3.1963 \(\int \frac{x^5}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b^2 \log \left (a x^3+b\right )}{3 a^3}-\frac{b x^3}{3 a^2}+\frac{x^6}{6 a} \]

[Out]

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

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Rubi [A]  time = 0.0814482, antiderivative size = 40, 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 \[ \frac{b^2 \log \left (a x^3+b\right )}{3 a^3}-\frac{b x^3}{3 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b/x^3),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0112327, size = 40, normalized size = 1. \[ \frac{b^2 \log \left (a x^3+b\right )}{3 a^3}-\frac{b x^3}{3 a^2}+\frac{x^6}{6 a} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b/x^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \[ -{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{x}^{6}}{6\,a}}+{\frac{{b}^{2}\ln \left ( a{x}^{3}+b \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a+b/x^3),x)

[Out]

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

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Maxima [A]  time = 1.42467, size = 46, normalized size = 1.15 \[ \frac{b^{2} \log \left (a x^{3} + b\right )}{3 \, a^{3}} + \frac{a x^{6} - 2 \, b x^{3}}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22924, size = 45, normalized size = 1.12 \[ \frac{a^{2} x^{6} - 2 \, a b x^{3} + 2 \, b^{2} \log \left (a x^{3} + b\right )}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.37638, size = 32, normalized size = 0.8 \[ \frac{x^{6}}{6 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} \log{\left (a x^{3} + b \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a+b/x**3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.225942, size = 47, normalized size = 1.18 \[ \frac{b^{2}{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, a^{3}} + \frac{a x^{6} - 2 \, b x^{3}}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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