3.1.11 \(\int x^2 (a+b \log (c x^n))^2 \log (1+e x) \, dx\) [11]

Optimal. Leaf size=396 \[ \frac {2 a b n x}{3 e^2}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {19 b^2 n^2 x^2}{108 e}-\frac {2}{27} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 \log (1+e x)}{27 e^3}+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3} \]

[Out]

2/3*a*b*n*x/e^2-26/27*b^2*n^2*x/e^2+19/108*b^2*n^2*x^2/e-2/27*b^2*n^2*x^3+2/3*b^2*n*x*ln(c*x^n)/e^2+2/9*b*n*x*
(a+b*ln(c*x^n))/e^2-5/18*b*n*x^2*(a+b*ln(c*x^n))/e+4/27*b*n*x^3*(a+b*ln(c*x^n))-1/3*x*(a+b*ln(c*x^n))^2/e^2+1/
6*x^2*(a+b*ln(c*x^n))^2/e-1/9*x^3*(a+b*ln(c*x^n))^2+2/27*b^2*n^2*ln(e*x+1)/e^3+2/27*b^2*n^2*x^3*ln(e*x+1)-2/9*
b*n*(a+b*ln(c*x^n))*ln(e*x+1)/e^3-2/9*b*n*x^3*(a+b*ln(c*x^n))*ln(e*x+1)+1/3*(a+b*ln(c*x^n))^2*ln(e*x+1)/e^3+1/
3*x^3*(a+b*ln(c*x^n))^2*ln(e*x+1)-2/9*b^2*n^2*polylog(2,-e*x)/e^3+2/3*b*n*(a+b*ln(c*x^n))*polylog(2,-e*x)/e^3-
2/3*b^2*n^2*polylog(3,-e*x)/e^3

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Rubi [A]
time = 0.20, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2442, 45, 2424, 2332, 2341, 2421, 6724, 2423, 2438} \begin {gather*} \frac {2 b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 b^2 n^2 \text {PolyLog}(2,-e x)}{9 e^3}-\frac {2 b^2 n^2 \text {PolyLog}(3,-e x)}{3 e^3}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 a b n x}{3 e^2}+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b^2 n^2 \log (e x+1)}{27 e^3}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {2}{27} b^2 n^2 x^3 \log (e x+1)+\frac {19 b^2 n^2 x^2}{108 e}-\frac {2}{27} b^2 n^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*n*x)/(3*e^2) - (26*b^2*n^2*x)/(27*e^2) + (19*b^2*n^2*x^2)/(108*e) - (2*b^2*n^2*x^3)/27 + (2*b^2*n*x*Log
[c*x^n])/(3*e^2) + (2*b*n*x*(a + b*Log[c*x^n]))/(9*e^2) - (5*b*n*x^2*(a + b*Log[c*x^n]))/(18*e) + (4*b*n*x^3*(
a + b*Log[c*x^n]))/27 - (x*(a + b*Log[c*x^n])^2)/(3*e^2) + (x^2*(a + b*Log[c*x^n])^2)/(6*e) - (x^3*(a + b*Log[
c*x^n])^2)/9 + (2*b^2*n^2*Log[1 + e*x])/(27*e^3) + (2*b^2*n^2*x^3*Log[1 + e*x])/27 - (2*b*n*(a + b*Log[c*x^n])
*Log[1 + e*x])/(9*e^3) - (2*b*n*x^3*(a + b*Log[c*x^n])*Log[1 + e*x])/9 + ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(
3*e^3) + (x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x])/3 - (2*b^2*n^2*PolyLog[2, -(e*x)])/(9*e^3) + (2*b*n*(a + b*Lo
g[c*x^n])*PolyLog[2, -(e*x)])/(3*e^3) - (2*b^2*n^2*PolyLog[3, -(e*x)])/(3*e^3)

Rule 45

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 2332

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

Rule 2341

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2423

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 2424

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 2438

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]

Rule 2442

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 6724

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]

