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 5 a^{3} c^{4} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + a^{2} c^{4} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 3 a c^{4} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{4} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**7*c**4*Piecewise((x**8*sqrt(-a**2*x**2 + 1)/9 - x**6*sqrt(-a**2*x**2 + 1)/(63*a**2) - 2*x**4*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*x**2*sqrt(-a**2*x**2 + 1)/(315*a**6) - 16*sqrt(-a**2*x**2 + 1)/(315*a**8), Ne(a, 0)), (x**8/8, True)) + 3*a**6*c**4*Piecewise((I*a**2*x**9/(8*sqrt(a**2*x**2 - 1)) - 7*I*x**7/(48*sqrt(a**2*x**2 - 1)) - I*x**5/(192*a**2*sqrt(a**2*x**2 - 1)) - 5*I*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*x**7/(48*sqrt(-a**2*x**2 + 1)) + x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*x**3/(384*a**4*sqrt(-a**2*x**2 + 1)) - 5*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*asin(a*x)/(128*a**7), True)) - a**5*c**4*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 5*a**4*c**4*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) + 5*a**3*c**4*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + a**2*c**4*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - 3*a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**4*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1255,1,632,0,13.784605," ","integrate((-a**2*c*x**2+c)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","a^{5} c^{3} \left(\begin{cases} \frac{x^{6} \sqrt{- a^{2} x^{2} + 1}}{7} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{35 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{105 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{105 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right) - 3 a^{4} c^{3} \left(\begin{cases} \frac{i a^{2} x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{5}}{24 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{48 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{16 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{16 a^{5}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{5}}{24 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{48 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{16 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{16 a^{5}} & \text{otherwise} \end{cases}\right) + 2 a^{3} c^{3} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 2 a^{2} c^{3} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 3 a c^{3} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{3} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"a**5*c**3*Piecewise((x**6*sqrt(-a**2*x**2 + 1)/7 - x**4*sqrt(-a**2*x**2 + 1)/(35*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)/(105*a**4) - 8*sqrt(-a**2*x**2 + 1)/(105*a**6), Ne(a, 0)), (x**6/6, True)) - 3*a**4*c**3*Piecewise((I*a**2*x**7/(6*sqrt(a**2*x**2 - 1)) - 5*I*x**5/(24*sqrt(a**2*x**2 - 1)) - I*x**3/(48*a**2*sqrt(a**2*x**2 - 1)) + I*x/(16*a**4*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(16*a**5), Abs(a**2*x**2) > 1), (-a**2*x**7/(6*sqrt(-a**2*x**2 + 1)) + 5*x**5/(24*sqrt(-a**2*x**2 + 1)) + x**3/(48*a**2*sqrt(-a**2*x**2 + 1)) - x/(16*a**4*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(16*a**5), True)) + 2*a**3*c**3*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 2*a**2*c**3*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - 3*a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**3*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1256,1,340,0,8.103245," ","integrate((-a**2*c*x**2+c)**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","- a^{3} c^{2} \left(\begin{cases} \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right) + 3 a^{2} c^{2} \left(\begin{cases} \frac{i a^{2} x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{3 i x^{3}}{8 \sqrt{a^{2} x^{2} - 1}} + \frac{i x}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{8 a^{3}} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\- \frac{a^{2} x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{3 x^{3}}{8 \sqrt{- a^{2} x^{2} + 1}} - \frac{x}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{\operatorname{asin}{\left(a x \right)}}{8 a^{3}} & \text{otherwise} \end{cases}\right) - 3 a c^{2} \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left(- a^{2} x^{2} + 1\right)^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right) + c^{2} \left(\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left(a x \right)}}{2 a} & \text{for}\: \left|{a^{2} x^{2}}\right| > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left(a x \right)}}{2 a} & \text{otherwise} \end{cases}\right)"," ",0,"-a**3*c**2*Piecewise((x**4*sqrt(-a**2*x**2 + 1)/5 - x**2*sqrt(-a**2*x**2 + 1)/(15*a**2) - 2*sqrt(-a**2*x**2 + 1)/(15*a**4), Ne(a, 0)), (x**4/4, True)) + 3*a**2*c**2*Piecewise((I*a**2*x**5/(4*sqrt(a**2*x**2 - 1)) - 3*I*x**3/(8*sqrt(a**2*x**2 - 1)) + I*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(8*a**3), Abs(a**2*x**2) > 1), (-a**2*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*x**3/(8*sqrt(-a**2*x**2 + 1)) - x/(8*a**2*sqrt(-a**2*x**2 + 1)) + asin(a*x)/(8*a**3), True)) - 3*a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-(-a**2*x**2 + 1)**(3/2)/(3*a**2), True)) + c**2*Piecewise((I*a**2*x**3/(2*sqrt(a**2*x**2 - 1)) - I*x/(2*sqrt(a**2*x**2 - 1)) - I*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (x*sqrt(-a**2*x**2 + 1)/2 + asin(a*x)/(2*a), True))","C",0
