3.12 \(\int x (a+b \log (c x^n))^2 \log (1+e x) \, dx\)

Optimal. Leaf size=327 \[ -\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{b^2 n^2 \text{PolyLog}(2,-e x)}{2 e^2}+\frac{b^2 n^2 \text{PolyLog}(3,-e x)}{e^2}+\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{1}{2} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac{1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{a b n x}{e}-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b^2 n^2 \log (e x+1)}{4 e^2}+\frac{1}{4} b^2 n^2 x^2 \log (e x+1)+\frac{7 b^2 n^2 x}{4 e}-\frac{3}{8} b^2 n^2 x^2 \]

[Out]

-((a*b*n*x)/e) + (7*b^2*n^2*x)/(4*e) - (3*b^2*n^2*x^2)/8 - (b^2*n*x*Log[c*x^n])/e - (b*n*x*(a + b*Log[c*x^n]))
/(2*e) + (b*n*x^2*(a + b*Log[c*x^n]))/2 + (x*(a + b*Log[c*x^n])^2)/(2*e) - (x^2*(a + b*Log[c*x^n])^2)/4 - (b^2
*n^2*Log[1 + e*x])/(4*e^2) + (b^2*n^2*x^2*Log[1 + e*x])/4 + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(2*e^2) - (b
*n*x^2*(a + b*Log[c*x^n])*Log[1 + e*x])/2 - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(2*e^2) + (x^2*(a + b*Log[c*x^
n])^2*Log[1 + e*x])/2 + (b^2*n^2*PolyLog[2, -(e*x)])/(2*e^2) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/e^2
+ (b^2*n^2*PolyLog[3, -(e*x)])/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.218963, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{b^2 n^2 \text{PolyLog}(2,-e x)}{2 e^2}+\frac{b^2 n^2 \text{PolyLog}(3,-e x)}{e^2}+\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{1}{2} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac{1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{a b n x}{e}-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b^2 n^2 \log (e x+1)}{4 e^2}+\frac{1}{4} b^2 n^2 x^2 \log (e x+1)+\frac{7 b^2 n^2 x}{4 e}-\frac{3}{8} b^2 n^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*b*n*x)/e) + (7*b^2*n^2*x)/(4*e) - (3*b^2*n^2*x^2)/8 - (b^2*n*x*Log[c*x^n])/e - (b*n*x*(a + b*Log[c*x^n]))
/(2*e) + (b*n*x^2*(a + b*Log[c*x^n]))/2 + (x*(a + b*Log[c*x^n])^2)/(2*e) - (x^2*(a + b*Log[c*x^n])^2)/4 - (b^2
*n^2*Log[1 + e*x])/(4*e^2) + (b^2*n^2*x^2*Log[1 + e*x])/4 + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(2*e^2) - (b
*n*x^2*(a + b*Log[c*x^n])*Log[1 + e*x])/2 - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(2*e^2) + (x^2*(a + b*Log[c*x^
n])^2*Log[1 + e*x])/2 + (b^2*n^2*PolyLog[2, -(e*x)])/(2*e^2) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/e^2
+ (b^2*n^2*PolyLog[3, -(e*x)])/e^2

