Optimal. Leaf size=263 \[ \frac {9}{2} a^4 b^2 c^2 \log ^2(f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right )+3 a b^2 c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right )-\frac {3 a^3 b^2 c^2 \log ^2(f) (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac {3 a^2 b^2 c^2 \log ^2(f) (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{2 \left (-c \log (f) (a+b x)^3\right )^{2/3}}-\frac {3 a^2 b c \log (f) f^{c (a+b x)^3}}{2 x}-\frac {b^2 c \log (f) (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac {f^{c (a+b x)^3}}{2 x^2} \]
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Rubi [A] time = 0.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {f^{c (a+b x)^3}}{x^3} \, dx &=-\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*}
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Mathematica [A] time = 1.48, size = 0, normalized size = 0.00 \[ \int \frac {f^{c (a+b x)^3}}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {f^{\left (b x +a \right )^{3} c}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f^{c\,{\left (a+b\,x\right )}^3}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c \left (a + b x\right )^{3}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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