3.353 \(\int \frac{x^5}{a-b x^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a \log \left (a-b x^3\right )}{3 b^2}-\frac{x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0492844, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{a \log \left (a-b x^3\right )}{3 b^2}-\frac{x^3}{3 b} \]

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 = 0.00812949, size = 28, normalized size = 1. \[ -\frac{a \log \left (a-b x^3\right )}{3 b^2}-\frac{x^3}{3 b} \]

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 = 26, 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/b^2*ln(b*x^3-a)

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Maxima [A]  time = 1.43349, size = 34, normalized size = 1.21 \[ -\frac{x^{3}}{3 \, b} - \frac{a \log \left (b x^{3} - a\right )}{3 \, b^{2}} \]

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 = 0.241833, size = 31, 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 = 1.26995, size = 22, normalized size = 0.79 \[ - \frac{a \log{\left (- a + b x^{3} \right )}}{3 b^{2}} - \frac{x^{3}}{3 b} \]

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 = 0.241904, size = 35, normalized size = 1.25 \[ -\frac{x^{3}}{3 \, b} - \frac{a{\rm ln}\left ({\left | b x^{3} - a \right |}\right )}{3 \, b^{2}} \]

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