1257,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","c \left(\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left(- \frac{2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right)\, dx + \int \frac{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right)"," ",0,"c*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))","F",0
1258,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c),x)","\frac{\int \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx}{c}"," ",0,"Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x)/c","F",0
1259,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{1}{a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{2}}"," ",0,"Integral(1/(a**3*x**3*sqrt(-a**2*x**2 + 1) + 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**2","F",0
1260,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**3,x)","\frac{\int \frac{1}{- a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 2 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}}"," ",0,"Integral(1/(-a**5*x**5*sqrt(-a**2*x**2 + 1) - 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) + 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**3","F",0
1261,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**4,x)","\frac{\int \frac{1}{a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} + 3 a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 5 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + 3 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}}"," ",0,"Integral(1/(a**7*x**7*sqrt(-a**2*x**2 + 1) + 3*a**6*x**6*sqrt(-a**2*x**2 + 1) + a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) - 5*a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**4","F",0
1262,0,0,0,0.000000," ","integrate(x**3*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{3} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**3*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1)**3, x)","F",0
1263,0,0,0,0.000000," ","integrate(x**2*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x^{2} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x**2*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1)**3, x)","F",0
1264,0,0,0,0.000000," ","integrate(x*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{x \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral(x*(-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1)**3, x)","F",0
1265,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(a*x + 1)**3, x)","F",0
1266,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(x*(a*x + 1)**3), x)","F",0
1267,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**2,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x^{2} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**2*(a*x + 1)**3), x)","F",0
1268,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**3,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x^{3} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**3*(a*x + 1)**3), x)","F",0
1269,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**4,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x^{4} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**4*(a*x + 1)**3), x)","F",0
1270,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}{x^{5} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**5*(a*x + 1)**3), x)","F",0
1271,-1,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(9/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1272,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(7/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(7/2)/(a*x + 1)**3, x)","F",0
1273,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(5/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(5/2)/(a*x + 1)**3, x)","F",0
1274,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**(3/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(3/2)/(a*x + 1)**3, x)","F",0
1275,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/(sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x + 1)**3), x)","F",0
1276,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)**3), x)","F",0
1277,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1)*(a*x + 1))**(5/2)*(a*x + 1)**3), x)","F",0
1278,0,0,0,0.000000," ","integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(7/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{7}{2}} \left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)**3), x)","F",0
