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 = 0.62, 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 12
Rule 37
Rule 2554
Rule 6742
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*}
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Mathematica [A] time = 0.52, 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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fricas [B] time = 0.91, size = 274, normalized size = 2.14 \[ \frac {{\left (A a c m - B a c n + A a c + {\left (A b d m - B b d n + A b d\right )} x^{2} + {\left (A b c + A a d + {\left (A b c + A a d\right )} m - {\left (B b c + B a d\right )} n\right )} x + {\left (B a c m + B a c + {\left (B b d m + B b d\right )} x^{2} + {\left (B b c + B a d + {\left (B b c + B a d\right )} m\right )} x\right )} \log \relax (e) + {\left ({\left (B b d m + B b d\right )} n x^{2} + {\left (B b c + B a d + {\left (B b c + B a d\right )} m\right )} n x + {\left (B a c m + B a c\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{m} e^{\left (-{\left (m + 2\right )} \log \left (b g x + a g\right ) + {\left (m + 2\right )} \log \left (\frac {b x + a}{d x + c}\right ) - {\left (m + 2\right )} \log \left (\frac {i}{g}\right )\right )}}{{\left (b c - a d\right )} m^{2} + b c - a d + 2 \, {\left (b c - a d\right )} m} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )} {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 8.82, size = 0, normalized size = 0.00 \[ \int \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right ) \left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-m -2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )} {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^m\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (c\,i+d\,i\,x\right )}^{m+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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