Optimal. Leaf size=223 \[ -\frac{(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (m+1) (c+d x) (b c-a d)}-\frac{2 B n (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1)^2 (c+d x) (b c-a d)}-\frac{2 B^2 n^2 (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{i^2 (m+1)^3 (c+d x) (b c-a d)} \]
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Rubi [F] time = 1.15911, 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 = {} \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (A^2 (219 c+219 d x)^m (a g+b g x)^{-2-m}+2 A B (219 c+219 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^2 (219 c+219 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=A^2 \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \, dx+(2 A B) \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+B^2 \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac{A^2 (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m}}{219 (b c-a d) g (1+m)}-\frac{2 A B (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{219 (b c-a d) g (1+m)}-(2 A B) \int \frac{219^m n (c+d x)^m (a g+b g x)^{-2-m}}{-1-m} \, dx+B^2 \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac{A^2 (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m}}{219 (b c-a d) g (1+m)}-\frac{2 A B (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{219 (b c-a d) g (1+m)}+B^2 \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+\frac{\left (2\ 219^m A B n\right ) \int (c+d x)^m (a g+b g x)^{-2-m} \, dx}{1+m}\\ &=-\frac{2\ 219^m A B n (c+d x)^{1+m} (a g+b g x)^{-1-m}}{(b c-a d) g (1+m)^2}-\frac{A^2 (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m}}{219 (b c-a d) g (1+m)}-\frac{2 A B (219 c+219 d x)^{1+m} (a g+b g x)^{-1-m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{219 (b c-a d) g (1+m)}+B^2 \int (219 c+219 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ \end{align*}
Mathematica [A] time = 2.03633, size = 134, normalized size = 0.6 \[ -\frac{(c+d x) (g (a+b x))^{-m-1} (i (c+d x))^m \left (2 B (m+1) (A m+A+B n) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^2 (m+1)^2 \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A^2 (m+1)^2+2 A B (m+1) n+2 B^2 n^2\right )}{g (m+1)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.479, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{-2-m} \left ( dix+ci \right ) ^{m} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}{\left (b g x + a g\right )}^{-m - 2}{\left (d i x + c i\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.646078, size = 2186, normalized size = 9.8 \begin{align*} \text{result too large to display} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}{\left (b g x + a g\right )}^{-m - 2}{\left (d i x + c i\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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