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

Optimal. Leaf size=41 \[ \frac{\log ^{p+1}\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.108104, antiderivative size = 41, 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 = {2505} \[ \frac{\log ^{p+1}\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2505

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(
b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && NeQ[s, -1]

Rubi steps

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

Mathematica [A]  time = 0.0263713, size = 40, normalized size = 0.98 \[ \frac{\log ^{p+1}\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(p+1) (b c n-a d n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.861, 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 ) ^{p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{p}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*((b*x + a)/(d*x + c))^n)^p/((b*x + a)*(d*x + c)), x)

________________________________________________________________________________________

Fricas [A]  time = 0.537488, size = 153, normalized size = 3.73 \begin{align*} \frac{{\left (n \log \left (\frac{b x + a}{d x + c}\right ) + \log \left (e\right )\right )}{\left (n \log \left (\frac{b x + a}{d x + c}\right ) + \log \left (e\right )\right )}^{p}}{{\left (b c - a d\right )} n p +{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.27801, size = 63, normalized size = 1.54 \begin{align*} \frac{{\left (n \log \left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right ) + 1\right )}^{p + 1}}{{\left (b c n - a d n\right )}{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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