3.134 \(\int \frac{x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=273 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^7 \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac{x \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{8 b^6}-\frac{x^3 \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{24 a b^5}+\frac{x^5 \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{40 a^2 b^4}+\frac{f x^7}{7 b^3} \]

[Out]

((15*b^3*c - 35*a*b^2*d + 63*a^2*b*e - 99*a^3*f)*x)/(8*b^6) - ((15*b^3*c - 35*a*
b^2*d + 63*a^2*b*e - 99*a^3*f)*x^3)/(24*a*b^5) + ((15*b^3*c - 35*a*b^2*d + 63*a^
2*b*e - 99*a^3*f)*x^5)/(40*a^2*b^4) + (f*x^7)/(7*b^3) + ((c - (a*(b^2*d - a*b*e
+ a^2*f))/b^3)*x^7)/(4*a*(a + b*x^2)^2) - ((3*b^3*c - 7*a*b^2*d + 11*a^2*b*e - 1
5*a^3*f)*x^7)/(8*a^2*b^3*(a + b*x^2)) - (Sqrt[a]*(15*b^3*c - 35*a*b^2*d + 63*a^2
*b*e - 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Rubi [A]  time = 0.966788, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^7 \left (-15 a^3 f+11 a^2 b e-7 a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac{x \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{8 b^6}-\frac{x^3 \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{24 a b^5}+\frac{x^5 \left (-99 a^3 f+63 a^2 b e-35 a b^2 d+15 b^3 c\right )}{40 a^2 b^4}+\frac{f x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]

[Out]

((15*b^3*c - 35*a*b^2*d + 63*a^2*b*e - 99*a^3*f)*x)/(8*b^6) - ((15*b^3*c - 35*a*
b^2*d + 63*a^2*b*e - 99*a^3*f)*x^3)/(24*a*b^5) + ((15*b^3*c - 35*a*b^2*d + 63*a^
2*b*e - 99*a^3*f)*x^5)/(40*a^2*b^4) + (f*x^7)/(7*b^3) + ((c - (a*(b^2*d - a*b*e
+ a^2*f))/b^3)*x^7)/(4*a*(a + b*x^2)^2) - ((3*b^3*c - 7*a*b^2*d + 11*a^2*b*e - 1
5*a^3*f)*x^7)/(8*a^2*b^3*(a + b*x^2)) - (Sqrt[a]*(15*b^3*c - 35*a*b^2*d + 63*a^2
*b*e - 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.291528, size = 232, normalized size = 0.85 \[ \frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (99 a^3 f-63 a^2 b e+35 a b^2 d-15 b^3 c\right )}{8 b^{13/2}}+\frac{a x \left (-21 a^3 f+17 a^2 b e-13 a b^2 d+9 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac{a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 b^6 \left (a+b x^2\right )^2}+\frac{x \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{b^6}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^3,x]

[Out]

((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x)/b^6 + ((b^2*d - 3*a*b*e + 6*a^2*f
)*x^3)/(3*b^5) + ((b*e - 3*a*f)*x^5)/(5*b^4) + (f*x^7)/(7*b^3) + (a^2*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f)*x)/(4*b^6*(a + b*x^2)^2) + (a*(9*b^3*c - 13*a*b^2*d
+ 17*a^2*b*e - 21*a^3*f)*x)/(8*b^6*(a + b*x^2)) + (Sqrt[a]*(-15*b^3*c + 35*a*b^
2*d - 63*a^2*b*e + 99*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Maple [A]  time = 0.019, size = 343, normalized size = 1.3 \[{\frac{f{x}^{7}}{7\,{b}^{3}}}-{\frac{3\,{x}^{5}af}{5\,{b}^{4}}}+{\frac{{x}^{5}e}{5\,{b}^{3}}}+2\,{\frac{{x}^{3}{a}^{2}f}{{b}^{5}}}-{\frac{a{x}^{3}e}{{b}^{4}}}+{\frac{{x}^{3}d}{3\,{b}^{3}}}-10\,{\frac{{a}^{3}fx}{{b}^{6}}}+6\,{\frac{{a}^{2}ex}{{b}^{5}}}-3\,{\frac{adx}{{b}^{4}}}+{\frac{cx}{{b}^{3}}}-{\frac{21\,{a}^{4}{x}^{3}f}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{17\,{a}^{3}{x}^{3}e}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}d}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}c}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}fx}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{a}^{4}ex}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,{a}^{3}dx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}cx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,f{a}^{4}}{8\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{a}^{3}e}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}d}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ac}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^3,x)

[Out]

