Optimal. Leaf size=34 \[ \text{Unintegrable}\left (\frac{(f+g x)^2}{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A},x\right ) \]
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Rubi [A] time = 0.188514, 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 = {} \[ \int \frac{(f+g x)^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\int \left (\frac{f^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}+\frac{2 f g x}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}+\frac{g^2 x^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}\right ) \, dx\\ &=f^2 \int \frac{1}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx+(2 f g) \int \frac{x}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx+g^2 \int \frac{x^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.414442, size = 0, normalized size = 0. \[ \int \frac{(f+g x)^2}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.396, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, 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{{\left (g x + f\right )}^{2}}{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}\,{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{g^{2} x^{2} + 2 \, f g x + f^{2}}{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (g x + f\right )}^{2}}{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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