Optimal. Leaf size=52 \[ \frac {e \left (\frac {a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^2} \]
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Rubi [A] time = 2.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6741, 12, 6692, 34} \[ \frac {e \left (\frac {a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 34
Rule 6692
Rule 6741
Rubi steps
\begin {align*} \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx &=\int \frac {e \left (\frac {a+b x}{c+d x}\right )^n \left (1+e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx\\ &=e \int \frac {\left (\frac {a+b x}{c+d x}\right )^n \left (1+e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1+x}{(1-x)^3} \, dx,x,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ &=\frac {e \left (\frac {a+b x}{c+d x}\right )^n}{(b c-a d) n \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 52, normalized size = 1.00 \[ -\frac {e \left (\frac {a+b x}{c+d x}\right )^n}{n (a d-b c) \left (e \left (\frac {a+b x}{c+d x}\right )^n-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.10, size = 87, normalized size = 1.67 \[ \frac {e \left (\frac {b x + a}{d x + c}\right )^{n}}{{\left (b c - a d\right )} e^{2} n \left (\frac {b x + a}{d x + c}\right )^{2 \, n} - 2 \, {\left (b c - a d\right )} e n \left (\frac {b x + a}{d x + c}\right )^{n} + {\left (b c - a d\right )} n} \]
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 = 0.28, size = 57, normalized size = 1.10 \[ -\frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -b c \right ) \left (e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 211, normalized size = 4.06 \[ \frac {1}{2} \, {\left (\frac {{\left (b x + a\right )}^{2 \, n} e}{{\left (b c e^{2} n - a d e^{2} n\right )} {\left (b x + a\right )}^{2 \, n} + {\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} - 2 \, {\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}} - \frac {{\left (b x + a\right )}^{2 \, n} e - 2 \, e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}{{\left (b c e^{2} n - a d e^{2} n\right )} {\left (b x + a\right )}^{2 \, n} + {\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} - 2 \, {\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}\right )} e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 81, normalized size = 1.56 \[ -\frac {e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n}{n\,\left (a\,d-b\,c\right )\,\left (e^2\,{\left (\frac {a}{c+d\,x}+\frac {b\,x}{c+d\,x}\right )}^{2\,n}-2\,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n+1\right )} \]
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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