Optimal. Leaf size=128 \[ \frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1) (c+d x) (b c-a d)}-\frac{B n (a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{i^2 (m+1)^2 (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.615875, antiderivative size = 168, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 4, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.085, Rules used = {6742, 37, 2554, 12} \[ \frac{A (a g+b g x)^{m+1} (c i+d i x)^{-m-1}}{g i (m+1) (b c-a d)}+\frac{B (a g+b g x)^{m+1} (c i+d i x)^{-m-1} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g i (m+1) (b c-a d)}-\frac{B n (a g+b g x)^{m+1} (c i+d i x)^{-m-1}}{g i (m+1)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 37
Rule 2554
Rule 12
Rubi steps
\begin{align*} \int (214 c+214 d x)^{-2-m} (a g+b g x)^m \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (A (214 c+214 d x)^{-2-m} (a g+b g x)^m+B (214 c+214 d x)^{-2-m} (a g+b g x)^m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=A \int (214 c+214 d x)^{-2-m} (a g+b g x)^m \, dx+B \int (214 c+214 d x)^{-2-m} (a g+b g x)^m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=\frac{A (214 c+214 d x)^{-1-m} (a g+b g x)^{1+m}}{214 (b c-a d) g (1+m)}+\frac{B (214 c+214 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 )}{214 (b c-a d) g (1+m)}-B \int \frac{214^{-2-m} n (c+d x)^{-2-m} (a g+b g x)^m}{1+m} \, dx\\ &=\frac{A (214 c+214 d x)^{-1-m} (a g+b g x)^{1+m}}{214 (b c-a d) g (1+m)}+\frac{B (214 c+214 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 )}{214 (b c-a d) g (1+m)}-\frac{\left (214^{-2-m} B n\right ) \int (c+d x)^{-2-m} (a g+b g x)^m \, dx}{1+m}\\ &=-\frac{214^{-2-m} B n (c+d x)^{-1-m} (a g+b g x)^{1+m}}{(b c-a d) g (1+m)^2}+\frac{A (214 c+214 d x)^{-1-m} (a g+b g x)^{1+m}}{214 (b c-a d) g (1+m)}+\frac{B (214 c+214 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 )}{214 (b c-a d) g (1+m)}\\ \end{align*}
Mathematica [A] time = 0.522248, size = 78, normalized size = 0.61 \[ \frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m-1} \left (B (m+1) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A m+A-B n\right )}{i (m+1)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.637, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{m} \left ( dix+ci \right ) ^{-2-m} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, 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 )}{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.573072, size = 647, normalized size = 5.05 \begin{align*} \frac{{\left (A a c m - B a c n + A a c +{\left (A b d m - B b d n + A b d\right )} x^{2} +{\left (A b c + A a d +{\left (A b c + A a d\right )} m -{\left (B b c + B a d\right )} n\right )} x +{\left (B a c m + B a c +{\left (B b d m + B b d\right )} x^{2} +{\left (B b c + B a d +{\left (B b c + B a d\right )} m\right )} x\right )} \log \left (e\right ) +{\left ({\left (B b d m + B b d\right )} n x^{2} +{\left (B b c + B a d +{\left (B b c + B a d\right )} m\right )} n x +{\left (B a c m + B a c\right )} n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )}{\left (b g x + a g\right )}^{m} e^{\left (-{\left (m + 2\right )} \log \left (b g x + a g\right ) +{\left (m + 2\right )} \log \left (\frac{b x + a}{d x + c}\right ) -{\left (m + 2\right )} \log \left (\frac{i}{g}\right )\right )}}{{\left (b c - a d\right )} m^{2} + b c - a d + 2 \,{\left (b c - a d\right )} m} \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 )}{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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