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

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

[Out]

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

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Rubi [A]  time = 0.37, 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, 6686} \[ \frac {1}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \]

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)^2),x]

[Out]

1/((b*c - a*d)*n*(1 - e*((a + b*x)/(c + d*x))^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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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 = 0.10, size = 35, normalized size = 0.97 \[ \frac {1}{n (a d-b c) \left (e \left (\frac {a+b x}{c+d x}\right )^n-1\right )} \]

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)^2),x]

[Out]

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

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fricas [A]  time = 1.53, size = 42, normalized size = 1.17 \[ -\frac {1}{{\left (b c - a d\right )} e n \left (\frac {b x + a}{d x + c}\right )^{n} - {\left (b c - a d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^2/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

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

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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.

[In]

integrate(e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c))^n)^2/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.16, size = 56, normalized size = 1.56 \[ \frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -b c \right ) \left (e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}-1\right )} \]

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)^2/(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)

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maxima [A]  time = 0.33, size = 52, normalized size = 1.44 \[ -\frac {{\left (b x + a\right )}^{n} e}{{\left (b c e n - a d e n\right )} {\left (b x + a\right )}^{n} - {\left (b c n - a d n\right )} {\left (d x + c\right )}^{n}} \]

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)^2/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 35, normalized size = 0.97 \[ \frac {1}{n\,\left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n-1\right )\,\left (a\,d-b\,c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

Timed out

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