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

Optimal. Leaf size=61 \[ -\frac{2 f^{a+\frac{b}{x}}}{b^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)} \]

[Out]

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

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Rubi [A]  time = 0.0968518, antiderivative size = 61, 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 \[ -\frac{2 f^{a+\frac{b}{x}}}{b^3 \log ^3(f)}+\frac{2 f^{a+\frac{b}{x}}}{b^2 x \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^2 \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0674, size = 49, normalized size = 0.8 \[ - \frac{f^{a + \frac{b}{x}}}{b x^{2} \log{\left (f \right )}} + \frac{2 f^{a + \frac{b}{x}}}{b^{2} x \log{\left (f \right )}^{2}} - \frac{2 f^{a + \frac{b}{x}}}{b^{3} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0151326, size = 41, normalized size = 0.67 \[ -\frac{f^{a+\frac{b}{x}} \left (b^2 \log ^2(f)-2 b x \log (f)+2 x^2\right )}{b^3 x^2 \log ^3(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)/x^4,x]

[Out]

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

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Maple [A]  time = 0.019, size = 73, normalized size = 1.2 \[{\frac{1}{{x}^{3}} \left ( -2\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+2\,{\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.823041, size = 30, normalized size = 0.49 \[ -\frac{f^{a} \Gamma \left (3, -\frac{b \log \left (f\right )}{x}\right )}{b^{3} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f^a*gamma(3, -b*log(f)/x)/(b^3*log(f)^3)

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Fricas [A]  time = 0.268704, size = 58, normalized size = 0.95 \[ -\frac{{\left (b^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right ) + 2 \, x^{2}\right )} f^{\frac{a x + b}{x}}}{b^{3} x^{2} \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.295813, size = 39, normalized size = 0.64 \[ \frac{f^{a + \frac{b}{x}} \left (- b^{2} \log{\left (f \right )}^{2} + 2 b x \log{\left (f \right )} - 2 x^{2}\right )}{b^{3} x^{2} \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)/x**4,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x}}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(a + b/x)/x^4, x)