3.242 \(\int \frac{(a+b x)^2}{(c+d x)^4 \log (e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-3/n} \text{Ei}\left (\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^3 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0731964, antiderivative size = 75, 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 = {2510} \[ \frac{(a+b x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-3/n} \text{Ei}\left (\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/((c + d*x)^4*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

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{(a+b x)^2}{(c+d x)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{(a+b x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-3/n} \text{Ei}\left (\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0240413, size = 75, normalized size = 1. \[ \frac{(a+b x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-3/n} \text{Ei}\left (\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/((c + d*x)^4*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.643, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{4}} \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((b*x+a)^2/(d*x+c)^4/ln(e*((b*x+a)/(d*x+c))^n),x)

[Out]

int((b*x+a)^2/(d*x+c)^4/ln(e*((b*x+a)/(d*x+c))^n),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2}}{{\left (d x + c\right )}^{4} \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((b*x+a)^2/(d*x+c)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^2/((d*x + c)^4*log(e*((b*x + a)/(d*x + c))^n)), x)

________________________________________________________________________________________

Fricas [A]  time = 0.475508, size = 177, normalized size = 2.36 \begin{align*} \frac{\logintegral \left (\frac{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} e^{\frac{3}{n}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{{\left (b c - a d\right )} e^{\frac{3}{n}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(d*x+c)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="fricas")

[Out]

log_integral((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*e^(3/n)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/((b*
c - a*d)*e^(3/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((b*x+a)**2/(d*x+c)**4/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{{\left (b x + a\right )}^{2}}{{\left (d x + c\right )}^{4} \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((b*x+a)^2/(d*x+c)^4/log(e*((b*x+a)/(d*x+c))^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^2/((d*x + c)^4*log(e*((b*x + a)/(d*x + c))^n)), x)