3.1.10 \(\int x^3 (a+b \log (c x^n))^2 \log (1+e x) \, dx\) [10]

Optimal. Leaf size=456 \[ -\frac {a b n x}{2 e^3}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {37 b^2 n^2 x^3}{864 e}-\frac {3}{128} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log (1+e x)}{32 e^4}+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {b^2 n^2 \text {Li}_2(-e x)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4} \]

[Out]

-1/2*a*b*n*x/e^3+21/32*b^2*n^2*x/e^3-7/64*b^2*n^2*x^2/e^2+37/864*b^2*n^2*x^3/e-3/128*b^2*n^2*x^4-1/2*b^2*n*x*l
n(c*x^n)/e^3-1/8*b*n*x*(a+b*ln(c*x^n))/e^3+3/16*b*n*x^2*(a+b*ln(c*x^n))/e^2-7/72*b*n*x^3*(a+b*ln(c*x^n))/e+1/1
6*b*n*x^4*(a+b*ln(c*x^n))+1/4*x*(a+b*ln(c*x^n))^2/e^3-1/8*x^2*(a+b*ln(c*x^n))^2/e^2+1/12*x^3*(a+b*ln(c*x^n))^2
/e-1/16*x^4*(a+b*ln(c*x^n))^2-1/32*b^2*n^2*ln(e*x+1)/e^4+1/32*b^2*n^2*x^4*ln(e*x+1)+1/8*b*n*(a+b*ln(c*x^n))*ln
(e*x+1)/e^4-1/8*b*n*x^4*(a+b*ln(c*x^n))*ln(e*x+1)-1/4*(a+b*ln(c*x^n))^2*ln(e*x+1)/e^4+1/4*x^4*(a+b*ln(c*x^n))^
2*ln(e*x+1)+1/8*b^2*n^2*polylog(2,-e*x)/e^4-1/2*b*n*(a+b*ln(c*x^n))*polylog(2,-e*x)/e^4+1/2*b^2*n^2*polylog(3,
-e*x)/e^4

________________________________________________________________________________________

Rubi [A]
time = 0.24, antiderivative size = 456, normalized size of antiderivative = 1.00, number of steps used = 15, 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 {b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {b^2 n^2 \text {PolyLog}(2,-e x)}{8 e^4}+\frac {b^2 n^2 \text {PolyLog}(3,-e x)}{2 e^4}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {a b n x}{2 e^3}-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b^2 n^2 \log (e x+1)}{32 e^4}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {1}{32} b^2 n^2 x^4 \log (e x+1)+\frac {37 b^2 n^2 x^3}{864 e}-\frac {3}{128} b^2 n^2 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x],x]

[Out]

-1/2*(a*b*n*x)/e^3 + (21*b^2*n^2*x)/(32*e^3) - (7*b^2*n^2*x^2)/(64*e^2) + (37*b^2*n^2*x^3)/(864*e) - (3*b^2*n^
2*x^4)/128 - (b^2*n*x*Log[c*x^n])/(2*e^3) - (b*n*x*(a + b*Log[c*x^n]))/(8*e^3) + (3*b*n*x^2*(a + b*Log[c*x^n])
)/(16*e^2) - (7*b*n*x^3*(a + b*Log[c*x^n]))/(72*e) + (b*n*x^4*(a + b*Log[c*x^n]))/16 + (x*(a + b*Log[c*x^n])^2
)/(4*e^3) - (x^2*(a + b*Log[c*x^n])^2)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^2)/(12*e) - (x^4*(a + b*Log[c*x^n])^2
)/16 - (b^2*n^2*Log[1 + e*x])/(32*e^4) + (b^2*n^2*x^4*Log[1 + e*x])/32 + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])
/(8*e^4) - (b*n*x^4*(a + b*Log[c*x^n])*Log[1 + e*x])/8 - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*e^4) + (x^4*(a
+ b*Log[c*x^n])^2*Log[1 + e*x])/4 + (b^2*n^2*PolyLog[2, -(e*x)])/(8*e^4) - (b*n*(a + b*Log[c*x^n])*PolyLog[2,
-(e*x)])/(2*e^4) + (b^2*n^2*PolyLog[3, -(e*x)])/(2*e^4)