1279,-1,0,0,0.000000," ","integrate(x**m*(-a**2*c*x**2+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1280,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**p/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)","\int \frac{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p}}{\left(a x + 1\right)^{3}}\, dx"," ",0,"Integral((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**p/(a*x + 1)**3, x)","F",0
1281,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*x**2+1)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1282,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*x**2+1)**(3/2),x)","\int \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}} \left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))*(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1283,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*x**2+1)**(1/2),x)","\int \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))*sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
1284,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(1/2),x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/sqrt(-(a*x - 1)*(a*x + 1)), x)","F",0
1285,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(3/2),x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/(-(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1286,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(5/2),x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\left(- \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/(-(a*x - 1)*(a*x + 1))**(5/2), x)","F",0
1287,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1288,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*x**2+1)**(9/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1289,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*c*x**2+c)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1290,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*c*x**2+c)**(3/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1291,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*(-a**2*c*x**2+c)**(1/2),x)","\int \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))*sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1292,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1293,0,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{\sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1294,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1295,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1296,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(9/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1297,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**3/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1298,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**2/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1299,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1300,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1301,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1302,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x**2/(-a**2*c*x**2+c)**(5/4),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1303,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**3/(-a**2*c*x**2+c)**(9/8),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1304,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x**2/(-a**2*c*x**2+c)**(9/8),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1305,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)*x/(-a**2*c*x**2+c)**(9/8),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1306,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/(-a**2*c*x**2+c)**(9/8),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1307,-1,0,0,0.000000," ","integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2)/x/(-a**2*c*x**2+c)**(9/8),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1308,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c),x)","- c \left(\int a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx\right)"," ",0,"-c*(Integral(a**2*x**2*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x)), x))","F",0
1309,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**2,x)","c^{2} \left(\int \left(- 2 a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx + \int a^{4} x^{4} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx\right)"," ",0,"c**2*(Integral(-2*a**2*x**2*exp(n*atanh(a*x)), x) + Integral(a**4*x**4*exp(n*atanh(a*x)), x) + Integral(exp(n*atanh(a*x)), x))","F",0
1310,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**3,x)","- c^{3} \left(\int 3 a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- 3 a^{4} x^{4} e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx + \int a^{6} x^{6} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx + \int \left(- e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx\right)"," ",0,"-c**3*(Integral(3*a**2*x**2*exp(n*atanh(a*x)), x) + Integral(-3*a**4*x**4*exp(n*atanh(a*x)), x) + Integral(a**6*x**6*exp(n*atanh(a*x)), x) + Integral(-exp(n*atanh(a*x)), x))","F",0
