Optimal. Leaf size=206 \[ \frac {e^{-\frac {A (1+m)}{B n}} (1+m) (a+b x) (g (a+b x))^m \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {1+m}{n}} (i (c+d x))^{-m} \text {Ei}\left (\frac {(1+m) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d) i^2 n^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B (b c-a d) i^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]
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Rubi [A]
time = 0.22, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {2563, 2343,
2347, 2209} \begin {gather*} \frac {(m+1) (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^2 i^2 n^2 (c+d x) (b c-a d)}-\frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2347
Rule 2563
Rubi steps
\begin {align*} \int \frac {(216 c+216 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 )^2} \, dx &=\int \frac {(216 c+216 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 )^2} \, dx\\ \end {align*}
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Mathematica [F]
time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m}}{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}\, dx\]
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 \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 = 0.40, size = 240, normalized size = 1.17 \begin {gather*} -\frac {{\left (B b d n x^{2} + B a c n + {\left (B b c + B a d\right )} n x\right )} {\left (i \, d x + i \, c\right )}^{-m - 2} e^{\left (m \log \left (i \, d x + i \, c\right ) + m \log \left (-i \, g\right ) + m \log \left (\frac {b x + a}{d x + c}\right )\right )} + {\left ({\left (B m + B\right )} n \log \left (\frac {b x + a}{d x + c}\right ) + {\left (A + B\right )} m + A + B\right )} {\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^{3} b c - B^{3} a d\right )} n^{3} \log \left (\frac {b x + a}{d x + c}\right ) + {\left ({\left (A B^{2} + B^{3}\right )} b c - {\left (A B^{2} + B^{3}\right )} a d\right )} n^{2}} \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 = -0.00 \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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