Optimal. Leaf size=261 \[ \frac{9}{2} a^4 b^2 c^2 \log ^2(f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^3}}{x},x\right )-\frac{3 a^3 b^2 c^2 \log ^2(f) (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{3 a^2 b^2 c^2 \log ^2(f) (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{2 \left (-c \log (f) (a+b x)^3\right )^{2/3}}+3 a b^2 c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^3}}{x},x\right )-\frac{b^2 c \log (f) (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{3 a^2 b c \log (f) f^{c (a+b x)^3}}{2 x}-\frac{f^{c (a+b x)^3}}{2 x^2} \]
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Rubi [A] time = 0.448682, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 1.38929, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^3}}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{3}}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{{\left (b x + a\right )}^{3} c}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{c \left (a + b x\right )^{3}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{{\left (b x + a\right )}^{3} c}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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