1311,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**4/(-a**2*c*x**2+c),x)","- \frac{\int \frac{x^{4} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"-Integral(x**4*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
1312,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c),x)","- \frac{\int \frac{x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"-Integral(x**3*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
1313,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2/(-a**2*c*x**2+c),x)","- \frac{\int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"-Integral(x**2*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
1314,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x/(-a**2*c*x**2+c),x)","- \frac{\int \frac{x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"-Integral(x*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
1315,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c),x)","\begin{cases} \tilde{\infty} x & \text{for}\: c = 0 \wedge n = 0 \\\tilde{\infty} \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\- \frac{\log{\left(x - \frac{1}{a} \right)}}{2 a c} + \frac{\log{\left(x + \frac{1}{a} \right)}}{2 a c} & \text{for}\: n = 0 \\\frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a c n} & \text{otherwise} \end{cases}"," ",0,"Piecewise((zoo*x, Eq(c, 0) & Eq(n, 0)), (zoo*Integral(exp(n*atanh(a*x)), x), Eq(c, 0)), (-log(x - 1/a)/(2*a*c) + log(x + 1/a)/(2*a*c), Eq(n, 0)), (exp(n*atanh(a*x))/(a*c*n), True))","F",0
1316,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x/(-a**2*c*x**2+c),x)","- \frac{\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{3} - x}\, dx}{c}"," ",0,"-Integral(exp(n*atanh(a*x))/(a**2*x**3 - x), x)/c","F",0
1317,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c),x)","- \frac{\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{4} - x^{2}}\, dx}{c}"," ",0,"-Integral(exp(n*atanh(a*x))/(a**2*x**4 - x**2), x)/c","F",0
1318,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**4/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{4} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}}"," ",0,"Integral(x**4*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2","F",0
1319,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}}"," ",0,"Integral(x**3*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2","F",0
1320,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2/(-a**2*c*x**2+c)**2,x)","\begin{cases} \tilde{\infty} x^{3} e^{- \infty n} & \text{for}\: a = - \frac{1}{x} \\\tilde{\infty} x^{3} e^{\infty n} & \text{for}\: a = \frac{1}{x} \\\tilde{\infty} \int x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\- \frac{a^{2} x^{2} \operatorname{atanh}{\left(a x \right)}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{3} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{2 a x \operatorname{atanh}{\left(a x \right)}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{3} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{3 a x}{4 a^{5} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{3} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{\operatorname{atanh}{\left(a x \right)}}{4 a^{5} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{3} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{2}{4 a^{5} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{3} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -2 \\\frac{a^{2} x^{2} \log{\left(x - \frac{1}{a} \right)}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac{a^{2} x^{2} \log{\left(x + \frac{1}{a} \right)}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac{2 a x}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} - \frac{\log{\left(x - \frac{1}{a} \right)}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} + \frac{\log{\left(x + \frac{1}{a} \right)}}{4 a^{5} c^{2} x^{2} - 4 a^{3} c^{2}} & \text{for}\: n = 0 \\\frac{\int \frac{x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text{for}\: n = 2 \\- \frac{a^{2} n^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} + \frac{2 a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} + \frac{2 a n x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} - \frac{2 e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{5} c^{2} n^{3} x^{2} - 4 a^{5} c^{2} n x^{2} - a^{3} c^{2} n^{3} + 4 a^{3} c^{2} n} & \text{otherwise} \end{cases}"," ",0,"Piecewise((zoo*x**3*exp(-oo*n), Eq(a, -1/x)), (zoo*x**3*exp(oo*n), Eq(a, 1/x)), (zoo*Integral(x**2*exp(n*atanh(a*x)), x), Eq(c, 0)), (-a**2*x**2*atanh(a*x)/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - 3*a*x/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))) - 2/(4*a**5*c**2*x**2*exp(2*atanh(a*x)) - 4*a**3*c**2*exp(2*atanh(a*x))), Eq(n, -2)), (a**2*x**2*log(x - 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) - a**2*x**2*log(x + 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) - 2*a*x/(4*a**5*c**2*x**2 - 4*a**3*c**2) - log(x - 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2) + log(x + 1/a)/(4*a**5*c**2*x**2 - 4*a**3*c**2), Eq(n, 0)), (Integral(x**2*exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (-a**2*n**2*x**2*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n) + 2*a**2*x**2*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n) + 2*a*n*x*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n) - 2*exp(n*atanh(a*x))/(a**5*c**2*n**3*x**2 - 4*a**5*c**2*n*x**2 - a**3*c**2*n**3 + 4*a**3*c**2*n), True))","F",0
