Optimal. Leaf size=36 \[ \frac {1}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \]
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Rubi [A] time = 0.37, 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, 6686} \[ \frac {1}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6686
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 )^2} \, 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 )^2} \, dx\\ &=\frac {1}{(b c-a d) n \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 35, normalized size = 0.97 \[ \frac {1}{n (a d-b c) \left (e \left (\frac {a+b x}{c+d x}\right )^n-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 42, normalized size = 1.17 \[ -\frac {1}{{\left (b c - a d\right )} e n \left (\frac {b x + a}{d x + c}\right )^{n} - {\left (b c - a d\right )} n} \]
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 = 0.16, size = 56, normalized size = 1.56 \[ \frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -b c \right ) \left (e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 52, normalized size = 1.44 \[ -\frac {{\left (b x + a\right )}^{n} e}{{\left (b c e n - a d e n\right )} {\left (b x + a\right )}^{n} - {\left (b c n - a d n\right )} {\left (d x + c\right )}^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 35, normalized size = 0.97 \[ \frac {1}{n\,\left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n-1\right )\,\left (a\,d-b\,c\right )} \]
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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