3.2.54 \(\int \frac {e (\frac {a+b x}{c+d x})^n}{(a+b x) (c+d x) (1-e (\frac {a+b x}{c+d x})^n)^2} \, dx\) [154]

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

[Out]

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

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Rubi [A]
time = 0.24, 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, 6818} \begin {gather*} \frac {1}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )} \end {gather*}

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 6818

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 = 0.36, size = 35, normalized size = 0.97 \begin {gather*} \frac {1}{(-b c+a d) n \left (-1+e \left (\frac {a+b x}{c+d x}\right )^n\right )} \end {gather*}

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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Maple [A]
time = 0.43, size = 36, normalized size = 1.00

method	result	size
risch	\(\frac {1}{n \left (a d -c b \right ) \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\)	\(36\)
norman	\(\frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -c b \right ) \left (-1+e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}\right )}\)	\(56\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 56, normalized size = 1.56 \begin {gather*} \frac {e^{\left (n \log \left (b x + a\right ) + 1\right )}}{{\left (b c n - a d n\right )} {\left (d x + c\right )}^{n} - {\left (b c n - a d n\right )} e^{\left (n \log \left (b x + a\right ) + 1\right )}} \end {gather*}

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]

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

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Fricas [A]
time = 0.38, size = 43, normalized size = 1.19 \begin {gather*} -\frac {1}{{\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(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)*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(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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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(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]

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

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

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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