Optimal. Leaf size=142 \[ \frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}-\frac {c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2239,
2250, 2245, 2241} \begin {gather*} \frac {a^2 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}-\frac {2 a (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}-\frac {c \log (f) \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps
\begin {align*} \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx &=\int \left (\frac {a^2 f^{\frac {c}{(a+b x)^3}}}{b^2}-\frac {2 a f^{\frac {c}{(a+b x)^3}} (a+b x)}{b^2}+\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^2}{b^2}\right ) \, dx\\ &=\frac {\int f^{\frac {c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{\frac {c}{(a+b x)^3}} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{\frac {c}{(a+b x)^3}} \, dx}{b^2}\\ &=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3}+\frac {(c \log (f)) \int \frac {f^{\frac {c}{(a+b x)^3}}}{a+b x} \, dx}{b^2}\\ &=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}-\frac {c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 127, normalized size = 0.89 \begin {gather*} \frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3-c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)+a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}-2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{2}\, dx\]
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 = 0.11, size = 194, normalized size = 1.37 \begin {gather*} \frac {3 \, a b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 3 \, a^{2} b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) + {\left (b^{3} x^{3} + a^{3}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{3 \, b^{3}} \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^{\frac {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^{\frac {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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