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]

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

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Rubi [A]  time = 0.32, 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, 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*}

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Mathematica [A]  time = 0.09, 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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fricas [A]  time = 1.60, size = 35, normalized size = 0.97 \[ -\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n} - 1\right )}{{\left (b c - a d\right )} n} \]

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)/(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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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)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.10, size = 37, normalized size = 1.03 \[ \frac {\ln \left (e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}-1\right )}{n \left (a d -b c \right )} \]

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)/(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 = 0.33, size = 58, normalized size = 1.61 \[ -e {\left (\frac {\log \left (-{\left (b x + a\right )}^{n} e + {\left (d x + c\right )}^{n}\right )}{b c e n - a d e n} - \frac {\log \left (d x + c\right )}{b c e - a d e}\right )} \]

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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mupad [B]  time = 0.27, size = 33, normalized size = 0.92 \[ \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n-1\right )}{a\,d\,n-b\,c\,n} \]

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)*(a + b*x)*(c + d*x)),x)

[Out]

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

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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)/(b*x+a)/(d*x+c),x)

[Out]

Timed out

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