Optimal. Leaf size=120 \[ \frac {f^{c (a+b x)^3}}{3 b^3 c \log (f)}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^3 \sqrt [3]{-c (a+b x)^3 \log (f)}} \]
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Rubi [A]
time = 0.06, 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 = {2258, 2239,
2250, 2240} \begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2240
Rule 2250
Rule 2258
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 = 0.42, size = 111, normalized size = 0.92 \begin {gather*} \frac {\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 (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 )}{\sqrt [3]{-c (a+b x)^3 \log (f)}}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int f^{c \left (b x +a \right )^{3}} x^{2}\, dx\]
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 [A]
time = 0.09, size = 155, normalized size = 1.29 \begin {gather*} \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 {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 )^{3}} 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 )}^3}\,x^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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