Optimal. Leaf size=36 \[ \frac {1}{(b c-a d) n \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \]
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Rubi [A]
time = 0.24, 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, 6818}
\begin {gather*} \frac {1}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
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 = 0.36, size = 35, normalized size = 0.97 \begin {gather*} \frac {1}{(-b c+a d) n \left (-1+e \left (\frac {a+b x}{c+d x}\right )^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 36, normalized size = 1.00
method | result | size |
risch | \(\frac {1}{n \left (a d -c b \right ) \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\) | \(36\) |
norman | \(\frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -c b \right ) \left (-1+e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}\right )}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 56, normalized size = 1.56 \begin {gather*} \frac {e^{\left (n \log \left (b x + a\right ) + 1\right )}}{{\left (b c n - a d n\right )} {\left (d x + c\right )}^{n} - {\left (b c n - a d n\right )} e^{\left (n \log \left (b x + a\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 43, normalized size = 1.19 \begin {gather*} -\frac {1}{{\left (b c - a d\right )} n \left (\frac {b x + a}{d x + c}\right )^{n} e - {\left (b c - a d\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 0.19, size = 35, normalized size = 0.97 \begin {gather*} \frac {1}{n\,\left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n-1\right )\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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