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

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 = 0.0537031, antiderivative size = 35, 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 \[ \frac{b \log \left (a+b x^3\right )}{3 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

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 = 8.32935, 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 = 0.0116304, 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 = 32, 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*ln(b*x^3+a)/a^2

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Maxima [A]  time = 1.43135, size = 45, normalized size = 1.29 \[ \frac{b \log \left (b x^{3} + a\right )}{3 \, a^{2}} - \frac{b \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]

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 = 0.218282, 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 = 1.79027, 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 = 0.220272, size = 57, normalized size = 1.63 \[ \frac{b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac{b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} - a}{3 \, a^{2} x^{3}} \]

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)