3.1976 \(\int \frac{x^5}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]

[Out]

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

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Rubi [A]  time = 0.114911, antiderivative size = 56, 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 \[ \frac{b^3}{3 a^4 \left (a x^3+b\right )}+\frac{b^2 \log \left (a x^3+b\right )}{a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x^3)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0317439, size = 49, normalized size = 0.88 \[ \frac{a^2 x^6+\frac{2 b^3}{a x^3+b}+6 b^2 \log \left (a x^3+b\right )-4 a b x^3}{6 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.008, size = 51, normalized size = 0.9 \[ -{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{3}}{3\,{a}^{4} \left ( a{x}^{3}+b \right ) }}+{\frac{{b}^{2}\ln \left ( a{x}^{3}+b \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x^3)^2,x)

[Out]

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

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Maxima [A]  time = 1.43814, size = 72, normalized size = 1.29 \[ \frac{b^{3}}{3 \,{\left (a^{5} x^{3} + a^{4} b\right )}} + \frac{b^{2} \log \left (a x^{3} + b\right )}{a^{4}} + \frac{a x^{6} - 4 \, b x^{3}}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(a + b/x^3)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22113, size = 95, normalized size = 1.7 \[ \frac{a^{3} x^{9} - 3 \, a^{2} b x^{6} - 4 \, a b^{2} x^{3} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{3} + b^{3}\right )} \log \left (a x^{3} + b\right )}{6 \,{\left (a^{5} x^{3} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.84176, size = 53, normalized size = 0.95 \[ \frac{b^{3}}{3 a^{5} x^{3} + 3 a^{4} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{b^{2} \log{\left (a x^{3} + b \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x**3)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.236479, size = 73, normalized size = 1.3 \[ \frac{b^{2}{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{a^{4}} + \frac{b^{3}}{3 \,{\left (a x^{3} + b\right )} a^{4}} + \frac{a^{2} x^{6} - 4 \, a b x^{3}}{6 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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