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2377

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-(2 b n) \int \left (\frac{a+b \log \left (c x^n\right )}{2 e}-\frac{1}{4} x \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2 x}+\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)\right ) \, dx\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{1}{2} (b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e^2}-\frac{(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac{a b n x}{e}-\frac{1}{8} b^2 n^2 x^2-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e^2}-\frac{\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e}+\left (b^2 n^2\right ) \int \left (\frac{1}{2 e}-\frac{x}{4}-\frac{\log (1+e x)}{2 e^2 x}+\frac{1}{2} x \log (1+e x)\right ) \, dx+\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{e^2}\\ &=-\frac{a b n x}{e}+\frac{3 b^2 n^2 x}{2 e}-\frac{1}{4} b^2 n^2 x^2-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e^2}+\frac{b^2 n^2 \text{Li}_3(-e x)}{e^2}+\frac{1}{2} \left (b^2 n^2\right ) \int x \log (1+e x) \, dx-\frac{\left (b^2 n^2\right ) \int \frac{\log (1+e x)}{x} \, dx}{2 e^2}\\ &=-\frac{a b n x}{e}+\frac{3 b^2 n^2 x}{2 e}-\frac{1}{4} b^2 n^2 x^2-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b^2 n^2 x^2 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e^2}+\frac{b^2 n^2 \text{Li}_3(-e x)}{e^2}-\frac{1}{4} \left (b^2 e n^2\right ) \int \frac{x^2}{1+e x} \, dx\\ &=-\frac{a b n x}{e}+\frac{3 b^2 n^2 x}{2 e}-\frac{1}{4} b^2 n^2 x^2-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} b^2 n^2 x^2 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e^2}+\frac{b^2 n^2 \text{Li}_3(-e x)}{e^2}-\frac{1}{4} \left (b^2 e n^2\right ) \int \left (-\frac{1}{e^2}+\frac{x}{e}+\frac{1}{e^2 (1+e x)}\right ) \, dx\\ &=-\frac{a b n x}{e}+\frac{7 b^2 n^2 x}{4 e}-\frac{3}{8} b^2 n^2 x^2-\frac{b^2 n x \log \left (c x^n\right )}{e}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 n^2 \log (1+e x)}{4 e^2}+\frac{1}{4} b^2 n^2 x^2 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac{1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{2 e^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{e^2}+\frac{b^2 n^2 \text{Li}_3(-e x)}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.128752, size = 416, normalized size = 1.27 \[ \frac{4 b n \text{PolyLog}(2,-e x) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+8 b^2 n^2 \text{PolyLog}(3,-e x)-2 a^2 e^2 x^2+4 a^2 e^2 x^2 \log (e x+1)+4 a^2 e x-4 a^2 \log (e x+1)-4 a b e^2 x^2 \log \left (c x^n\right )+8 a b e^2 x^2 \log (e x+1) \log \left (c x^n\right )+8 a b e x \log \left (c x^n\right )-8 a b \log (e x+1) \log \left (c x^n\right )+4 a b e^2 n x^2-4 a b e^2 n x^2 \log (e x+1)-12 a b e n x+4 a b n \log (e x+1)-2 b^2 e^2 x^2 \log ^2\left (c x^n\right )+4 b^2 e^2 x^2 \log (e x+1) \log ^2\left (c x^n\right )+4 b^2 e^2 n x^2 \log \left (c x^n\right )-4 b^2 e^2 n x^2 \log (e x+1) \log \left (c x^n\right )+4 b^2 e x \log ^2\left (c x^n\right )-4 b^2 \log (e x+1) \log ^2\left (c x^n\right )-12 b^2 e n x \log \left (c x^n\right )+4 b^2 n \log (e x+1) \log \left (c x^n\right )-3 b^2 e^2 n^2 x^2+2 b^2 e^2 n^2 x^2 \log (e x+1)+14 b^2 e n^2 x-2 b^2 n^2 \log (e x+1)}{8 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*e*x - 12*a*b*e*n*x + 14*b^2*e*n^2*x - 2*a^2*e^2*x^2 + 4*a*b*e^2*n*x^2 - 3*b^2*e^2*n^2*x^2 + 8*a*b*e*x*L
og[c*x^n] - 12*b^2*e*n*x*Log[c*x^n] - 4*a*b*e^2*x^2*Log[c*x^n] + 4*b^2*e^2*n*x^2*Log[c*x^n] + 4*b^2*e*x*Log[c*
x^n]^2 - 2*b^2*e^2*x^2*Log[c*x^n]^2 - 4*a^2*Log[1 + e*x] + 4*a*b*n*Log[1 + e*x] - 2*b^2*n^2*Log[1 + e*x] + 4*a
^2*e^2*x^2*Log[1 + e*x] - 4*a*b*e^2*n*x^2*Log[1 + e*x] + 2*b^2*e^2*n^2*x^2*Log[1 + e*x] - 8*a*b*Log[c*x^n]*Log
[1 + e*x] + 4*b^2*n*Log[c*x^n]*Log[1 + e*x] + 8*a*b*e^2*x^2*Log[c*x^n]*Log[1 + e*x] - 4*b^2*e^2*n*x^2*Log[c*x^
n]*Log[1 + e*x] - 4*b^2*Log[c*x^n]^2*Log[1 + e*x] + 4*b^2*e^2*x^2*Log[c*x^n]^2*Log[1 + e*x] + 4*b*n*(-2*a + b*
n - 2*b*Log[c*x^n])*PolyLog[2, -(e*x)] + 8*b^2*n^2*PolyLog[3, -(e*x)])/(8*e^2)

________________________________________________________________________________________

Maple [F]  time = 0.239, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( ex+1 \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b^{2} e^{2} x^{2} - 2 \, b^{2} e x - 2 \,{\left (b^{2} e^{2} x^{2} - b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{4 \, e^{2}} + \frac{-\frac{1}{4} \, b^{2} e^{2} n^{2} x^{2} + \frac{1}{2} \, b^{2} e^{2} n x^{2} \log \left (x^{n}\right ) + \frac{1}{2} \,{\left (2 \, x^{2} \log \left (e x + 1\right ) - e{\left (\frac{e x^{2} - 2 \, x}{e^{2}} + \frac{2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} b^{2} e^{2} \log \left (c\right )^{2} + 2 \, b^{2} e n^{2} x +{\left (2 \, x^{2} \log \left (e x + 1\right ) - e{\left (\frac{e x^{2} - 2 \, x}{e^{2}} + \frac{2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a b e^{2} \log \left (c\right ) - 2 \, b^{2} e n x \log \left (x^{n}\right ) + \frac{1}{2} \,{\left (2 \, x^{2} \log \left (e x + 1\right ) - e{\left (\frac{e x^{2} - 2 \, x}{e^{2}} + \frac{2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a^{2} e^{2} + \int \frac{2 \,{\left (b^{2} n +{\left (2 \, a b e^{2} -{\left (e^{2} n - 2 \, e^{2} \log \left (c\right )\right )} b^{2}\right )} x^{2}\right )} \log \left (e x + 1\right ) \log \left (x^{n}\right )}{x}\,{d x}}{2 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*e^2*x^2 - 2*b^2*e*x - 2*(b^2*e^2*x^2 - b^2)*log(e*x + 1))*log(x^n)^2/e^2 + 1/2*integrate((2*(b^2*e^2
*log(c)^2 + 2*a*b*e^2*log(c) + a^2*e^2)*x^2*log(e*x + 1) + (b^2*e^2*n*x^2 - 2*b^2*e*n*x + 2*(b^2*n + (2*a*b*e^
2 - (e^2*n - 2*e^2*log(c))*b^2)*x^2)*log(e*x + 1))*log(x^n))/x, x)/e^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x*log(c*x^n)^2*log(e*x + 1) + 2*a*b*x*log(c*x^n)*log(e*x + 1) + a^2*x*log(e*x + 1), x)

________________________________________________________________________________________

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(x*(a+b*ln(c*x**n))**2*ln(e*x+1),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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