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

Optimal. Leaf size=37 \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

[Out]

-((f^a*Gamma[-n^(-1), -(b*x^n*Log[f])]*(-(b*x^n*Log[f]))^n^(-1))/(n*x))

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Rubi [A]  time = 0.0406282, antiderivative size = 37, 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 \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^n)/x^2,x]

[Out]

-((f^a*Gamma[-n^(-1), -(b*x^n*Log[f])]*(-(b*x^n*Log[f]))^n^(-1))/(n*x))

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Rubi in Sympy [A]  time = 3.78294, size = 36, normalized size = 0.97 \[ - \frac{f^{a} \left (- b x^{n} \log{\left (f \right )}\right )^{\frac{1}{n}} \Gamma{\left (- \frac{1}{n},- b x^{n} \log{\left (f \right )} \right )}}{n x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f**a*(-b*x**n*log(f))**(1/n)*Gamma(-1/n, -b*x**n*log(f))/(n*x)

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Mathematica [A]  time = 0.0146427, size = 37, normalized size = 1. \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^n)/x^2,x]

[Out]

-((f^a*Gamma[-n^(-1), -(b*x^n*Log[f])]*(-(b*x^n*Log[f]))^n^(-1))/(n*x))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{{f}^{a+b{x}^{n}}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*x^n)/x^2,x)

[Out]

int(f^(a+b*x^n)/x^2,x)

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Maxima [A]  time = 0.943157, size = 50, normalized size = 1.35 \[ -\frac{\left (-b x^{n} \log \left (f\right )\right )^{\left (\frac{1}{n}\right )} f^{a} \Gamma \left (-\frac{1}{n}, -b x^{n} \log \left (f\right )\right )}{n x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-b*x^n*log(f))^(1/n)*f^a*gamma(-1/n, -b*x^n*log(f))/(n*x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f^{b x^{n} + a}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(f^(b*x^n + a)/x^2, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b*x**n)/x**2,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{b x^{n} + a}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(f^(b*x^n + a)/x^2, x)