3.155 \(\int \frac{e (\frac{a+b x}{c+d x})^n+e^2 (\frac{a+b x}{c+d x})^{2 n}}{(a+b x) (c+d x) (1-e (\frac{a+b x}{c+d x})^n)^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{e \left (\frac{a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^2} \]

[Out]

(e*((a + b*x)/(c + d*x))^n)/((b*c - a*d)*n*(1 - e*((a + b*x)/(c + d*x))^n)^2)

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Rubi [A]  time = 2.02456, antiderivative size = 52, normalized size of antiderivative = 1., 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.

[In]

Int[(e*((a + b*x)/(c + d*x))^n + e^2*((a + b*x)/(c + d*x))^(2*n))/((a + b*x)*(c + d*x)*(1 - e*((a + b*x)/(c +
d*x))^n)^3),x]

[Out]

(e*((a + b*x)/(c + d*x))^n)/((b*c - a*d)*n*(1 - e*((a + b*x)/(c + d*x))^n)^2)

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

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

Rule 6692

Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((a_.) + (b_.)*(y_))^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u,
x]}, Dist[q, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y], x] /;  !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n}, x]
&& EqQ[v, y]

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(d*x*(a + b*x)^(m + 1))/(b*(m + 2)), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

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*}

Mathematica [A]  time = 0.265491, size = 52, normalized size = 1. \[ -\frac{e \left (\frac{a+b x}{c+d x}\right )^n}{n (a d-b c) \left (e \left (\frac{a+b x}{c+d x}\right )^n-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*((a + b*x)/(c + d*x))^n + e^2*((a + b*x)/(c + d*x))^(2*n))/((a + b*x)*(c + d*x)*(1 - e*((a + b*x)
/(c + d*x))^n)^3),x]

[Out]

-((e*((a + b*x)/(c + d*x))^n)/((-(b*c) + a*d)*n*(-1 + e*((a + b*x)/(c + d*x))^n)^2))

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Maple [A]  time = 0.276, size = 57, normalized size = 1.1 \begin{align*} -{\frac{e}{n \left ( ad-bc \right ) }{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }} \left ( e{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }}-1 \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^3/(b*x+a)/(d*x+c),x)

[Out]

-e/n/(a*d-b*c)*exp(n*ln((b*x+a)/(d*x+c)))/(e*exp(n*ln((b*x+a)/(d*x+c)))-1)^2

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Maxima [B]  time = 1.17927, size = 285, normalized size = 5.48 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^3/(b*x+a)/(d*x+c),x, alg
orithm="maxima")

[Out]

1/2*((b*x + a)^(2*n)*e/((b*c*e^2*n - a*d*e^2*n)*(b*x + a)^(2*n) + (b*c*n - a*d*n)*(d*x + c)^(2*n) - 2*(b*c*e*n
- a*d*e*n)*e^(n*log(b*x + a) + n*log(d*x + c))) - ((b*x + a)^(2*n)*e - 2*e^(n*log(b*x + a) + n*log(d*x + c)))
/((b*c*e^2*n - a*d*e^2*n)*(b*x + a)^(2*n) + (b*c*n - a*d*n)*(d*x + c)^(2*n) - 2*(b*c*e*n - a*d*e*n)*e^(n*log(b
*x + a) + n*log(d*x + c))))*e

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Fricas [A]  time = 2.27585, size = 182, normalized size = 3.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^3/(b*x+a)/(d*x+c),x, alg
orithm="fricas")

[Out]

e*((b*x + a)/(d*x + c))^n/((b*c - a*d)*e^2*n*((b*x + a)/(d*x + c))^(2*n) - 2*(b*c - a*d)*e*n*((b*x + a)/(d*x +
c))^n + (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(-(1+e*((b*x+a)/(d*x+c))**n)*e*((b*x+a)/(d*x+c))**n/(-1+e*((b*x+a)/(d*x+c))**n)**3/(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{{\left (e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )} 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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^3/(b*x+a)/(d*x+c),x, alg
orithm="giac")

[Out]

integrate(-(e*((b*x + a)/(d*x + c))^n + 1)*e*((b*x + a)/(d*x + c))^n/((b*x + a)*(d*x + c)*(e*((b*x + a)/(d*x +
c))^n - 1)^3), x)