Optimal. Leaf size=34 \[ \text{Unintegrable}\left (\frac{(f+g x)^2}{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]
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Rubi [A] time = 0.207001, 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 = {} \[ \int \frac{(f+g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \left (\frac{f^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{2 f g x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{g^2 x^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}\right ) \, dx\\ &=f^2 \int \frac{1}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+(2 f g) \int \frac{x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+g^2 \int \frac{x^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.935178, size = 0, normalized size = 0. \[ \int \frac{(f+g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.418, 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 ) ^{-2}}\, 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*} -\frac{b d g^{2} x^{4} + a c f^{2} +{\left (a d g^{2} +{\left (2 \, d f g + c g^{2}\right )} b\right )} x^{3} +{\left ({\left (2 \, d f g + c g^{2}\right )} a +{\left (d f^{2} + 2 \, c f g\right )} b\right )} x^{2} +{\left (b c f^{2} +{\left (d f^{2} + 2 \, c f g\right )} a\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) +{\left (b c n - a d n\right )} A B +{\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{2}} + \int \frac{4 \, b d g^{2} x^{3} + b c f^{2} + 3 \,{\left (a d g^{2} +{\left (2 \, d f g + c g^{2}\right )} b\right )} x^{2} +{\left (d f^{2} + 2 \, c f g\right )} a + 2 \,{\left ({\left (2 \, d f g + c g^{2}\right )} a +{\left (d f^{2} + 2 \, c f g\right )} b\right )} x}{{\left (b c n - a d n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (b c n - a d n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) +{\left (b c n - a d n\right )} A B +{\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{2}}\,{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^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A^{2}}, 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}}{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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