3.153 \(\int \frac{e (\frac{a+b x}{c+d x})^n}{(a+b x) (c+d x) (1-e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{\log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

[Out]

-(Log[1 - e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n))

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Rubi [A]  time = 0.316718, antiderivative size = 36, normalized size of antiderivative = 1., 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.

[In]

Int[(e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)*(1 - e*((a + b*x)/(c + d*x))^n)),x]

[Out]

-(Log[1 - e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

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*}

Mathematica [A]  time = 0.0914007, 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.

[In]

Integrate[(e*((a + b*x)/(c + d*x))^n)/((a + b*x)*(c + d*x)*(1 - e*((a + b*x)/(c + d*x))^n)),x]

[Out]

-((e*Log[1 - e*((a + b*x)/(c + d*x))^n])/(b*c*e*n - a*d*e*n))

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Maple [A]  time = 0.121, size = 37, normalized size = 1. \begin{align*}{\frac{1}{n \left ( ad-bc \right ) }\ln \left ( e{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

[Out]

1/n/(a*d-b*c)*ln(e*exp(n*ln((b*x+a)/(d*x+c)))-1)

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Maxima [A]  time = 1.01397, size = 78, normalized size = 2.17 \begin{align*} -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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-e*(log(-(b*x + a)^n*e + (d*x + c)^n)/(b*c*e*n - a*d*e*n) - log(d*x + c)/(b*c*e - a*d*e))

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Fricas [A]  time = 2.34046, size = 72, normalized size = 2. \begin{align*} -\frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n} - 1\right )}{{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

-log(e*((b*x + a)/(d*x + c))^n - 1)/((b*c - a*d)*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-e*((b*x+a)/(d*x+c))**n/(-1+e*((b*x+a)/(d*x+c))**n)/(b*x+a)/(d*x+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{e \left (\frac{b x + a}{d x + c}\right )^{n}}{{\left (b x + a\right )}{\left (d x + c\right )}{\left (e \left (\frac{b x + a}{d x + c}\right )^{n} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(-e*((b*x + a)/(d*x + c))^n/((b*x + a)*(d*x + c)*(e*((b*x + a)/(d*x + c))^n - 1)), x)