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^3 \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}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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 )}{4 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac {1}{16} x^3 \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)}{4 e^4 x}+\frac {1}{4} x^3 \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}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {1}{8} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{2} (b n) \int x^3 \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}{2 e^4}-\frac {(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e^3}+\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 e^2}-\frac {(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 e}\\ &=-\frac {a b n x}{2 e^3}-\frac {b^2 n^2 x^2}{16 e^2}+\frac {b^2 n^2 x^3}{54 e}-\frac {1}{128} b^2 n^2 x^4-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{2 e^4}-\frac {\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 e^3}+\frac {1}{2} \left (b^2 n^2\right ) \int \left (\frac {1}{4 e^3}-\frac {x}{8 e^2}+\frac {x^2}{12 e}-\frac {x^3}{16}-\frac {\log (1+e x)}{4 e^4 x}+\frac {1}{4} x^3 \log (1+e x)\right ) \, dx+\frac {\left (b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{2 e^4}\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}+\frac {1}{8} \left (b^2 n^2\right ) \int x^3 \log (1+e x) \, dx-\frac {\left (b^2 n^2\right ) \int \frac {\log (1+e x)}{x} \, dx}{8 e^4}\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}-\frac {1}{32} \left (b^2 e n^2\right ) \int \frac {x^4}{1+e x} \, dx\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}-\frac {1}{32} \left (b^2 e n^2\right ) \int \left (-\frac {1}{e^4}+\frac {x}{e^3}-\frac {x^2}{e^2}+\frac {x^3}{e}+\frac {1}{e^4 (1+e x)}\right ) \, dx\\ &=-\frac {a b n x}{2 e^3}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {37 b^2 n^2 x^3}{864 e}-\frac {3}{128} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log (1+e x)}{32 e^4}+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 594, normalized size = 1.30 \begin {gather*} \frac {864 a^2 e x-2160 a b e n x+2268 b^2 e n^2 x-432 a^2 e^2 x^2+648 a b e^2 n x^2-378 b^2 e^2 n^2 x^2+288 a^2 e^3 x^3-336 a b e^3 n x^3+148 b^2 e^3 n^2 x^3-216 a^2 e^4 x^4+216 a b e^4 n x^4-81 b^2 e^4 n^2 x^4+1728 a b e x \log \left (c x^n\right )-2160 b^2 e n x \log \left (c x^n\right )-864 a b e^2 x^2 \log \left (c x^n\right )+648 b^2 e^2 n x^2 \log \left (c x^n\right )+576 a b e^3 x^3 \log \left (c x^n\right )-336 b^2 e^3 n x^3 \log \left (c x^n\right )-432 a b e^4 x^4 \log \left (c x^n\right )+216 b^2 e^4 n x^4 \log \left (c x^n\right )+864 b^2 e x \log ^2\left (c x^n\right )-432 b^2 e^2 x^2 \log ^2\left (c x^n\right )+288 b^2 e^3 x^3 \log ^2\left (c x^n\right )-216 b^2 e^4 x^4 \log ^2\left (c x^n\right )-864 a^2 \log (1+e x)+432 a b n \log (1+e x)-108 b^2 n^2 \log (1+e x)+864 a^2 e^4 x^4 \log (1+e x)-432 a b e^4 n x^4 \log (1+e x)+108 b^2 e^4 n^2 x^4 \log (1+e x)-1728 a b \log \left (c x^n\right ) \log (1+e x)+432 b^2 n \log \left (c x^n\right ) \log (1+e x)+1728 a b e^4 x^4 \log \left (c x^n\right ) \log (1+e x)-432 b^2 e^4 n x^4 \log \left (c x^n\right ) \log (1+e x)-864 b^2 \log ^2\left (c x^n\right ) \log (1+e x)+864 b^2 e^4 x^4 \log ^2\left (c x^n\right ) \log (1+e x)+432 b n \left (-4 a+b n-4 b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+1728 b^2 n^2 \text {Li}_3(-e x)}{3456 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x],x]

