3.1975 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0661373, antiderivative size = 35, 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 \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.21433, size = 34, normalized size = 0.97 \[ - \frac{a \log{\left (x^{3} \right )}}{3 b^{2}} + \frac{a \log{\left (a x^{3} + b \right )}}{3 b^{2}} - \frac{1}{3 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0127865, size = 35, normalized size = 1. \[ \frac{a \log \left (a x^3+b\right )}{3 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)*x^7),x]

[Out]

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

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Maple [A]  time = 0.008, size = 32, normalized size = 0.9 \[ -{\frac{1}{3\,b{x}^{3}}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( a{x}^{3}+b \right ) }{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.42775, size = 45, normalized size = 1.29 \[ \frac{a \log \left (a x^{3} + b\right )}{3 \, b^{2}} - \frac{a \log \left (x^{3}\right )}{3 \, b^{2}} - \frac{1}{3 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225132, size = 45, normalized size = 1.29 \[ \frac{a x^{3} \log \left (a x^{3} + b\right ) - 3 \, a x^{3} \log \left (x\right ) - b}{3 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.86828, size = 31, normalized size = 0.89 \[ - \frac{a \log{\left (x \right )}}{b^{2}} + \frac{a \log{\left (x^{3} + \frac{b}{a} \right )}}{3 b^{2}} - \frac{1}{3 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222301, size = 57, normalized size = 1.63 \[ \frac{a{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} - \frac{a{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} + \frac{a x^{3} - b}{3 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^7),x, algorithm="giac")

[Out]

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