Optimal. Leaf size=39 \[ -\frac {f^a x^3 \Gamma \left (\frac {3}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-3/n}}{n} \]
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Rubi [A]
time = 0.02, 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 = {2250}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.04, size = 212, normalized size = 5.44
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {3}{n}} \ln \left (f \right )^{-\frac {3}{n}} \left (\frac {n \,x^{3} \left (-b \right )^{\frac {3}{n}} \ln \left (f \right )^{\frac {3}{n}} \left (\ln \left (f \right ) x^{n} b n +n +3\right ) \Gamma \left (1-\frac {3}{n}\right ) \Gamma \left (\frac {3+n}{n}+1\right ) L_{-\frac {3}{n}}^{\left (\frac {3+n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right )}{3 \left (3+n \right ) \Gamma \left (-\frac {3}{n}+\frac {3+n}{n}+1\right )}-\frac {n^{2} x^{3+n} \left (-b \right )^{\frac {3}{n}} \ln \left (f \right )^{1+\frac {3}{n}} b L_{-\frac {3}{n}}^{\left (\frac {3+n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (1-\frac {3}{n}\right ) \Gamma \left (\frac {3+n}{n}+1\right )}{3 \left (3+n \right ) \Gamma \left (-\frac {3}{n}+\frac {3+n}{n}+1\right )}\right )}{n}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 41, normalized size = 1.05 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 {gather*} \begin {cases} - \frac {b f^{a} f^{b x^{n}} n x^{3} x^{n} \log {\left (f \right )}}{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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {f^a\,x^3\,{\mathrm {e}}^{\frac {b\,x^n\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{\frac {1}{2}-\frac {3}{2\,n},\frac {3}{2\,n}}\left (b\,x^n\,\ln \left (f\right )\right )}{3\,{\left (b\,x^n\,\ln \left (f\right )\right )}^{\frac {3}{2\,n}+\frac {1}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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