3.161 \(\int \frac{f^{a+\frac{b}{x^3}}}{x^4} \, dx\)

Optimal. Leaf size=20 \[ -\frac{f^{a+\frac{b}{x^3}}}{3 b \log (f)} \]

[Out]

-f^(a + b/x^3)/(3*b*Log[f])

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Rubi [A]  time = 0.0360995, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{f^{a+\frac{b}{x^3}}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x^3)/x^4,x]

[Out]

-f^(a + b/x^3)/(3*b*Log[f])

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Rubi in Sympy [A]  time = 3.40104, size = 15, normalized size = 0.75 \[ - \frac{f^{a + \frac{b}{x^{3}}}}{3 b \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**(a + b/x**3)/(3*b*log(f))

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Mathematica [A]  time = 0.00623711, size = 20, normalized size = 1. \[ -\frac{f^{a+\frac{b}{x^3}}}{3 b \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x^3)/x^4,x]

[Out]

-f^(a + b/x^3)/(3*b*Log[f])

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \[ -{\frac{1}{3\,b\ln \left ( f \right ) }{f}^{a+{\frac{b}{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x^3)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.766235, size = 24, normalized size = 1.2 \[ -\frac{f^{a + \frac{b}{x^{3}}}}{3 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^4,x, algorithm="maxima")

[Out]

-1/3*f^(a + b/x^3)/(b*log(f))

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Fricas [A]  time = 0.244846, size = 30, normalized size = 1.5 \[ -\frac{f^{\frac{a x^{3} + b}{x^{3}}}}{3 \, b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^4,x, algorithm="fricas")

[Out]

-1/3*f^((a*x^3 + b)/x^3)/(b*log(f))

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Sympy [A]  time = 0.215427, size = 29, normalized size = 1.45 \[ \begin{cases} - \frac{f^{a + \frac{b}{x^{3}}}}{3 b \log{\left (f \right )}} & \text{for}\: 3 b \log{\left (f \right )} \neq 0 \\- \frac{1}{3 x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x**3)/x**4,x)

[Out]

Piecewise((-f**(a + b/x**3)/(3*b*log(f)), Ne(3*b*log(f), 0)), (-1/(3*x**3), True
))

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GIAC/XCAS [A]  time = 0.245394, size = 24, normalized size = 1.2 \[ -\frac{f^{a + \frac{b}{x^{3}}}}{3 \, b{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x^3)/x^4,x, algorithm="giac")

[Out]

-1/3*f^(a + b/x^3)/(b*ln(f))