Optimal. Leaf size=37 \[ \text{Unintegrable}\left (\frac{(c g+d g x)^2}{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]
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Rubi [A] time = 0.232878, 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 = {} \[ \int \frac{(c g+d g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(c g+d 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{c^2 g^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{2 c d g^2 x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{d^2 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\\ &=\left (c^2 g^2\right ) \int \frac{1}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+\left (2 c d g^2\right ) \int \frac{x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+\left (d^2 g^2\right ) \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 = 0.989418, size = 0, normalized size = 0. \[ \int \frac{(c g+d g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.392, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dgx+cg \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^{3} g^{2} x^{4} + a c^{3} g^{2} +{\left (3 \, b c d^{2} g^{2} + a d^{3} g^{2}\right )} x^{3} + 3 \,{\left (b c^{2} d g^{2} + a c d^{2} g^{2}\right )} x^{2} +{\left (b c^{3} g^{2} + 3 \, a c^{2} d g^{2}\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^{3} g^{2} x^{3} + b c^{3} g^{2} + 3 \, a c^{2} d g^{2} + 3 \,{\left (3 \, b c d^{2} g^{2} + a d^{3} g^{2}\right )} x^{2} + 6 \,{\left (b c^{2} d g^{2} + a c d^{2} g^{2}\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{d^{2} g^{2} x^{2} + 2 \, c d g^{2} x + c^{2} g^{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 (d g x + c g\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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