Optimal. Leaf size=133 \[ 3 a^2 b c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right )-\frac {f^{c (a+b x)^3}}{x}-\frac {a b c \log (f) (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{\sqrt [3]{-c \log (f) (a+b x)^3}}-\frac {b c \log (f) (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{\left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.30, 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 = {} \[ \int \frac {f^{c (a+b x)^3}}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {f^{c (a+b x)^3}}{x^2} \, dx &=-\frac {f^{c (a+b x)^3}}{x}+(3 b c \log (f)) \int \frac {f^{c (a+b x)^3} (a+b x)^2}{x} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{x}+(3 b c \log (f)) \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}}{x}+\left (3 a^2 b c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\left (3 b^2 c \log (f)\right ) \int f^{c (a+b x)^3} (a+b x) \, dx+\left (3 a b^2 c \log (f)\right ) \int f^{c (a+b x)^3} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{x}-\frac {b c (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{\left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a b c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{\sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a^2 b c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 1.11, size = 0, normalized size = 0.00 \[ \int \frac {f^{c (a+b x)^3}}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{x^{2}}, x\right ) \]
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 \[ \int \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {f^{\left (b x +a \right )^{3} c}}{x^{2}}\, dx \]
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 \[ \int \frac {f^{{\left (b x + a\right )}^{3} c}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{c\,{\left (a+b\,x\right )}^3}}{x^2} \,d x \]
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 \[ \int \frac {f^{c \left (a + b x\right )^{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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