Optimal. Leaf size=37 \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]
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Rubi [A] time = 0.0238832, 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, Rules used = {2218} \[ -\frac{f^a \left (-b \log (f) x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-b \log (f) x^n\right )}{n x} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{f^{a+b x^n}}{x^2} \, dx &=-\frac{f^a \Gamma \left (-\frac{1}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{\frac{1}{n}}}{n x}\\ \end{align*}
Mathematica [A] time = 0.0038872, 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.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{a+b{x}^{n}}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20209, size = 50, normalized size = 1.35 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{f^{b x^{n} + a}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{n} + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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