3.247 \(\int \frac{c+d x}{(a+b x)^3 \log (e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=75 \[ \frac{(c+d x)^2 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{2/n} \text{Ei}\left (-\frac{2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

[Out]

((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a
*d)*n*(a + b*x)^2)

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Rubi [A]  time = 0.0472205, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {2510} \[ \frac{(c+d x)^2 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{2/n} \text{Ei}\left (-\frac{2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/((a + b*x)^3*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a
*d)*n*(a + b*x)^2)

Rule 2510

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

Rubi steps

\begin{align*} \int \frac{c+d x}{(a+b x)^3 \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 )^{2/n} (c+d x)^2 \text{Ei}\left (-\frac{2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (a+b x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0165232, size = 75, normalized size = 1. \[ \frac{(c+d x)^2 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{2/n} \text{Ei}\left (-\frac{2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/((a + b*x)^3*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a
*d)*n*(a + b*x)^2)

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Maple [F]  time = 0.439, size = 0, normalized size = 0. \begin{align*} \int{\frac{dx+c}{ \left ( bx+a \right ) ^{3}} \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((d*x+c)/(b*x+a)^3/ln(e*((b*x+a)/(d*x+c))^n),x)

[Out]

int((d*x+c)/(b*x+a)^3/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{d x + c}{{\left (b x + a\right )}^{3} \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((d*x+c)/(b*x+a)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="maxima")

[Out]

integrate((d*x + c)/((b*x + a)^3*log(e*((b*x + a)/(d*x + c))^n)), x)

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Fricas [A]  time = 0.511544, size = 136, normalized size = 1.81 \begin{align*} \frac{e^{\frac{2}{n}} \logintegral \left (\frac{d^{2} x^{2} + 2 \, c d x + c^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\frac{2}{n}}}\right )}{{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fricas")

[Out]

e^(2/n)*log_integral((d^2*x^2 + 2*c*d*x + c^2)/((b^2*x^2 + 2*a*b*x + a^2)*e^(2/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((d*x+c)/(b*x+a)**3/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{d x + c}{{\left (b x + a\right )}^{3} \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((d*x+c)/(b*x+a)^3/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="giac")

[Out]

integrate((d*x + c)/((b*x + a)^3*log(e*((b*x + a)/(d*x + c))^n)), x)