Integrand size = 15, antiderivative size = 15 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=-\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {3 a^2 b^2 c^2 (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}-\frac {3 a^3 b^2 c^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+3 a b^2 c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right )+\frac {9}{2} a^4 b^2 c^2 \log ^2(f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right ) \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int \frac {f^{c (a+b x)^3}}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} (3 b c \log (f)) \int \frac {f^{c (a+b x)^3} (a+b x)^2}{x^2} \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} (3 b c \log (f)) \int \left (b^2 f^{c (a+b x)^3}+\frac {a^2 f^{c (a+b x)^3}}{x^2}+\frac {2 a b f^{c (a+b x)^3}}{x}\right ) \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} \left (3 a^2 b c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x^2} \, dx+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (3 b^3 c \log (f)\right ) \int f^{c (a+b x)^3} \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3} (a+b x)^2}{x} \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^2 c^2 \log ^2(f)\right ) \int \left (a b f^{c (a+b x)^3}+\frac {a^2 f^{c (a+b x)^3}}{x}+b f^{c (a+b x)^3} (a+b x)\right ) \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^4 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^3 c^2 \log ^2(f)\right ) \int f^{c (a+b x)^3} (a+b x) \, dx+\frac {1}{2} \left (9 a^3 b^3 c^2 \log ^2(f)\right ) \int f^{c (a+b x)^3} \, dx \\ & = -\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {3 a^2 b^2 c^2 (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}-\frac {3 a^3 b^2 c^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^4 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx \\ \end{align*}
Not integrable
Time = 1.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int \frac {f^{c (a+b x)^3}}{x^3} \, dx \]
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Not integrable
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
\[\int \frac {f^{c \left (b x +a \right )^{3}}}{x^{3}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.67 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{3}} \,d x } \]
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Not integrable
Time = 0.69 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int \frac {f^{c \left (a + b x\right )^{3}}}{x^{3}}\, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{3}} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int { \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{3}} \,d x } \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx=\int \frac {f^{c\,{\left (a+b\,x\right )}^3}}{x^3} \,d x \]
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