3.776 \(\int \frac{x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=195 \[ \frac{(7 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{5/2} b^{9/2}}+\frac{\sqrt{x} (7 a B+3 A b)}{128 a^2 b^4 (a+b x)}-\frac{\sqrt{x} (7 a B+3 A b)}{64 a b^4 (a+b x)^2}-\frac{x^{3/2} (7 a B+3 A b)}{48 a b^3 (a+b x)^3}-\frac{x^{5/2} (7 a B+3 A b)}{40 a b^2 (a+b x)^4}+\frac{x^{7/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

((A*b - a*B)*x^(7/2))/(5*a*b*(a + b*x)^5) - ((3*A*b + 7*a*B)*x^(5/2))/(40*a*b^2*
(a + b*x)^4) - ((3*A*b + 7*a*B)*x^(3/2))/(48*a*b^3*(a + b*x)^3) - ((3*A*b + 7*a*
B)*Sqrt[x])/(64*a*b^4*(a + b*x)^2) + ((3*A*b + 7*a*B)*Sqrt[x])/(128*a^2*b^4*(a +
b*x)) + ((3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(5/2)*b^(9/2
))

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Rubi [A]  time = 0.228144, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(7 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{5/2} b^{9/2}}+\frac{\sqrt{x} (7 a B+3 A b)}{128 a^2 b^4 (a+b x)}-\frac{\sqrt{x} (7 a B+3 A b)}{64 a b^4 (a+b x)^2}-\frac{x^{3/2} (7 a B+3 A b)}{48 a b^3 (a+b x)^3}-\frac{x^{5/2} (7 a B+3 A b)}{40 a b^2 (a+b x)^4}+\frac{x^{7/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((A*b - a*B)*x^(7/2))/(5*a*b*(a + b*x)^5) - ((3*A*b + 7*a*B)*x^(5/2))/(40*a*b^2*
(a + b*x)^4) - ((3*A*b + 7*a*B)*x^(3/2))/(48*a*b^3*(a + b*x)^3) - ((3*A*b + 7*a*
B)*Sqrt[x])/(64*a*b^4*(a + b*x)^2) + ((3*A*b + 7*a*B)*Sqrt[x])/(128*a^2*b^4*(a +
b*x)) + ((3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(5/2)*b^(9/2
))

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Rubi in Sympy [A]  time = 54.6765, size = 177, normalized size = 0.91 \[ \frac{x^{\frac{7}{2}} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} - \frac{x^{\frac{5}{2}} \left (3 A b + 7 B a\right )}{40 a b^{2} \left (a + b x\right )^{4}} - \frac{x^{\frac{3}{2}} \left (3 A b + 7 B a\right )}{48 a b^{3} \left (a + b x\right )^{3}} - \frac{\sqrt{x} \left (3 A b + 7 B a\right )}{64 a b^{4} \left (a + b x\right )^{2}} + \frac{\sqrt{x} \left (3 A b + 7 B a\right )}{128 a^{2} b^{4} \left (a + b x\right )} + \frac{\left (3 A b + 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{5}{2}} b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

x**(7/2)*(A*b - B*a)/(5*a*b*(a + b*x)**5) - x**(5/2)*(3*A*b + 7*B*a)/(40*a*b**2*
(a + b*x)**4) - x**(3/2)*(3*A*b + 7*B*a)/(48*a*b**3*(a + b*x)**3) - sqrt(x)*(3*A
*b + 7*B*a)/(64*a*b**4*(a + b*x)**2) + sqrt(x)*(3*A*b + 7*B*a)/(128*a**2*b**4*(a
+ b*x)) + (3*A*b + 7*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(128*a**(5/2)*b**(9/2))

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Mathematica [A]  time = 0.346919, size = 146, normalized size = 0.75 \[ \frac{\frac{15 (7 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{\sqrt{b} \sqrt{x} \left (-105 a^5 B-5 a^4 b (9 A+98 B x)-14 a^3 b^2 x (15 A+64 B x)-2 a^2 b^3 x^2 (192 A+395 B x)+105 a b^4 x^3 (2 A+B x)+45 A b^5 x^4\right )}{a^2 (a+b x)^5}}{1920 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((Sqrt[b]*Sqrt[x]*(-105*a^5*B + 45*A*b^5*x^4 + 105*a*b^4*x^3*(2*A + B*x) - 14*a^
3*b^2*x*(15*A + 64*B*x) - 5*a^4*b*(9*A + 98*B*x) - 2*a^2*b^3*x^2*(192*A + 395*B*
x)))/(a^2*(a + b*x)^5) + (15*(3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/
a^(5/2))/(1920*b^(9/2))

