Optimal. Leaf size=120 \[ -\frac{a^2 (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^3 \sqrt [3]{-c \log (f) (a+b x)^3}}+\frac{2 a (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^3 \left (-c \log (f) (a+b x)^3\right )^{2/3}}+\frac{f^{c (a+b x)^3}}{3 b^3 c \log (f)} \]
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Rubi [A] time = 0.0889143, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2226, 2208, 2218, 2209} \[ -\frac{a^2 (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^3 \sqrt [3]{-c \log (f) (a+b x)^3}}+\frac{2 a (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^3 \left (-c \log (f) (a+b x)^3\right )^{2/3}}+\frac{f^{c (a+b x)^3}}{3 b^3 c \log (f)} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
Rule 2209
Rubi steps
\begin{align*} \int f^{c (a+b x)^3} x^2 \, dx &=\int \left (\frac{a^2 f^{c (a+b x)^3}}{b^2}-\frac{2 a f^{c (a+b x)^3} (a+b x)}{b^2}+\frac{f^{c (a+b x)^3} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac{\int f^{c (a+b x)^3} (a+b x)^2 \, dx}{b^2}-\frac{(2 a) \int f^{c (a+b x)^3} (a+b x) \, dx}{b^2}+\frac{a^2 \int f^{c (a+b x)^3} \, dx}{b^2}\\ &=\frac{f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac{2 a (a+b x)^2 \Gamma \left (\frac{2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac{a^2 (a+b x) \Gamma \left (\frac{1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c (a+b x)^3 \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.199875, size = 111, normalized size = 0.92 \[ \frac{-\frac{a^2 (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{\sqrt [3]{-c \log (f) (a+b x)^3}}+\frac{2 a (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{\left (-c \log (f) (a+b x)^3\right )^{2/3}}+\frac{f^{c (a+b x)^3}}{c \log (f)}}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{3}}{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 )}^{3} c} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55489, size = 373, normalized size = 3.11 \begin{align*} \frac{\left (-b^{3} c \log \left (f\right )\right )^{\frac{2}{3}} a^{2} \Gamma \left (\frac{1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) - 2 \, \left (-b^{3} c \log \left (f\right )\right )^{\frac{1}{3}} a b \Gamma \left (\frac{2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) + b^{2} f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{3 \, b^{5} c \log \left (f\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 )^{3}} 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 )}^{3} c} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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