3.1857 \(\int \frac{x^5}{\left (a+\frac{b}{x^2}\right )^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b^4}{2 a^5 \left (a x^2+b\right )}-\frac{2 b^3 \log \left (a x^2+b\right )}{a^5}+\frac{3 b^2 x^2}{2 a^4}-\frac{b x^4}{2 a^3}+\frac{x^6}{6 a^2} \]

[Out]

(3*b^2*x^2)/(2*a^4) - (b*x^4)/(2*a^3) + x^6/(6*a^2) - b^4/(2*a^5*(b + a*x^2)) -
(2*b^3*Log[b + a*x^2])/a^5

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Rubi [A]  time = 0.140501, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^4}{2 a^5 \left (a x^2+b\right )}-\frac{2 b^3 \log \left (a x^2+b\right )}{a^5}+\frac{3 b^2 x^2}{2 a^4}-\frac{b x^4}{2 a^3}+\frac{x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x^2)^2,x]

[Out]

(3*b^2*x^2)/(2*a^4) - (b*x^4)/(2*a^3) + x^6/(6*a^2) - b^4/(2*a^5*(b + a*x^2)) -
(2*b^3*Log[b + a*x^2])/a^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{6}}{6 a^{2}} - \frac{b \int ^{x^{2}} x\, dx}{a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{b^{4}}{2 a^{5} \left (a x^{2} + b\right )} - \frac{2 b^{3} \log{\left (a x^{2} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6/(6*a**2) - b*Integral(x, (x, x**2))/a**3 + 3*b**2*x**2/(2*a**4) - b**4/(2*a
**5*(a*x**2 + b)) - 2*b**3*log(a*x**2 + b)/a**5

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Mathematica [A]  time = 0.0382696, size = 60, normalized size = 0.86 \[ \frac{a^3 x^6-3 a^2 b x^4-\frac{3 b^4}{a x^2+b}-12 b^3 \log \left (a x^2+b\right )+9 a b^2 x^2}{6 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x^2)^2,x]

[Out]

(9*a*b^2*x^2 - 3*a^2*b*x^4 + a^3*x^6 - (3*b^4)/(b + a*x^2) - 12*b^3*Log[b + a*x^
2])/(6*a^5)

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Maple [A]  time = 0.014, size = 63, normalized size = 0.9 \[{\frac{3\,{b}^{2}{x}^{2}}{2\,{a}^{4}}}-{\frac{b{x}^{4}}{2\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}-{\frac{{b}^{4}}{2\,{a}^{5} \left ( a{x}^{2}+b \right ) }}-2\,{\frac{{b}^{3}\ln \left ( a{x}^{2}+b \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x^2)^2,x)

[Out]

3/2*b^2*x^2/a^4-1/2*b*x^4/a^3+1/6*x^6/a^2-1/2*b^4/a^5/(a*x^2+b)-2*b^3*ln(a*x^2+b
)/a^5

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Maxima [A]  time = 1.44029, size = 88, normalized size = 1.26 \[ -\frac{b^{4}}{2 \,{\left (a^{6} x^{2} + a^{5} b\right )}} - \frac{2 \, b^{3} \log \left (a x^{2} + b\right )}{a^{5}} + \frac{a^{2} x^{6} - 3 \, a b x^{4} + 9 \, b^{2} x^{2}}{6 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x^2)^2,x, algorithm="maxima")

[Out]

-1/2*b^4/(a^6*x^2 + a^5*b) - 2*b^3*log(a*x^2 + b)/a^5 + 1/6*(a^2*x^6 - 3*a*b*x^4
+ 9*b^2*x^2)/a^4

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Fricas [A]  time = 0.223944, size = 109, normalized size = 1.56 \[ \frac{a^{4} x^{8} - 2 \, a^{3} b x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a b^{3} x^{2} - 3 \, b^{4} - 12 \,{\left (a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{6 \,{\left (a^{6} x^{2} + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(a^4*x^8 - 2*a^3*b*x^6 + 6*a^2*b^2*x^4 + 9*a*b^3*x^2 - 3*b^4 - 12*(a*b^3*x^2
+ b^4)*log(a*x^2 + b))/(a^6*x^2 + a^5*b)

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Sympy [A]  time = 1.6935, size = 66, normalized size = 0.94 \[ - \frac{b^{4}}{2 a^{6} x^{2} + 2 a^{5} b} + \frac{x^{6}}{6 a^{2}} - \frac{b x^{4}}{2 a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{2 b^{3} \log{\left (a x^{2} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x**2)**2,x)

[Out]

-b**4/(2*a**6*x**2 + 2*a**5*b) + x**6/(6*a**2) - b*x**4/(2*a**3) + 3*b**2*x**2/(
2*a**4) - 2*b**3*log(a*x**2 + b)/a**5

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GIAC/XCAS [A]  time = 0.21858, size = 108, normalized size = 1.54 \[ -\frac{2 \, b^{3}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac{a^{4} x^{6} - 3 \, a^{3} b x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, a^{6}} + \frac{4 \, a b^{3} x^{2} + 3 \, b^{4}}{2 \,{\left (a x^{2} + b\right )} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b^3*ln(abs(a*x^2 + b))/a^5 + 1/6*(a^4*x^6 - 3*a^3*b*x^4 + 9*a^2*b^2*x^2)/a^6
+ 1/2*(4*a*b^3*x^2 + 3*b^4)/((a*x^2 + b)*a^5)