Rubi steps

\begin {align*} \int x^2 \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}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \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 )}{3 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^2 \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)}{3 e^3 x}+\frac {1}{3} x^2 \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}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {1}{9} (2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{3} (2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{3 e^3}+\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e^2}-\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e}\\ &=\frac {2 a b n x}{3 e^2}+\frac {b^2 n^2 x^2}{12 e}-\frac {2}{81} b^2 n^2 x^3+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{3 e^2}+\frac {1}{3} \left (2 b^2 n^2\right ) \int \left (-\frac {1}{3 e^2}+\frac {x}{6 e}-\frac {x^2}{9}+\frac {\log (1+e x)}{3 e^3 x}+\frac {1}{3} x^2 \log (1+e x)\right ) \, dx-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{3 e^3}\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}+\frac {1}{9} \left (2 b^2 n^2\right ) \int x^2 \log (1+e x) \, dx+\frac {\left (2 b^2 n^2\right ) \int \frac {\log (1+e x)}{x} \, dx}{9 e^3}\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}-\frac {1}{27} \left (2 b^2 e n^2\right ) \int \frac {x^3}{1+e x} \, dx\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}-\frac {1}{27} \left (2 b^2 e n^2\right ) \int \left (\frac {1}{e^3}-\frac {x}{e^2}+\frac {x^2}{e}-\frac {1}{e^3 (1+e x)}\right ) \, dx\\ &=\frac {2 a b n x}{3 e^2}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {19 b^2 n^2 x^2}{108 e}-\frac {2}{27} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 \log (1+e x)}{27 e^3}+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 506, normalized size = 1.28 \begin {gather*} \frac {-36 a^2 e x+96 a b e n x-104 b^2 e n^2 x+18 a^2 e^2 x^2-30 a b e^2 n x^2+19 b^2 e^2 n^2 x^2-12 a^2 e^3 x^3+16 a b e^3 n x^3-8 b^2 e^3 n^2 x^3-72 a b e x \log \left (c x^n\right )+96 b^2 e n x \log \left (c x^n\right )+36 a b e^2 x^2 \log \left (c x^n\right )-30 b^2 e^2 n x^2 \log \left (c x^n\right )-24 a b e^3 x^3 \log \left (c x^n\right )+16 b^2 e^3 n x^3 \log \left (c x^n\right )-36 b^2 e x \log ^2\left (c x^n\right )+18 b^2 e^2 x^2 \log ^2\left (c x^n\right )-12 b^2 e^3 x^3 \log ^2\left (c x^n\right )+36 a^2 \log (1+e x)-24 a b n \log (1+e x)+8 b^2 n^2 \log (1+e x)+36 a^2 e^3 x^3 \log (1+e x)-24 a b e^3 n x^3 \log (1+e x)+8 b^2 e^3 n^2 x^3 \log (1+e x)+72 a b \log \left (c x^n\right ) \log (1+e x)-24 b^2 n \log \left (c x^n\right ) \log (1+e x)+72 a b e^3 x^3 \log \left (c x^n\right ) \log (1+e x)-24 b^2 e^3 n x^3 \log \left (c x^n\right ) \log (1+e x)+36 b^2 \log ^2\left (c x^n\right ) \log (1+e x)+36 b^2 e^3 x^3 \log ^2\left (c x^n\right ) \log (1+e x)+24 b n \left (3 a-b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)-72 b^2 n^2 \text {Li}_3(-e x)}{108 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*a^2*e*x + 96*a*b*e*n*x - 104*b^2*e*n^2*x + 18*a^2*e^2*x^2 - 30*a*b*e^2*n*x^2 + 19*b^2*e^2*n^2*x^2 - 12*a^
2*e^3*x^3 + 16*a*b*e^3*n*x^3 - 8*b^2*e^3*n^2*x^3 - 72*a*b*e*x*Log[c*x^n] + 96*b^2*e*n*x*Log[c*x^n] + 36*a*b*e^
2*x^2*Log[c*x^n] - 30*b^2*e^2*n*x^2*Log[c*x^n] - 24*a*b*e^3*x^3*Log[c*x^n] + 16*b^2*e^3*n*x^3*Log[c*x^n] - 36*
b^2*e*x*Log[c*x^n]^2 + 18*b^2*e^2*x^2*Log[c*x^n]^2 - 12*b^2*e^3*x^3*Log[c*x^n]^2 + 36*a^2*Log[1 + e*x] - 24*a*
b*n*Log[1 + e*x] + 8*b^2*n^2*Log[1 + e*x] + 36*a^2*e^3*x^3*Log[1 + e*x] - 24*a*b*e^3*n*x^3*Log[1 + e*x] + 8*b^
2*e^3*n^2*x^3*Log[1 + e*x] + 72*a*b*Log[c*x^n]*Log[1 + e*x] - 24*b^2*n*Log[c*x^n]*Log[1 + e*x] + 72*a*b*e^3*x^
3*Log[c*x^n]*Log[1 + e*x] - 24*b^2*e^3*n*x^3*Log[c*x^n]*Log[1 + e*x] + 36*b^2*Log[c*x^n]^2*Log[1 + e*x] + 36*b
^2*e^3*x^3*Log[c*x^n]^2*Log[1 + e*x] + 24*b*n*(3*a - b*n + 3*b*Log[c*x^n])*PolyLog[2, -(e*x)] - 72*b^2*n^2*Pol
yLog[3, -(e*x)])/(108*e^3)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*b^2*x^3*e^3 - 3*b^2*x^2*e^2 + 6*b^2*x*e - 6*(b^2*x^3*e^3 + b^2)*log(x*e + 1))*e^(-3)*log(x^n)^2 + 1/9
*e^(-3)*integrate((9*(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*x^3*e^3*log(x*e + 1) + (2*b^2*n*x^3*e^3 - 3*b^2*n*x^2
*e^2 + 6*b^2*n*x*e - 6*((b^2*(n - 3*log(c)) - 3*a*b)*x^3*e^3 + b^2*n)*log(x*e + 1))*log(x^n))/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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