3.356 \(\int \frac{1}{x^4 \left (a-b x^3\right )} \, dx\)

Optimal. Leaf size=35 \[ -\frac{b \log \left (a-b x^3\right )}{3 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) + (b*Log[x])/a^2 - (b*Log[a - b*x^3])/(3*a^2)

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Rubi [A]  time = 0.0547398, antiderivative size = 35, 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 \[ -\frac{b \log \left (a-b x^3\right )}{3 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a - b*x^3)),x]

[Out]

-1/(3*a*x^3) + (b*Log[x])/a^2 - (b*Log[a - b*x^3])/(3*a^2)

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Rubi in Sympy [A]  time = 8.91449, size = 34, normalized size = 0.97 \[ - \frac{1}{3 a x^{3}} + \frac{b \log{\left (x^{3} \right )}}{3 a^{2}} - \frac{b \log{\left (a - b x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.0146619, size = 35, normalized size = 1. \[ -\frac{b \log \left (a-b x^3\right )}{3 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a - b*x^3)),x]

[Out]

-1/(3*a*x^3) + (b*Log[x])/a^2 - (b*Log[a - b*x^3])/(3*a^2)

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Maple [A]  time = 0.009, size = 33, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{b\ln \left ( x \right ) }{{a}^{2}}}-{\frac{b\ln \left ( b{x}^{3}-a \right ) }{3\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 1.45052, size = 47, normalized size = 1.34 \[ -\frac{b \log \left (b x^{3} - a\right )}{3 \, a^{2}} + \frac{b \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^3 - a)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.257425, size = 45, normalized size = 1.29 \[ -\frac{b x^{3} \log \left (b x^{3} - a\right ) - 3 \, b x^{3} \log \left (x\right ) + a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^3 - a)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.83954, size = 31, normalized size = 0.89 \[ - \frac{1}{3 a x^{3}} + \frac{b \log{\left (x \right )}}{a^{2}} - \frac{b \log{\left (- \frac{a}{b} + x^{3} \right )}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.25075, size = 55, normalized size = 1.57 \[ -\frac{b{\rm ln}\left ({\left | b x^{3} - a \right |}\right )}{3 \, a^{2}} + \frac{b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{b x^{3} + a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^3 - a)*x^4),x, algorithm="giac")

[Out]

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