Integrand size = 15, antiderivative size = 269 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=-\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {a c^3 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4} \]
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Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2258, 2237, 2241, 2245, 2250} \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\frac {a^3 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^4}-\frac {a^3 (a+b x) f^{\frac {c}{a+b x}}}{b^4}-\frac {3 a^2 c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}+\frac {3 a^2 (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4}+\frac {3 a^2 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}+\frac {c^4 \log ^4(f) \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^4}+\frac {a c^3 \log ^3(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{2 b^4}-\frac {a c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{2 b^4}-\frac {a (a+b x)^3 f^{\frac {c}{a+b x}}}{b^4}-\frac {a c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{2 b^4} \]
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Rule 2237
Rule 2241
Rule 2245
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 f^{\frac {c}{a+b x}}}{b^3}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)}{b^3}-\frac {3 a f^{\frac {c}{a+b x}} (a+b x)^2}{b^3}+\frac {f^{\frac {c}{a+b x}} (a+b x)^3}{b^3}\right ) \, dx \\ & = \frac {\int f^{\frac {c}{a+b x}} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{\frac {c}{a+b x}} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{\frac {c}{a+b x}} \, dx}{b^3} \\ & = -\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {(a c \log (f)) \int f^{\frac {c}{a+b x}} (a+b x) \, dx}{b^3}+\frac {\left (3 a^2 c \log (f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{2 b^3}-\frac {\left (a^3 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{b^3} \\ & = -\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {\left (a c^2 \log ^2(f)\right ) \int f^{\frac {c}{a+b x}} \, dx}{2 b^3}+\frac {\left (3 a^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 b^3} \\ & = -\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4}-\frac {\left (a c^3 \log ^3(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 b^3} \\ & = -\frac {a^3 f^{\frac {c}{a+b x}} (a+b x)}{b^4}+\frac {3 a^2 f^{\frac {c}{a+b x}} (a+b x)^2}{2 b^4}-\frac {a f^{\frac {c}{a+b x}} (a+b x)^3}{b^4}+\frac {3 a^2 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{2 b^4}-\frac {a c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{2 b^4}+\frac {a^3 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^4}-\frac {a c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{2 b^4}-\frac {3 a^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 b^4}+\frac {a c^3 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{2 b^4}+\frac {c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.67 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=-\frac {a f^{\frac {c}{a+b x}} \left (6 a^3-26 a^2 c \log (f)+11 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )}{24 b^4}+\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (24 a^3-36 a^2 c \log (f)+12 a c^2 \log ^2(f)-c^3 \log ^3(f)\right )+b f^{\frac {c}{a+b x}} x \left (6 b^3 x^3+2 c \left (9 a^2-3 a b x+b^2 x^2\right ) \log (f)+c^2 (-10 a+b x) \log ^2(f)+c^3 \log ^3(f)\right )}{24 b^4} \]
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Time = 0.14 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} c^{3} x}{24 b^{3}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} c^{2} x^{2}}{24 b^{2}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) c \,x^{3}}{12 b}+\frac {f^{\frac {c}{b x +a}} x^{4}}{4}+\frac {\operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) c^{4} \ln \left (f \right )^{4}}{24 b^{4}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} a \,c^{3}}{24 b^{4}}-\frac {5 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a \,c^{2} x}{12 b^{3}}-\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) a c \,x^{2}}{4 b^{2}}-\frac {\ln \left (f \right )^{3} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a \,c^{3}}{2 b^{4}}-\frac {11 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a^{2} c^{2}}{24 b^{4}}+\frac {3 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} c x}{4 b^{3}}+\frac {3 \ln \left (f \right )^{2} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{2} c^{2}}{2 b^{4}}+\frac {13 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{3} c}{12 b^{4}}-\frac {\ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{3} c}{b^{4}}-\frac {f^{\frac {c}{b x +a}} a^{4}}{4 b^{4}}\) | \(359\) |
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Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.64 \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\frac {{\left (6 \, b^{4} x^{4} - 6 \, a^{4} + {\left (b c^{3} x + a c^{3}\right )} \log \left (f\right )^{3} + {\left (b^{2} c^{2} x^{2} - 10 \, a b c^{2} x - 11 \, a^{2} c^{2}\right )} \log \left (f\right )^{2} + 2 \, {\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} + 9 \, a^{2} b c x + 13 \, a^{3} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{4} \log \left (f\right )^{4} - 12 \, a c^{3} \log \left (f\right )^{3} + 36 \, a^{2} c^{2} \log \left (f\right )^{2} - 24 \, a^{3} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{24 \, b^{4}} \]
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\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int f^{\frac {c}{a + b x}} x^{3}\, dx \]
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\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int { f^{\frac {c}{b x + a}} x^{3} \,d x } \]
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\[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int { f^{\frac {c}{b x + a}} x^{3} \,d x } \]
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Timed out. \[ \int f^{\frac {c}{a+b x}} x^3 \, dx=\int f^{\frac {c}{a+b\,x}}\,x^3 \,d x \]
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