Optimal. Leaf size=125 \[ \frac {e^{-\frac {A (1+m)}{B n}} (a+b x) (g (a+b x))^m \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {1+m}{n}} (i (c+d x))^{-m} \text {Ei}\left (\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B (b c-a d) i^2 n (c+d x)} \]
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Rubi [A]
time = 0.19, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2563, 2347,
2209} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2347
Rule 2563
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 = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m}}{A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 83, normalized size = 0.66 \begin {gather*} -\frac {{\rm Ei}\left (\frac {{\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A + B\right )} m + A + B}{B n}\right ) e^{\left (\frac {B m n \log \left (-i \, g\right ) - {\left (A + B\right )} m - A - B}{B n}\right )}}{{\left (B b c - B a d\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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