Integrand size = 15, antiderivative size = 142 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}-\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \]
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Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2239, 2250, 2245, 2241} \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {a^2 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}-\frac {c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3} \]
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Rule 2239
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)+a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}-2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \]
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\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{2}d x\]
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none
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.37 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {3 \, a b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 3 \, a^{2} b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) + {\left (b^{3} x^{3} + a^{3}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{3 \, b^{3}} \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{2}\, dx \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \]
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Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^2 \,d x \]
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