1/7*f*x^7/b^3-3/5/b^4*x^5*a*f+1/5/b^3*x^5*e+2/b^5*x^3*a^2*f-1/b^4*x^3*a*e+1/3/b^
3*x^3*d-10/b^6*a^3*f*x+6/b^5*a^2*e*x-3/b^4*a*d*x+1/b^3*c*x-21/8*a^4/b^5/(b*x^2+a
)^2*x^3*f+17/8*a^3/b^4/(b*x^2+a)^2*x^3*e-13/8*a^2/b^3/(b*x^2+a)^2*x^3*d+9/8*a/b^
2/(b*x^2+a)^2*x^3*c-19/8*a^5/b^6/(b*x^2+a)^2*f*x+15/8*a^4/b^5/(b*x^2+a)^2*e*x-11
/8*a^3/b^4/(b*x^2+a)^2*d*x+7/8*a^2/b^3/(b*x^2+a)^2*c*x+99/8*a^4/b^6/(a*b)^(1/2)*
arctan(x*b/(a*b)^(1/2))*f-63/8*a^3/b^5/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e+35/
8*a^2/b^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d-15/8*a/b^3/(a*b)^(1/2)*arctan(x*
b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235125, size = 1, normalized size = 0. \[ \left [\frac{240 \, b^{5} f x^{11} + 48 \,{\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 16 \,{\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 112 \,{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 350 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f +{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 210 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{1680 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac{120 \, b^{5} f x^{11} + 24 \,{\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 8 \,{\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 56 \,{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 175 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f +{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) + 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{840 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[1/1680*(240*b^5*f*x^11 + 48*(7*b^5*e - 11*a*b^4*f)*x^9 + 16*(35*b^5*d - 63*a*b^
4*e + 99*a^2*b^3*f)*x^7 + 112*(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2
*f)*x^5 + 350*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)*x^3 - 105*
(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + (15*b^5*c - 35*a*b^4*d +
63*a^2*b^3*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e -
99*a^4*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2
10*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x
^2 + a^2*b^6), 1/840*(120*b^5*f*x^11 + 24*(7*b^5*e - 11*a*b^4*f)*x^9 + 8*(35*b^5
*d - 63*a*b^4*e + 99*a^2*b^3*f)*x^7 + 56*(15*b^5*c - 35*a*b^4*d + 63*a^2*b^3*e -
99*a^3*b^2*f)*x^5 + 175*(15*a*b^4*c - 35*a^2*b^3*d + 63*a^3*b^2*e - 99*a^4*b*f)
*x^3 - 105*(15*a^2*b^3*c - 35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f + (15*b^5*c - 35
*a*b^4*d + 63*a^2*b^3*e - 99*a^3*b^2*f)*x^4 + 2*(15*a*b^4*c - 35*a^2*b^3*d + 63*
a^3*b^2*e - 99*a^4*b*f)*x^2)*sqrt(a/b)*arctan(x/sqrt(a/b)) + 105*(15*a^2*b^3*c -
35*a^3*b^2*d + 63*a^4*b*e - 99*a^5*f)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)]

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Sympy [A]  time = 25.271, size = 311, normalized size = 1.14 \[ - \frac{\sqrt{- \frac{a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log{\left (- b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{16} + \frac{\sqrt{- \frac{a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log{\left (b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{16} - \frac{x^{3} \left (21 a^{4} b f - 17 a^{3} b^{2} e + 13 a^{2} b^{3} d - 9 a b^{4} c\right ) + x \left (19 a^{5} f - 15 a^{4} b e + 11 a^{3} b^{2} d - 7 a^{2} b^{3} c\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac{f x^{7}}{7 b^{3}} - \frac{x^{5} \left (3 a f - b e\right )}{5 b^{4}} + \frac{x^{3} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{3 b^{5}} - \frac{x \left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**3,x)

[Out]

-sqrt(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b**2*d - 15*b**3*c)*log(-b**6*sq
rt(-a/b**13) + x)/16 + sqrt(-a/b**13)*(99*a**3*f - 63*a**2*b*e + 35*a*b**2*d - 1
5*b**3*c)*log(b**6*sqrt(-a/b**13) + x)/16 - (x**3*(21*a**4*b*f - 17*a**3*b**2*e
+ 13*a**2*b**3*d - 9*a*b**4*c) + x*(19*a**5*f - 15*a**4*b*e + 11*a**3*b**2*d - 7
*a**2*b**3*c))/(8*a**2*b**6 + 16*a*b**7*x**2 + 8*b**8*x**4) + f*x**7/(7*b**3) -
x**5*(3*a*f - b*e)/(5*b**4) + x**3*(6*a**2*f - 3*a*b*e + b**2*d)/(3*b**5) - x*(1
0*a**3*f - 6*a**2*b*e + 3*a*b**2*d - b**3*c)/b**6

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GIAC/XCAS [A]  time = 0.218793, size = 338, normalized size = 1.24 \[ -\frac{{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d - 99 \, a^{4} f + 63 \, a^{3} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{6}} + \frac{9 \, a b^{4} c x^{3} - 13 \, a^{2} b^{3} d x^{3} - 21 \, a^{4} b f x^{3} + 17 \, a^{3} b^{2} x^{3} e + 7 \, a^{2} b^{3} c x - 11 \, a^{3} b^{2} d x - 19 \, a^{5} f x + 15 \, a^{4} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{15 \, b^{18} f x^{7} - 63 \, a b^{17} f x^{5} + 21 \, b^{18} x^{5} e + 35 \, b^{18} d x^{3} + 210 \, a^{2} b^{16} f x^{3} - 105 \, a b^{17} x^{3} e + 105 \, b^{18} c x - 315 \, a b^{17} d x - 1050 \, a^{3} b^{15} f x + 630 \, a^{2} b^{16} x e}{105 \, b^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

-1/8*(15*a*b^3*c - 35*a^2*b^2*d - 99*a^4*f + 63*a^3*b*e)*arctan(b*x/sqrt(a*b))/(
sqrt(a*b)*b^6) + 1/8*(9*a*b^4*c*x^3 - 13*a^2*b^3*d*x^3 - 21*a^4*b*f*x^3 + 17*a^3
*b^2*x^3*e + 7*a^2*b^3*c*x - 11*a^3*b^2*d*x - 19*a^5*f*x + 15*a^4*b*x*e)/((b*x^2
+ a)^2*b^6) + 1/105*(15*b^18*f*x^7 - 63*a*b^17*f*x^5 + 21*b^18*x^5*e + 35*b^18*
d*x^3 + 210*a^2*b^16*f*x^3 - 105*a*b^17*x^3*e + 105*b^18*c*x - 315*a*b^17*d*x -
1050*a^3*b^15*f*x + 630*a^2*b^16*x*e)/b^21