3.245 \(\int \frac{1}{(a+b x) (c+d x) \log (e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=33 \[ \frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0749734, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2504} \[ \frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

Rule 2504

Int[(u_)/Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)], x_Symbol] :> With[{h
= Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]])/(p*r*(b*c - a*d)), x] /
; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x) (c+d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 0.069878, size = 34, normalized size = 1.03 \[ -\frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

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Maple [F]  time = 0.815, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) } \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.74459, size = 50, normalized size = 1.52 \begin{align*} \frac{\log \left (-\log \left ({\left (b x + a\right )}^{n}\right ) + \log \left ({\left (d x + c\right )}^{n}\right ) - \log \left (e\right )\right )}{b c n - a d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.486093, size = 78, normalized size = 2.36 \begin{align*} \frac{\log \left (n \log \left (\frac{b x + a}{d x + c}\right ) + \log \left (e\right )\right )}{{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.17881, size = 43, normalized size = 1.3 \begin{align*} \frac{\log \left (n \log \left (\frac{b x + a}{d x + c}\right ) + 1\right )}{b c n - a d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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