Integrand size = 15, antiderivative size = 166 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b^2 c f^{\frac {c}{a+b x}} \log (f)}{2 a^3}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {b^2 c^2 f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log ^2(f)}{2 a^4} \]
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Time = 0.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2255, 6874, 2254, 2241, 2260, 2209, 2240} \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {b^2 c^2 \log ^2(f) f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{2 a^4}+\frac {b^2 c \log (f) f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a (a+b x)}\right )}{a^3}+\frac {b^2 c \log (f) f^{\frac {c}{a+b x}}}{2 a^3}+\frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}+\frac {b c \log (f) f^{\frac {c}{a+b x}}}{2 a^2 x}-\frac {f^{\frac {c}{a+b x}}}{2 x^2} \]
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Rule 2209
Rule 2240
Rule 2241
Rule 2254
Rule 2255
Rule 2260
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{\frac {c}{a+b x}}}{2 x^2}-\frac {1}{2} (b c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{x^2 (a+b x)^2} \, dx \\ & = -\frac {f^{\frac {c}{a+b x}}}{2 x^2}-\frac {1}{2} (b c \log (f)) \int \left (\frac {f^{\frac {c}{a+b x}}}{a^2 x^2}-\frac {2 b f^{\frac {c}{a+b x}}}{a^3 x}+\frac {b^2 f^{\frac {c}{a+b x}}}{a^2 (a+b x)^2}+\frac {2 b^2 f^{\frac {c}{a+b x}}}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {f^{\frac {c}{a+b x}}}{2 x^2}-\frac {(b c \log (f)) \int \frac {f^{\frac {c}{a+b x}}}{x^2} \, dx}{2 a^2}+\frac {\left (b^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{x} \, dx}{a^3}-\frac {\left (b^3 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{a^3}-\frac {\left (b^3 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{(a+b x)^2} \, dx}{2 a^2} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{a^3}+\frac {\left (b^2 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)} \, dx}{a^2}+\frac {\left (b^3 c \log (f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{a^3}+\frac {\left (b^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)^2} \, dx}{2 a^2} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {\left (b^2 c \log (f)\right ) \text {Subst}\left (\int \frac {f^{\frac {c}{a}-\frac {b c x}{a}}}{x} \, dx,x,\frac {x}{a+b x}\right )}{a^3}+\frac {\left (b^2 c^2 \log ^2(f)\right ) \int \left (\frac {f^{\frac {c}{a+b x}}}{a^2 x}-\frac {b f^{\frac {c}{a+b x}}}{a (a+b x)^2}-\frac {b f^{\frac {c}{a+b x}}}{a^2 (a+b x)}\right ) \, dx}{2 a^2} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {\left (b^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{x} \, dx}{2 a^4}-\frac {\left (b^3 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 a^4}-\frac {\left (b^3 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{(a+b x)^2} \, dx}{2 a^3} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b^2 c f^{\frac {c}{a+b x}} \log (f)}{2 a^3}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {b^2 c^2 \text {Ei}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{2 a^4}+\frac {\left (b^2 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{x (a+b x)} \, dx}{2 a^3}+\frac {\left (b^3 c^2 \log ^2(f)\right ) \int \frac {f^{\frac {c}{a+b x}}}{a+b x} \, dx}{2 a^4} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b^2 c f^{\frac {c}{a+b x}} \log (f)}{2 a^3}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {\left (b^2 c^2 \log ^2(f)\right ) \text {Subst}\left (\int \frac {f^{\frac {c}{a}-\frac {b c x}{a}}}{x} \, dx,x,\frac {x}{a+b x}\right )}{2 a^4} \\ & = \frac {b^2 f^{\frac {c}{a+b x}}}{2 a^2}-\frac {f^{\frac {c}{a+b x}}}{2 x^2}+\frac {b^2 c f^{\frac {c}{a+b x}} \log (f)}{2 a^3}+\frac {b c f^{\frac {c}{a+b x}} \log (f)}{2 a^2 x}+\frac {b^2 c f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^3}+\frac {b^2 c^2 f^{\frac {c}{a}} \text {Ei}\left (-\frac {b c x \log (f)}{a (a+b x)}\right ) \log ^2(f)}{2 a^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.69 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {b^2 f^{\frac {c}{a+b x}} (2 a+c \log (f))}{2 a^3}+\frac {b^2 c f^{\frac {c}{a}} \operatorname {ExpIntegralEi}\left (-\frac {b c x \log (f)}{a^2+a b x}\right ) \log (f) (2 a+c \log (f))-\frac {a^2 f^{\frac {c}{a+b x}} \left (a^2+b^2 x^2-b c x \log (f)\right )}{x^2}}{2 a^4} \]
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Time = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {-f^{\frac {c}{a}} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right ) c^{2} b^{2} \ln \left (f \right )^{2} x^{2}-2 \ln \left (f \right ) f^{\frac {c}{a}} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}+\frac {c \ln \left (f \right )}{a}\right ) a \,b^{2} c \,x^{2}+f^{\frac {c}{b x +a}} \ln \left (f \right ) a \,b^{2} c \,x^{2}+f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} b c x +f^{\frac {c}{b x +a}} a^{2} b^{2} x^{2}-f^{\frac {c}{b x +a}} a^{4}}{2 a^{4} x^{2}}\) | \(174\) |
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Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.66 \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\frac {{\left (b^{2} c^{2} x^{2} \log \left (f\right )^{2} + 2 \, a b^{2} c x^{2} \log \left (f\right )\right )} f^{\frac {c}{a}} {\rm Ei}\left (-\frac {b c x \log \left (f\right )}{a b x + a^{2}}\right ) + {\left (a^{2} b^{2} x^{2} - a^{4} + {\left (a b^{2} c x^{2} + a^{2} b c x\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}}}{2 \, a^{4} x^{2}} \]
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\[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\int \frac {f^{\frac {c}{a + b x}}}{x^{3}}\, dx \]
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\[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\int { \frac {f^{\frac {c}{b x + a}}}{x^{3}} \,d x } \]
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\[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\int { \frac {f^{\frac {c}{b x + a}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {f^{\frac {c}{a+b x}}}{x^3} \, dx=\int \frac {f^{\frac {c}{a+b\,x}}}{x^3} \,d x \]
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