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

Optimal. Leaf size=71 \[ \frac{(c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0289577, antiderivative size = 71, 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} \[ \frac{(c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-(Log[e*((a + b*x)/(c + d*x))^n]/n)])/((b*c - a*d)
*n*(a + b*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}{(a+b x)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} (c+d x) \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 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0688921, size = 71, normalized size = 1. \[ \frac{(c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.44, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \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/(b*x+a)^2/ln(e*((b*x+a)/(d*x+c))^n),x)

[Out]

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

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

[Out]

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

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Fricas [A]  time = 0.487622, size = 93, normalized size = 1.31 \begin{align*} \frac{e^{\left (\frac{1}{n}\right )} \logintegral \left (\frac{d x + c}{{\left (b x + a\right )} e^{\left (\frac{1}{n}\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)^2/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

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