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

Optimal. Leaf size=27 \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0440447, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{\int ^{x^{3}} \frac{1}{b}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*log(a + b*x**3)/(3*b**2) + Integral(1/b, (x, x**3))/3

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Mathematica [A]  time = 0.00711098, size = 27, normalized size = 1. \[ \frac{x^3}{3 b}-\frac{a \log \left (a+b x^3\right )}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,b}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.44499, size = 31, normalized size = 1.15 \[ \frac{x^{3}}{3 \, b} - \frac{a \log \left (b x^{3} + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3/b - 1/3*a*log(b*x^3 + a)/b^2

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Fricas [A]  time = 0.223957, size = 30, normalized size = 1.11 \[ \frac{b x^{3} - a \log \left (b x^{3} + a\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^3 - a*log(b*x^3 + a))/b^2

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Sympy [A]  time = 1.25572, size = 20, normalized size = 0.74 \[ - \frac{a \log{\left (a + b x^{3} \right )}}{3 b^{2}} + \frac{x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220352, size = 32, normalized size = 1.19 \[ \frac{x^{3}}{3 \, b} - \frac{a{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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