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Maple [A]  time = 0.026, size = 154, normalized size = 0.8 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 3\,Ab+7\,Ba \right ){x}^{9/2}}{256\,{a}^{2}}}+{\frac{ \left ( 21\,Ab-79\,Ba \right ){x}^{7/2}}{384\,ab}}-1/30\,{\frac{ \left ( 3\,Ab+7\,Ba \right ){x}^{5/2}}{{b}^{2}}}-{\frac{7\,a \left ( 3\,Ab+7\,Ba \right ){x}^{3/2}}{384\,{b}^{3}}}-{\frac{ \left ( 3\,Ab+7\,Ba \right ){a}^{2}\sqrt{x}}{256\,{b}^{4}}} \right ) }+{\frac{3\,A}{128\,{a}^{2}{b}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{7\,B}{128\,a{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*(1/256*(3*A*b+7*B*a)/a^2*x^(9/2)+1/384*(21*A*b-79*B*a)/a/b*x^(7/2)-1/30/b^2*(3
*A*b+7*B*a)*x^(5/2)-7/384*a/b^3*(3*A*b+7*B*a)*x^(3/2)-1/256*(3*A*b+7*B*a)*a^2/b^
4*x^(1/2))/(b*x+a)^5+3/128/a^2/b^3/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+7
/128/a/b^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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((B*x + A)*x^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310931, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (105 \, B a^{5} + 45 \, A a^{4} b - 15 \,{\left (7 \, B a b^{4} + 3 \, A b^{5}\right )} x^{4} + 10 \,{\left (79 \, B a^{2} b^{3} - 21 \, A a b^{4}\right )} x^{3} + 128 \,{\left (7 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{2} + 70 \,{\left (7 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 15 \,{\left (7 \, B a^{6} + 3 \, A a^{5} b +{\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \,{\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \,{\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{3840 \,{\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )} \sqrt{-a b}}, -\frac{{\left (105 \, B a^{5} + 45 \, A a^{4} b - 15 \,{\left (7 \, B a b^{4} + 3 \, A b^{5}\right )} x^{4} + 10 \,{\left (79 \, B a^{2} b^{3} - 21 \, A a b^{4}\right )} x^{3} + 128 \,{\left (7 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{2} + 70 \,{\left (7 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 15 \,{\left (7 \, B a^{6} + 3 \, A a^{5} b +{\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \,{\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \,{\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{1920 \,{\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(2*(105*B*a^5 + 45*A*a^4*b - 15*(7*B*a*b^4 + 3*A*b^5)*x^4 + 10*(79*B*a^
2*b^3 - 21*A*a*b^4)*x^3 + 128*(7*B*a^3*b^2 + 3*A*a^2*b^3)*x^2 + 70*(7*B*a^4*b +
3*A*a^3*b^2)*x)*sqrt(-a*b)*sqrt(x) - 15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A*
b^6)*x^5 + 5*(7*B*a^2*b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^4)*x^3
+ 10*(7*B*a^4*b^2 + 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)*x)*log((2*a*b
*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a^2*b^9*x^5 + 5*a^3*b^8*x^4 + 10*
a^4*b^7*x^3 + 10*a^5*b^6*x^2 + 5*a^6*b^5*x + a^7*b^4)*sqrt(-a*b)), -1/1920*((105
*B*a^5 + 45*A*a^4*b - 15*(7*B*a*b^4 + 3*A*b^5)*x^4 + 10*(79*B*a^2*b^3 - 21*A*a*b
^4)*x^3 + 128*(7*B*a^3*b^2 + 3*A*a^2*b^3)*x^2 + 70*(7*B*a^4*b + 3*A*a^3*b^2)*x)*
sqrt(a*b)*sqrt(x) + 15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A*b^6)*x^5 + 5*(7*B
*a^2*b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^4)*x^3 + 10*(7*B*a^4*b^2
+ 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x)
)))/((a^2*b^9*x^5 + 5*a^3*b^8*x^4 + 10*a^4*b^7*x^3 + 10*a^5*b^6*x^2 + 5*a^6*b^5*
x + a^7*b^4)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\left (a + b x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Integral(x**(5/2)*(A + B*x)/(a + b*x)**6, x)

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GIAC/XCAS [A]  time = 0.272443, size = 211, normalized size = 1.08 \[ \frac{{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{2} b^{4}} + \frac{105 \, B a b^{4} x^{\frac{9}{2}} + 45 \, A b^{5} x^{\frac{9}{2}} - 790 \, B a^{2} b^{3} x^{\frac{7}{2}} + 210 \, A a b^{4} x^{\frac{7}{2}} - 896 \, B a^{3} b^{2} x^{\frac{5}{2}} - 384 \, A a^{2} b^{3} x^{\frac{5}{2}} - 490 \, B a^{4} b x^{\frac{3}{2}} - 210 \, A a^{3} b^{2} x^{\frac{3}{2}} - 105 \, B a^{5} \sqrt{x} - 45 \, A a^{4} b \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")

[Out]

1/128*(7*B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^4) + 1/1920*(
105*B*a*b^4*x^(9/2) + 45*A*b^5*x^(9/2) - 790*B*a^2*b^3*x^(7/2) + 210*A*a*b^4*x^(
7/2) - 896*B*a^3*b^2*x^(5/2) - 384*A*a^2*b^3*x^(5/2) - 490*B*a^4*b*x^(3/2) - 210
*A*a^3*b^2*x^(3/2) - 105*B*a^5*sqrt(x) - 45*A*a^4*b*sqrt(x))/((b*x + a)^5*a^2*b^
4)