Optimal. Leaf size=120 \[ -\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c \log (f) (a+b x)^3}}+\frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, 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 2208
Rule 2209
Rule 2218
Rule 2226
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*}
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Mathematica [A] time = 0.20, size = 111, normalized size = 0.92 \[ \frac {-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{\sqrt [3]{-c \log (f) (a+b x)^3}}+\frac {f^{c (a+b x)^3}}{c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{\left (-c \log (f) (a+b x)^3\right )^{2/3}}}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 155, normalized size = 1.29 \[ \frac {\left (-b^{3} c \log \relax (f)\right )^{\frac {2}{3}} a^{2} \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \relax (f)\right ) - 2 \, \left (-b^{3} c \log \relax (f)\right )^{\frac {1}{3}} a b \Gamma \left (\frac {2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \relax (f)\right ) + b^{2} f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{3 \, b^{5} c \log \relax (f)} \]
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 \[ \int f^{{\left (b x + a\right )}^{3} c} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{2} f^{\left (b x +a \right )^{3} c}\, dx \]
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 \[ \int f^{{\left (b x + a\right )}^{3} c} x^{2}\,{d x} \]
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 \[ \int f^{c\,{\left (a+b\,x\right )}^3}\,x^2 \,d x \]
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 \[ \int f^{c \left (a + b x\right )^{3}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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