Optimal. Leaf size=125 \[ \frac {(a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B i^2 n (c+d x) (b c-a d)} \]
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Rubi [F] time = 0.74, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\int \frac {(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.83, size = 98, normalized size = 0.78 \[ \frac {{\rm Ei}\left (\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right ) e^{\left (-\frac {{\left (B m + 2 \, B\right )} n \log \left (\frac {i}{g}\right ) + A m + {\left (B m + B\right )} \log \relax (e) + A}{B n}\right )}}{{\left (B b c - B a d\right )} g^{2} n} \]
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 \[ \int \frac {{\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-m -2}}{B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}\, dx \]
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 \[ \int \frac {{\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \]
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 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^m}{{\left (c\,i+d\,i\,x\right )}^{m+2}\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )} \,d x \]
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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