3.244 \(\int \frac{1}{(c+d x)^2 \log (e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=72 \[ \frac{(a+b x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0315986, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2493} \[ \frac{(a+b x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

Rule 2493

Int[1/(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^2),
x_Symbol] :> Simp[(b*(c + d*x)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^(1/(p*r))*ExpIntegralEi[-(Log[e*(f*(a + b*x)
^p*(c + d*x)^q)^r]/(p*r))])/(h*p*r*(b*c - a*d)*(g + h*x)), x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] &
& NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[b*g - a*h, 0]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{(a+b x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.0707403, size = 72, normalized size = 1. \[ \frac{(a+b x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2}} \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/(d*x+c)^2/ln(e*((b*x+a)/(d*x+c))^n),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.456603, size = 90, normalized size = 1.25 \begin{align*} \frac{\logintegral \left (\frac{{\left (b x + a\right )} e^{\left (\frac{1}{n}\right )}}{d x + c}\right )}{{\left (b c - a d\right )} e^{\left (\frac{1}{n}\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="giac")

[Out]

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