Optimal. Leaf size=154 \[ -\frac {(a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-3/n}}{b^3 n}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^3 n}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^3 n} \]
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Rubi [A]
time = 0.08, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2258, 2239,
2250} \begin {gather*} -\frac {a^2 (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}-\frac {(a+b x)^3 \left (-c \log (f) (a+b x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}+\frac {2 a (a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-c \log (f) (a+b x)^n\right )}{b^3 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2250
Rule 2258
Rubi steps
\begin {align*} \int f^{c (a+b x)^n} x^2 \, dx &=\int \left (\frac {a^2 f^{c (a+b x)^n}}{b^2}-\frac {2 a f^{c (a+b x)^n} (a+b x)}{b^2}+\frac {f^{c (a+b x)^n} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac {\int f^{c (a+b x)^n} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{c (a+b x)^n} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{c (a+b x)^n} \, dx}{b^2}\\ &=-\frac {(a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-3/n}}{b^3 n}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^3 n}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 136, normalized size = 0.88 \begin {gather*} -\frac {(a+b x) \left (-c (a+b x)^n \log (f)\right )^{-3/n} \left ((a+b x)^2 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right )+a \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}} \left (-2 (a+b x) \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )+a \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}}\right )\right )}{b^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int f^{c \left (b x +a \right )^{n}} x^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*} \int f^{c \left (a + b x\right )^{n}} x^{2}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,{\left (a+b\,x\right )}^n}\,x^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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