Optimal. Leaf size=154 \[ -\frac{a^2 (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}-\frac{(a+b x)^3 \left (-c \log (f) (a+b x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}+\frac{2 a (a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^3 n} \]
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Rubi [A] time = 0.108436, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2226, 2208, 2218} \[ -\frac{a^2 (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}-\frac{(a+b x)^3 \left (-c \log (f) (a+b x)^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-c \log (f) (a+b x)^n\right )}{b^3 n}+\frac{2 a (a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )}{b^3 n} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
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*}
Mathematica [A] time = 0.0689383, size = 136, normalized size = 0.88 \[ -\frac{(a+b x) \left (-c \log (f) (a+b x)^n\right )^{-3/n} \left (a \left (-c \log (f) (a+b x)^n\right )^{\frac{1}{n}} \left (a \left (-c \log (f) (a+b x)^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-c \log (f) (a+b x)^n\right )-2 (a+b x) \text{Gamma}\left (\frac{2}{n},-c \log (f) (a+b x)^n\right )\right )+(a+b x)^2 \text{Gamma}\left (\frac{3}{n},-c \log (f) (a+b x)^n\right )\right )}{b^3 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{n}}{x}^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{{\left (b x + a\right )}^{n} c} x^{2}\,{d 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 (f^{{\left (b x + a\right )}^{n} c} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{c \left (a + b x\right )^{n}} x^{2}\, dx \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^{{\left (b x + a\right )}^{n} c} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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