Optimal. Leaf size=36 \[ -\frac {\log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
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Rubi [A] time = 0.32, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 6684} \[ -\frac {\log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6684
Rubi steps
\begin {align*} \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=e \int \frac {\left (\frac {a+b x}{c+d x}\right )^n}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx\\ &=-\frac {\log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 38, normalized size = 1.06 \[ -\frac {e \log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c e n-a d e n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 35, normalized size = 0.97 \[ -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n} - 1\right )}{{\left (b c - a d\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.10, size = 37, normalized size = 1.03 \[ \frac {\ln \left (e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}-1\right )}{n \left (a d -b c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 58, normalized size = 1.61 \[ -e {\left (\frac {\log \left (-{\left (b x + a\right )}^{n} e + {\left (d x + c\right )}^{n}\right )}{b c e n - a d e n} - \frac {\log \left (d x + c\right )}{b c e - a d e}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 33, normalized size = 0.92 \[ \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n-1\right )}{a\,d\,n-b\,c\,n} \]
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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