Optimal. Leaf size=135 \[ 2 a^2 b^2 c^2 \log ^2(f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^2}}{x},x\right )+b^2 c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^2}}{x},x\right )+\sqrt{\pi } a b^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )-\frac{f^{c (a+b x)^2}}{2 x^2}-\frac{a b c \log (f) f^{c (a+b x)^2}}{x} \]
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Rubi [A] time = 0.0971859, 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 = {} \[ \int \frac{f^{c (a+b x)^2}}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{f^{c (a+b x)^2}}{x^3} \, dx &=-\frac{f^{c (a+b x)^2}}{2 x^2}+(a b c \log (f)) \int \frac{f^{c (a+b x)^2}}{x^2} \, dx+\left (b^2 c \log (f)\right ) \int \frac{f^{c (a+b x)^2}}{x} \, dx\\ &=-\frac{f^{c (a+b x)^2}}{2 x^2}-\frac{a b c f^{c (a+b x)^2} \log (f)}{x}+\left (b^2 c \log (f)\right ) \int \frac{f^{c (a+b x)^2}}{x} \, dx+\left (2 a^2 b^2 c^2 \log ^2(f)\right ) \int \frac{f^{c (a+b x)^2}}{x} \, dx+\left (2 a b^3 c^2 \log ^2(f)\right ) \int f^{c (a+b x)^2} \, dx\\ &=-\frac{f^{c (a+b x)^2}}{2 x^2}-\frac{a b c f^{c (a+b x)^2} \log (f)}{x}+a b^2 c^{3/2} \sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right ) \log ^{\frac{3}{2}}(f)+\left (b^2 c \log (f)\right ) \int \frac{f^{c (a+b x)^2}}{x} \, dx+\left (2 a^2 b^2 c^2 \log ^2(f)\right ) \int \frac{f^{c (a+b x)^2}}{x} \, dx\\ \end{align*}
Mathematica [A] time = 0.408973, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^2}}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{2}}}{{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 )}^{2} 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^{2} c x^{2} + 2 \, a b c x + a^{2} 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 )^{2}}}{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 )}^{2} c}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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