1321,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x/(-a**2*c*x**2+c)**2,x)","\begin{cases} \tilde{\infty} x^{2} e^{- \infty n} & \text{for}\: a = - \frac{1}{x} \\\tilde{\infty} x^{2} e^{\infty n} & \text{for}\: a = \frac{1}{x} \\\tilde{\infty} \int x e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\- \frac{a^{2} x^{2} \operatorname{atanh}{\left(a x \right)}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{2 a x \operatorname{atanh}{\left(a x \right)}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{a x}{4 a^{4} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{\operatorname{atanh}{\left(a x \right)}}{4 a^{4} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a^{2} c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -2 \\\frac{\int \frac{x e^{2 \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text{for}\: n = 2 \\\frac{a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} - \frac{a n x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} + \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} c^{2} n^{2} x^{2} - 4 a^{4} c^{2} x^{2} - a^{2} c^{2} n^{2} + 4 a^{2} c^{2}} & \text{otherwise} \end{cases}"," ",0,"Piecewise((zoo*x**2*exp(-oo*n), Eq(a, -1/x)), (zoo*x**2*exp(oo*n), Eq(a, 1/x)), (zoo*Integral(x*exp(n*atanh(a*x)), x), Eq(c, 0)), (-a**2*x**2*atanh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) + a*x/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(4*a**4*c**2*x**2*exp(2*atanh(a*x)) - 4*a**2*c**2*exp(2*atanh(a*x))), Eq(n, -2)), (Integral(x*exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (a**2*x**2*exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) - a*n*x*exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2) + exp(n*atanh(a*x))/(a**4*c**2*n**2*x**2 - 4*a**4*c**2*x**2 - a**2*c**2*n**2 + 4*a**2*c**2), True))","F",0
1322,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**2,x)","\begin{cases} \tilde{\infty} x e^{- \infty n} & \text{for}\: a = - \frac{1}{x} \\\tilde{\infty} x e^{\infty n} & \text{for}\: a = \frac{1}{x} \\\tilde{\infty} \int e^{n \operatorname{atanh}{\left(a x \right)}}\, dx & \text{for}\: c = 0 \\- \frac{a^{2} x^{2} \operatorname{atanh}{\left(a x \right)}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{2 a x \operatorname{atanh}{\left(a x \right)}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{a x}{4 a^{3} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} - \frac{\operatorname{atanh}{\left(a x \right)}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} + \frac{2}{4 a^{3} c^{2} x^{2} e^{2 \operatorname{atanh}{\left(a x \right)}} - 4 a c^{2} e^{2 \operatorname{atanh}{\left(a x \right)}}} & \text{for}\: n = -2 \\- \frac{a^{2} x^{2} \log{\left(x - \frac{1}{a} \right)}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac{a^{2} x^{2} \log{\left(x + \frac{1}{a} \right)}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac{2 a x}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac{\log{\left(x - \frac{1}{a} \right)}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac{\log{\left(x + \frac{1}{a} \right)}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} & \text{for}\: n = 0 \\\frac{\int \frac{e^{2 \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text{for}\: n = 2 \\- \frac{2 a^{2} x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac{2 a n x e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} - \frac{n^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac{2 e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} & \text{otherwise} \end{cases}"," ",0,"Piecewise((zoo*x*exp(-oo*n), Eq(a, -1/x)), (zoo*x*exp(oo*n), Eq(a, 1/x)), (zoo*Integral(exp(n*atanh(a*x)), x), Eq(c, 0)), (-a**2*x**2*atanh(a*x)/(4*a**3*c**2*x**2*exp(2*atanh(a*x)) - 4*a*c**2*exp(2*atanh(a*x))) - 2*a*x*atanh(a*x)/(4*a**3*c**2*x**2*exp(2*atanh(a*x)) - 4*a*c**2*exp(2*atanh(a*x))) + a*x/(4*a**3*c**2*x**2*exp(2*atanh(a*x)) - 4*a*c**2*exp(2*atanh(a*x))) - atanh(a*x)/(4*a**3*c**2*x**2*exp(2*atanh(a*x)) - 4*a*c**2*exp(2*atanh(a*x))) + 2/(4*a**3*c**2*x**2*exp(2*atanh(a*x)) - 4*a*c**2*exp(2*atanh(a*x))), Eq(n, -2)), (-a**2*x**2*log(x - 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2) + a**2*x**2*log(x + 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2) - 2*a*x/(4*a**3*c**2*x**2 - 4*a*c**2) + log(x - 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2) - log(x + 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2), Eq(n, 0)), (Integral(exp(2*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2, Eq(n, 2)), (-2*a**2*x**2*exp(n*atanh(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n) + 2*a*n*x*exp(n*atanh(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n) - n**2*exp(n*atanh(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n) + 2*exp(n*atanh(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n), True))","F",0
