Optimal. Leaf size=39 \[ -\frac{x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (f) x^n\right )}{n} \]
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Rubi [A] time = 0.0252264, antiderivative size = 39, 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} \[ -\frac{x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (f) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^n} x^2 \, dx &=-\frac{f^a x^3 \Gamma \left (\frac{3}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-3/n}}{n}\\ \end{align*}
Mathematica [A] time = 0.005968, size = 39, normalized size = 1. \[ -\frac{x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b \log (f) x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{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 = 1.24579, size = 55, normalized size = 1.41 \begin{align*} -\frac{f^{a} x^{3} \Gamma \left (\frac{3}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac{3}{n}} n} \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 (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 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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