3.2.55 \(\int \frac {e (\frac {a+b x}{c+d x})^n+e^2 (\frac {a+b x}{c+d x})^{2 n}}{(a+b x) (c+d x) (1-e (\frac {a+b x}{c+d x})^n)^3} \, dx\) [155]

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

[Out]

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

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Rubi [A]
time = 1.44, 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 = {6873, 12, 6824, 34} \begin {gather*} \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} \end {gather*}

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 12

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

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]

Rule 6824

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 6873

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

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 {\text {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 = 0.50, size = 52, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.51, size = 53, normalized size = 1.02

method	result	size
risch	\(-\frac {e \left (\frac {b x +a}{d x +c}\right )^{n}}{n \left (a d -c b \right ) \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}\)	\(53\)
norman	\(-\frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -c b \right ) \left (-1+e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}\right )^{2}}\)	\(57\)

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,method=_RE
TURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (53) = 106\).
time = 0.30, size = 207, normalized size = 3.98 \begin {gather*} -\frac {1}{2} \, {\left (\frac {e^{\left (2 \, n \log \left (b x + a\right ) + 1\right )} - 2 \, e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}{{\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} + {\left (b c n - a d n\right )} e^{\left (2 \, n \log \left (b x + a\right ) + 2\right )} - 2 \, {\left (b c n - a d n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right ) + 1\right )}} - \frac {e^{\left (2 \, n \log \left (b x + a\right ) + 1\right )}}{{\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} + {\left (b c n - a d n\right )} e^{\left (2 \, n \log \left (b x + a\right ) + 2\right )} - 2 \, {\left (b c n - a d n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right ) + 1\right )}}\right )} e \end {gather*}

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

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Fricas [A]
time = 0.37, size = 88, normalized size = 1.69 \begin {gather*} \frac {\left (\frac {b x + a}{d x + c}\right )^{n} e}{{\left (b c - a d\right )} n \left (\frac {b x + a}{d x + c}\right )^{2 \, n} e^{2} - 2 \, {\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.

[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]

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

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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.

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.25, size = 81, normalized size = 1.56 \begin {gather*} -\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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