1323,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{5} - 2 a^{2} x^{3} + x}\, dx}{c^{2}}"," ",0,"Integral(exp(n*atanh(a*x))/(a**4*x**5 - 2*a**2*x**3 + x), x)/c**2","F",0
1324,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{6} - 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}}"," ",0,"Integral(exp(n*atanh(a*x))/(a**4*x**6 - 2*a**2*x**4 + x**2), x)/c**2","F",0
1325,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**3,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1326,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**4,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1327,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3*(-a**2*c*x**2+c)**(1/2),x)","\int x^{3} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**3*sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)","F",0
1328,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2*(-a**2*c*x**2+c)**(1/2),x)","\int x^{2} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**2*sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)","F",0
1329,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x*(-a**2*c*x**2+c)**(1/2),x)","\int x \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)","F",0
1330,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(1/2),x)","\int \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x)), x)","F",0
1331,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(1/2)/x,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}}{x}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x))/x, x)","F",0
1332,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(1/2)/x**2,x)","\int \frac{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)} e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2}}\, dx"," ",0,"Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*exp(n*atanh(a*x))/x**2, x)","F",0
1333,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**(3/2),x)","\int \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*exp(n*atanh(a*x)), x)","F",0
1334,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**3*exp(n*atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1335,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x**2*exp(n*atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1336,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{x e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(x*exp(n*atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1337,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x)","F",0
1338,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
1339,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x**2*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
1340,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**3/(-a**2*c*x**2+c)**(1/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{3} \sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x**3*sqrt(-c*(a*x - 1)*(a*x + 1))), x)","F",0
1341,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**3*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1342,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1343,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1344,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)","F",0
1345,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1346,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x**2*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1347,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**3/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{3} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x**3*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1348,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**3/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{3} e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**3*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)","F",0
1349,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x^{2} e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x**2*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)","F",0
1350,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{x e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(x*exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)","F",0
1351,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)","F",0
1352,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
1353,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**2/(-a**2*c*x**2+c)**(5/2),x)","\int \frac{e^{n \operatorname{atanh}{\left(a x \right)}}}{x^{2} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"Integral(exp(n*atanh(a*x))/(x**2*(-c*(a*x - 1)*(a*x + 1))**(5/2)), x)","F",0
