3.122 \(\int \frac{f^{a+\frac{b}{x}}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{f^{a+\frac{b}{x}}}{b \log (f)} \]

[Out]

-(f^(a + b/x)/(b*Log[f]))

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Rubi [A]  time = 0.0304173, antiderivative size = 18, 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 \[ -\frac{f^{a+\frac{b}{x}}}{b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)/x^2,x]

[Out]

-(f^(a + b/x)/(b*Log[f]))

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Rubi in Sympy [A]  time = 3.42955, size = 12, normalized size = 0.67 \[ - \frac{f^{a + \frac{b}{x}}}{b \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x)/x**2,x)

[Out]

-f**(a + b/x)/(b*log(f))

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Mathematica [A]  time = 0.00520356, size = 18, normalized size = 1. \[ -\frac{f^{a+\frac{b}{x}}}{b \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)/x^2,x]

[Out]

-(f^(a + b/x)/(b*Log[f]))

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Maple [A]  time = 0.003, size = 19, normalized size = 1.1 \[ -{\frac{1}{b\ln \left ( f \right ) }{f}^{a+{\frac{b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)/x^2,x)

[Out]

-f^(a+b/x)/b/ln(f)

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Maxima [A]  time = 0.923425, size = 24, normalized size = 1.33 \[ -\frac{f^{a + \frac{b}{x}}}{b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^2,x, algorithm="maxima")

[Out]

-f^(a + b/x)/(b*log(f))

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Fricas [A]  time = 0.26732, size = 27, normalized size = 1.5 \[ -\frac{f^{\frac{a x + b}{x}}}{b \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^2,x, algorithm="fricas")

[Out]

-f^((a*x + b)/x)/(b*log(f))

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Sympy [A]  time = 0.228775, size = 20, normalized size = 1.11 \[ \begin{cases} - \frac{f^{a + \frac{b}{x}}}{b \log{\left (f \right )}} & \text{for}\: b \log{\left (f \right )} \neq 0 \\- \frac{1}{x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.24468, size = 24, normalized size = 1.33 \[ -\frac{f^{a + \frac{b}{x}}}{b{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x^2,x, algorithm="giac")

[Out]

-f^(a + b/x)/(b*ln(f))