Optimal. Leaf size=39 \[ -\frac {x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b x^n \log (f)\right )}{n} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ -\frac {x^3 f^a \left (-b \log (f) x^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{b x^{n} + a} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x^{n} + a} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{2} f^{b \,x^{n}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 41, normalized size = 1.05 \[ -\frac {f^{a} x^{3} \Gamma \left (\frac {3}{n}, -b x^{n} \log \relax (f)\right )}{\left (-b x^{n} \log \relax (f)\right )^{\frac {3}{n}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 54, normalized size = 1.38 \[ \frac {f^a\,x^3\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{\frac {1}{2}-\frac {3}{2\,n},\frac {3}{2\,n}}\left (b\,x^n\,\ln \relax (f)\right )}{3\,{\left (b\,x^n\,\ln \relax (f)\right )}^{\frac {3}{2\,n}+\frac {1}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x^{3} x^{n} \log {\relax (f )}}{3 n + 9} + \frac {f^{a} f^{b x^{n}} n x^{3}}{3 n + 9} + \frac {3 f^{a} f^{b x^{n}} x^{3}}{3 n + 9} & \text {for}\: n \neq -3 \\\int f^{a + \frac {b}{x^{3}}} x^{2}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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