3.2.53 \(\int \frac {e (\frac {a+b x}{c+d x})^n}{(a+b x) (c+d x) (1-e (\frac {a+b x}{c+d x})^n)} \, dx\) [153]

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

[Out]

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

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Rubi [A]
time = 0.21, 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, 6816} \begin {gather*} -\frac {\log \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \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)),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 6816

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 = 0.39, size = 38, normalized size = 1.06 \begin {gather*} -\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} \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)),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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Maple [A]
time = 0.45, size = 37, normalized size = 1.03

method	result	size
norman	\(\frac {\ln \left (-1+e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}\right )}{n \left (a d -c b \right )}\)	\(37\)
risch	\(-\frac {\ln \left (-d x -c \right )}{a d -c b}+\frac {\ln \left (b x +a \right )}{a d -c b}-\frac {\ln \left (\frac {b x +a}{d x +c}\right )}{a d -c b}+\frac {\ln \left (\left (\frac {b x +a}{d x +c}\right )^{n}-\frac {1}{e}\right )}{n \left (a d -c b \right )}\)	\(102\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 61, normalized size = 1.69 \begin {gather*} {\left (\frac {e^{\left (-1\right )} \log \left (d x + c\right )}{b c - a d} - \frac {e^{\left (-1\right )} \log \left ({\left (d x + c\right )}^{n} - e^{\left (n \log \left (b x + a\right ) + 1\right )}\right )}{b c n - a d n}\right )} e \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)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 36, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\left (\frac {b x + a}{d x + c}\right )^{n} e - 1\right )}{{\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)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

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

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