1354,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/x**3/(-a**2*c*x**2+c)**(5/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1355,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))/(-a**2*c*x**2+c)**(7/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1356,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m*(-a**2*c*x**2+c)**2,x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1357,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m*(-a**2*c*x**2+c),x)","- c \left(\int \left(- x^{m} e^{n \operatorname{atanh}{\left(a x \right)}}\right)\, dx + \int a^{2} x^{2} x^{m} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx\right)"," ",0,"-c*(Integral(-x**m*exp(n*atanh(a*x)), x) + Integral(a**2*x**2*x**m*exp(n*atanh(a*x)), x))","F",0
1358,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m/(-a**2*c*x**2+c),x)","- \frac{\int \frac{x^{m} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{2} x^{2} - 1}\, dx}{c}"," ",0,"-Integral(x**m*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c","F",0
1359,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m/(-a**2*c*x**2+c)**2,x)","\frac{\int \frac{x^{m} e^{n \operatorname{atanh}{\left(a x \right)}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}}"," ",0,"Integral(x**m*exp(n*atanh(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2","F",0
1360,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**m*(-a**2*c*x**2+c)**p,x)","\int x^{m} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x**m*(-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)","F",0
1361,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x*(-a**2*c*x**2+c)**p,x)","\int x \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral(x*(-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)","F",0
1362,0,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*(-a**2*c*x**2+c)**p,x)","\int \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{p} e^{n \operatorname{atanh}{\left(a x \right)}}\, dx"," ",0,"Integral((-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)","F",0
1363,-1,0,0,0.000000," ","integrate(exp(2*(1+p)*atanh(a*x))/((-a**2*x**2+1)**p),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1364,-1,0,0,0.000000," ","integrate(exp(2*(1+p)*atanh(a*x))/((-a**2*c*x**2+c)**p),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1365,0,0,0,0.000000," ","integrate(exp(2*p*atanh(a*x))*(-a**2*c*x**2+c)**p,x)","\begin{cases} \frac{x}{\sqrt{c}} & \text{for}\: a = 0 \wedge p = - \frac{1}{2} \\c^{p} x & \text{for}\: a = 0 \\\int \frac{e^{- \operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx & \text{for}\: p = - \frac{1}{2} \\\frac{a x \left(- a^{2} c x^{2} + c\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}}{2 a p + a} + \frac{\left(- a^{2} c x^{2} + c\right)^{p} e^{2 p \operatorname{atanh}{\left(a x \right)}}}{2 a p + a} & \text{otherwise} \end{cases}"," ",0,"Piecewise((x/sqrt(c), Eq(a, 0) & Eq(p, -1/2)), (c**p*x, Eq(a, 0)), (Integral(exp(-atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x), Eq(p, -1/2)), (a*x*(-a**2*c*x**2 + c)**p*exp(2*p*atanh(a*x))/(2*a*p + a) + (-a**2*c*x**2 + c)**p*exp(2*p*atanh(a*x))/(2*a*p + a), True))","F",0
1366,0,0,0,0.000000," ","integrate((-a**2*c*x**2+c)**p/exp(2*p*atanh(a*x)),x)","\begin{cases} \frac{x}{\sqrt{c}} & \text{for}\: a = 0 \wedge p = - \frac{1}{2} \\c^{p} x & \text{for}\: a = 0 \\\int \frac{e^{\operatorname{atanh}{\left(a x \right)}}}{\sqrt{- c \left(a x - 1\right) \left(a x + 1\right)}}\, dx & \text{for}\: p = - \frac{1}{2} \\\frac{a x \left(- a^{2} c x^{2} + c\right)^{p}}{2 a p e^{2 p \operatorname{atanh}{\left(a x \right)}} + a e^{2 p \operatorname{atanh}{\left(a x \right)}}} - \frac{\left(- a^{2} c x^{2} + c\right)^{p}}{2 a p e^{2 p \operatorname{atanh}{\left(a x \right)}} + a e^{2 p \operatorname{atanh}{\left(a x \right)}}} & \text{otherwise} \end{cases}"," ",0,"Piecewise((x/sqrt(c), Eq(a, 0) & Eq(p, -1/2)), (c**p*x, Eq(a, 0)), (Integral(exp(atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x), Eq(p, -1/2)), (a*x*(-a**2*c*x**2 + c)**p/(2*a*p*exp(2*p*atanh(a*x)) + a*exp(2*p*atanh(a*x))) - (-a**2*c*x**2 + c)**p/(2*a*p*exp(2*p*atanh(a*x)) + a*exp(2*p*atanh(a*x))), True))","F",0
1367,-1,0,0,0.000000," ","integrate(exp(n*atanh(a*x))*x**2*(-a**2*c*x**2+c)**(-1-1/2*n**2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1368,1,405,0,3.150562," ","integrate((a*x+1)**6/(-a**2*x**2+1)**3*x**2/(-a**2*c*x**2+c)**19,x)","\frac{- 6 a x + 1}{210 a^{39} c^{19} x^{36} - 1260 a^{38} c^{19} x^{35} + 14700 a^{36} c^{19} x^{33} - 22050 a^{35} c^{19} x^{32} - 70560 a^{34} c^{19} x^{31} + 188160 a^{33} c^{19} x^{30} + 151200 a^{32} c^{19} x^{29} - 819000 a^{31} c^{19} x^{28} + 58800 a^{30} c^{19} x^{27} + 2257920 a^{29} c^{19} x^{26} - 1375920 a^{28} c^{19} x^{25} - 4204200 a^{27} c^{19} x^{24} + 4586400 a^{26} c^{19} x^{23} + 5241600 a^{25} c^{19} x^{22} - 9129120 a^{24} c^{19} x^{21} - 3783780 a^{23} c^{19} x^{20} + 12612600 a^{22} c^{19} x^{19} - 12612600 a^{20} c^{19} x^{17} + 3783780 a^{19} c^{19} x^{16} + 9129120 a^{18} c^{19} x^{15} - 5241600 a^{17} c^{19} x^{14} - 4586400 