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

Optimal. Leaf size=39 \[ \frac{f^{a+\frac{b}{x}}}{b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0625852, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{f^{a+\frac{b}{x}}}{b^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.6541, size = 27, normalized size = 0.69 \[ - \frac{f^{a + \frac{b}{x}}}{b x \log{\left (f \right )}} + \frac{f^{a + \frac{b}{x}}}{b^{2} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**(a + b/x)/(b*x*log(f)) + f**(a + b/x)/(b**2*log(f)**2)

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Mathematica [A]  time = 0.0107066, size = 27, normalized size = 0.69 \[ \frac{f^{a+\frac{b}{x}} (x-b \log (f))}{b^2 x \log ^2(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)/x^3,x]

[Out]

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

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Maple [A]  time = 0.014, size = 49, normalized size = 1.3 \[{\frac{1}{{x}^{2}} \left ({\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.821745, size = 28, normalized size = 0.72 \[ \frac{f^{a} \Gamma \left (2, -\frac{b \log \left (f\right )}{x}\right )}{b^{2} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f^a*gamma(2, -b*log(f)/x)/(b^2*log(f)^2)

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Fricas [A]  time = 0.260734, size = 42, normalized size = 1.08 \[ -\frac{{\left (b \log \left (f\right ) - x\right )} f^{\frac{a x + b}{x}}}{b^{2} x \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.248367, size = 22, normalized size = 0.56 \[ \frac{f^{a + \frac{b}{x}} \left (- b \log{\left (f \right )} + x\right )}{b^{2} x \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)/x**3,x)

[Out]

f**(a + b/x)*(-b*log(f) + x)/(b**2*x*log(f)**2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(a + b/x)/x^3, x)