[Out]

(864*a^2*e*x - 2160*a*b*e*n*x + 2268*b^2*e*n^2*x - 432*a^2*e^2*x^2 + 648*a*b*e^2*n*x^2 - 378*b^2*e^2*n^2*x^2 +
288*a^2*e^3*x^3 - 336*a*b*e^3*n*x^3 + 148*b^2*e^3*n^2*x^3 - 216*a^2*e^4*x^4 + 216*a*b*e^4*n*x^4 - 81*b^2*e^4*
n^2*x^4 + 1728*a*b*e*x*Log[c*x^n] - 2160*b^2*e*n*x*Log[c*x^n] - 864*a*b*e^2*x^2*Log[c*x^n] + 648*b^2*e^2*n*x^2
*Log[c*x^n] + 576*a*b*e^3*x^3*Log[c*x^n] - 336*b^2*e^3*n*x^3*Log[c*x^n] - 432*a*b*e^4*x^4*Log[c*x^n] + 216*b^2
*e^4*n*x^4*Log[c*x^n] + 864*b^2*e*x*Log[c*x^n]^2 - 432*b^2*e^2*x^2*Log[c*x^n]^2 + 288*b^2*e^3*x^3*Log[c*x^n]^2
- 216*b^2*e^4*x^4*Log[c*x^n]^2 - 864*a^2*Log[1 + e*x] + 432*a*b*n*Log[1 + e*x] - 108*b^2*n^2*Log[1 + e*x] + 8
64*a^2*e^4*x^4*Log[1 + e*x] - 432*a*b*e^4*n*x^4*Log[1 + e*x] + 108*b^2*e^4*n^2*x^4*Log[1 + e*x] - 1728*a*b*Log
[c*x^n]*Log[1 + e*x] + 432*b^2*n*Log[c*x^n]*Log[1 + e*x] + 1728*a*b*e^4*x^4*Log[c*x^n]*Log[1 + e*x] - 432*b^2*
e^4*n*x^4*Log[c*x^n]*Log[1 + e*x] - 864*b^2*Log[c*x^n]^2*Log[1 + e*x] + 864*b^2*e^4*x^4*Log[c*x^n]^2*Log[1 + e
*x] + 432*b*n*(-4*a + b*n - 4*b*Log[c*x^n])*PolyLog[2, -(e*x)] + 1728*b^2*n^2*PolyLog[3, -(e*x)])/(3456*e^4)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))^2*ln(e*x+1),x)

[Out]

int(x^3*(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^3*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="maxima")

[Out]

-1/48*(3*b^2*x^4*e^4 - 4*b^2*x^3*e^3 + 6*b^2*x^2*e^2 - 12*b^2*x*e - 12*(b^2*x^4*e^4 - b^2)*log(x*e + 1))*e^(-4
)*log(x^n)^2 + 1/24*e^(-4)*integrate((24*(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*x^4*e^4*log(x*e + 1) + (3*b^2*n*x
^4*e^4 - 4*b^2*n*x^3*e^3 + 6*b^2*n*x^2*e^2 - 12*b^2*n*x*e - 12*((b^2*(n - 4*log(c)) - 4*a*b)*x^4*e^4 - 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^3*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="fricas")

[Out]

integral(b^2*x^3*log(c*x^n)^2*log(x*e + 1) + 2*a*b*x^3*log(c*x^n)*log(x*e + 1) + a^2*x^3*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**3*(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^3*(a+b*log(c*x^n))^2*log(e*x+1),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^2*x^3*log(x*e + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\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^3*log(e*x + 1)*(a + b*log(c*x^n))^2,x)

[Out]

int(x^3*log(e*x + 1)*(a + b*log(c*x^n))^2, x)

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