a^{16} c^{19} x^{13} + 4204200 a^{15} c^{19} x^{12} + 1375920 a^{14} c^{19} x^{11} - 2257920 a^{13} c^{19} x^{10} - 58800 a^{12} c^{19} x^{9} + 819000 a^{11} c^{19} x^{8} - 151200 a^{10} c^{19} x^{7} - 188160 a^{9} c^{19} x^{6} + 70560 a^{8} c^{19} x^{5} + 22050 a^{7} c^{19} x^{4} - 14700 a^{6} c^{19} x^{3} + 1260 a^{4} c^{19} x - 210 a^{3} c^{19}}"," ",0,"(-6*a*x + 1)/(210*a**39*c**19*x**36 - 1260*a**38*c**19*x**35 + 14700*a**36*c**19*x**33 - 22050*a**35*c**19*x**32 - 70560*a**34*c**19*x**31 + 188160*a**33*c**19*x**30 + 151200*a**32*c**19*x**29 - 819000*a**31*c**19*x**28 + 58800*a**30*c**19*x**27 + 2257920*a**29*c**19*x**26 - 1375920*a**28*c**19*x**25 - 4204200*a**27*c**19*x**24 + 4586400*a**26*c**19*x**23 + 5241600*a**25*c**19*x**22 - 9129120*a**24*c**19*x**21 - 3783780*a**23*c**19*x**20 + 12612600*a**22*c**19*x**19 - 12612600*a**20*c**19*x**17 + 3783780*a**19*c**19*x**16 + 9129120*a**18*c**19*x**15 - 5241600*a**17*c**19*x**14 - 4586400*a**16*c**19*x**13 + 4204200*a**15*c**19*x**12 + 1375920*a**14*c**19*x**11 - 2257920*a**13*c**19*x**10 - 58800*a**12*c**19*x**9 + 819000*a**11*c**19*x**8 - 151200*a**10*c**19*x**7 - 188160*a**9*c**19*x**6 + 70560*a**8*c**19*x**5 + 22050*a**7*c**19*x**4 - 14700*a**6*c**19*x**3 + 1260*a**4*c**19*x - 210*a**3*c**19)","B",0
1369,1,180,0,1.223657," ","integrate((a*x+1)**4/(-a**2*x**2+1)**2*x**2/(-a**2*c*x**2+c)**9,x)","- \frac{- 4 a x + 1}{60 a^{19} c^{9} x^{16} - 240 a^{18} c^{9} x^{15} + 1200 a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} - 2160 a^{14} c^{9} x^{11} + 3840 a^{13} c^{9} x^{10} + 1200 a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} + 1200 a^{10} c^{9} x^{7} + 3840 a^{9} c^{9} x^{6} - 2160 a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} + 1200 a^{6} c^{9} x^{3} - 240 a^{4} c^{9} x + 60 a^{3} c^{9}}"," ",0,"-(-4*a*x + 1)/(60*a**19*c**9*x**16 - 240*a**18*c**9*x**15 + 1200*a**16*c**9*x**13 - 1200*a**15*c**9*x**12 - 2160*a**14*c**9*x**11 + 3840*a**13*c**9*x**10 + 1200*a**12*c**9*x**9 - 5400*a**11*c**9*x**8 + 1200*a**10*c**9*x**7 + 3840*a**9*c**9*x**6 - 2160*a**8*c**9*x**5 - 1200*a**7*c**9*x**4 + 1200*a**6*c**9*x**3 - 240*a**4*c**9*x + 60*a**3*c**9)","B",0
1370,1,48,0,0.349336," ","integrate((a*x+1)**2/(-a**2*x**2+1)*x**2/(-a**2*c*x**2+c)**3,x)","\frac{- 2 a x + 1}{6 a^{7} c^{3} x^{4} - 12 a^{6} c^{3} x^{3} + 12 a^{4} c^{3} x - 6 a^{3} c^{3}}"," ",0,"(-2*a*x + 1)/(6*a**7*c**3*x**4 - 12*a**6*c**3*x**3 + 12*a**4*c**3*x - 6*a**3*c**3)","A",0
1371,1,49,0,0.356321," ","integrate(x**2/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**3,x)","\frac{- 2 a x - 1}{6 a^{7} c^{3} x^{4} + 12 a^{6} c^{3} x^{3} - 12 a^{4} c^{3} x - 6 a^{3} c^{3}}"," ",0,"(-2*a*x - 1)/(6*a**7*c**3*x**4 + 12*a**6*c**3*x**3 - 12*a**4*c**3*x - 6*a**3*c**3)","A",0
1372,1,182,0,1.214533," ","integrate(x**2/(a*x+1)**4*(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**9,x)","- \frac{- 4 a x - 1}{60 a^{19} c^{9} x^{16} + 240 a^{18} c^{9} x^{15} - 1200 a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} + 2160 a^{14} c^{9} x^{11} + 3840 a^{13} c^{9} x^{10} - 1200 a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} - 1200 a^{10} c^{9} x^{7} + 3840 a^{9} c^{9} x^{6} + 2160 a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} - 1200 a^{6} c^{9} x^{3} + 240 a^{4} c^{9} x + 60 a^{3} c^{9}}"," ",0,"-(-4*a*x - 1)/(60*a**19*c**9*x**16 + 240*a**18*c**9*x**15 - 1200*a**16*c**9*x**13 - 1200*a**15*c**9*x**12 + 2160*a**14*c**9*x**11 + 3840*a**13*c**9*x**10 - 1200*a**12*c**9*x**9 - 5400*a**11*c**9*x**8 - 1200*a**10*c**9*x**7 + 3840*a**9*c**9*x**6 + 2160*a**8*c**9*x**5 - 1200*a**7*c**9*x**4 - 1200*a**6*c**9*x**3 + 240*a**4*c**9*x + 60*a**3*c**9)","B",0
1373,-1,0,0,0.000000," ","integrate((a*x+1)**5/(-a**2*x**2+1)**(5/2)*x**2/(-a**2*c*x**2+c)**(27/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1374,-1,0,0,0.000000," ","integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2/(-a**2*c*x**2+c)**(11/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1375,0,0,0,0.000000," ","integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{2} \left(a x + 1\right)}{\sqrt{- \left(a x - 1\right) \left(a x + 1\right)} \left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}}}\, dx"," ",0,"Integral(x**2*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), x)","F",0
1376,0,0,0,0.000000," ","integrate(x**2/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**(3/2),x)","\int \frac{x^{2} \sqrt{- \left(a x - 1\right) \left(a x + 1\right)}}{\left(- c \left(a x - 1\right) \left(a x + 1\right)\right)^{\frac{3}{2}} \left(a x + 1\right)}\, dx"," ",0,"Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))/((-c*(a*x - 1)*(a*x + 1))**(3/2)*(a*x + 1)), x)","F",0
1377,-1,0,0,0.000000," ","integrate(x**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(11/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
1378,-1,0,0,0.000000," ","integrate(x**2/(a*x+1)**5*(-a**2*x**2+1)**(5/2)/(-a**2*c*x**2+c)**(27/2),x)","\text{Timed out}"," ",0,"Timed out","F(-1)",0
