Integrand size = 15, antiderivative size = 239 \[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\frac {2 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^5}-\frac {2 a^2 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^5}+\frac {a^4 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^5}-\frac {4 a^3 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^5}-\frac {4 a (a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^5}+\frac {(a+b x)^5 \Gamma \left (-\frac {5}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5} \]
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Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2239, 2250, 2245, 2241} \[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\frac {a^4 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac {4 a^3 (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac {2 a^2 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^5}+\frac {(a+b x)^5 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{5/3} \Gamma \left (-\frac {5}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^5}-\frac {4 a (a+b x)^4 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^5} \]
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Rule 2239
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4 f^{\frac {c}{(a+b x)^3}}}{b^4}-\frac {4 a^3 f^{\frac {c}{(a+b x)^3}} (a+b x)}{b^4}+\frac {6 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)^2}{b^4}-\frac {4 a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^4}{b^4}\right ) \, dx \\ & = \frac {\int f^{\frac {c}{(a+b x)^3}} (a+b x)^4 \, dx}{b^4}-\frac {(4 a) \int f^{\frac {c}{(a+b x)^3}} (a+b x)^3 \, dx}{b^4}+\frac {\left (6 a^2\right ) \int f^{\frac {c}{(a+b x)^3}} (a+b x)^2 \, dx}{b^4}-\frac {\left (4 a^3\right ) \int f^{\frac {c}{(a+b x)^3}} (a+b x) \, dx}{b^4}+\frac {a^4 \int f^{\frac {c}{(a+b x)^3}} \, dx}{b^4} \\ & = \frac {2 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^5}+\frac {a^4 (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^5}-\frac {4 a^3 (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^5}-\frac {4 a (a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^5}+\frac {(a+b x)^5 \Gamma \left (-\frac {5}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5}+\frac {\left (6 a^2 c \log (f)\right ) \int \frac {f^{\frac {c}{(a+b x)^3}}}{a+b x} \, dx}{b^4} \\ & = \frac {2 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^5}-\frac {2 a^2 c \text {Ei}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^5}+\frac {a^4 (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^5}-\frac {4 a^3 (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^5}-\frac {4 a (a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^5}+\frac {(a+b x)^5 \Gamma \left (-\frac {5}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.92 \[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\frac {6 a^2 f^{\frac {c}{(a+b x)^3}} (a+b x)^3-6 a^2 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)+a^4 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+4 a c (a+b x) \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \log (f) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}-4 a^3 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}+(a+b x)^5 \Gamma \left (-\frac {5}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{5/3}}{3 b^5} \]
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\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{4}d x\]
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Time = 0.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=-\frac {20 \, a^{2} c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) - {\left (20 \, a^{3} b^{2} - 3 \, b^{2} c \log \left (f\right )\right )} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) + 10 \, {\left (a^{4} b - 3 \, a b c \log \left (f\right )\right )} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (2 \, b^{5} x^{5} + 2 \, a^{5} + 3 \, {\left (b^{2} c x^{2} - 8 \, a b c x - 9 \, a^{2} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{10 \, b^{5}} \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{4}\, dx \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{4} \,d x } \]
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\[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{4} \,d x } \]
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Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^4 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^4 \,d x \]
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