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

Optimal. Leaf size=914 \[ \text{result too large to display} \]

[Out]

(-175*b^3*f*n^3)/(216*e*x^(3/2)) + (45*b^3*f^2*n^3)/(16*e^2*x) - (255*b^3*f^3*n^3)/(8*e^3*Sqrt[x]) + (3*b^3*f^
4*n^3*Log[e + f*Sqrt[x]])/(8*e^4) - (3*b^3*n^3*Log[d*(e + f*Sqrt[x])])/(8*x^2) - (3*b^3*f^4*n^3*Log[e + f*Sqrt
[x]]*Log[-((f*Sqrt[x])/e)])/(2*e^4) - (3*b^3*f^4*n^3*Log[x])/(16*e^4) + (3*b^3*f^4*n^3*Log[x]^2)/(16*e^4) - (3
7*b^2*f*n^2*(a + b*Log[c*x^n]))/(36*e*x^(3/2)) + (21*b^2*f^2*n^2*(a + b*Log[c*x^n]))/(8*e^2*x) - (63*b^2*f^3*n
^2*(a + b*Log[c*x^n]))/(4*e^3*Sqrt[x]) + (3*b^2*f^4*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) - (3*b^
2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) - (3*b^2*f^4*n^2*Log[x]*(a + b*Log[c*x^n]))/(8*e^4) -
(7*b*f*n*(a + b*Log[c*x^n])^2)/(12*e*x^(3/2)) + (9*b*f^2*n*(a + b*Log[c*x^n])^2)/(8*e^2*x) - (15*b*f^3*n*(a +
b*Log[c*x^n])^2)/(4*e^3*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) + (3*b*f^4*n*L
og[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) - (f^4*(a + b*Log[c*x^n])^3)/(8*e^4) - (f*(a + b*Log[c*x^n
])^3)/(6*e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4*e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) - (Log
[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/(2*x^2) + (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(2*e^4)
- (f^4*(a + b*Log[c*x^n])^4)/(16*b*e^4*n) - (3*b^3*f^4*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) + (3*b^2*f^4
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 + (3*b*f^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sq
rt[x])/e)])/e^4 - (6*b^3*f^4*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^4 - (12*b^2*f^4*n^2*(a + b*Log[c*x^n])*PolyLo
g[3, -((f*Sqrt[x])/e)])/e^4 + (24*b^3*f^4*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^4

________________________________________________________________________________________

Rubi [A]  time = 1.52042, antiderivative size = 914, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 19, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.679, Rules used = {2454, 2395, 44, 2377, 2305, 2304, 2375, 2337, 2374, 2383, 6589, 2376, 2394, 2315, 2301, 2366, 12, 2302, 30} \[ -\frac{\left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b e^4 n}+\frac{\log \left (\frac{\sqrt{x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{2 e^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 f^4}{8 e^4}+\frac{3 b^3 n^3 \log ^2(x) f^4}{16 e^4}+\frac{3 b n \log \left (\frac{\sqrt{x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 e^4}+\frac{3 b^3 n^3 \log \left (e+f \sqrt{x}\right ) f^4}{8 e^4}-\frac{3 b^3 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right ) f^4}{2 e^4}-\frac{3 b^3 n^3 \log (x) f^4}{16 e^4}+\frac{3 b^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 e^4}-\frac{3 b^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{8 e^4}-\frac{3 b^3 n^3 \text{PolyLog}\left (2,\frac{\sqrt{x} f}{e}+1\right ) f^4}{2 e^4}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) f^4}{e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) f^4}{e^4}-\frac{6 b^3 n^3 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) f^4}{e^4}-\frac{12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) f^4}{e^4}+\frac{24 b^3 n^3 \text{PolyLog}\left (4,-\frac{f \sqrt{x}}{e}\right ) f^4}{e^4}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 e^3 \sqrt{x}}-\frac{15 b n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 e^3 \sqrt{x}}-\frac{63 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 e^3 \sqrt{x}}-\frac{255 b^3 n^3 f^3}{8 e^3 \sqrt{x}}+\frac{\left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 e^2 x}+\frac{9 b n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 e^2 x}+\frac{21 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 e^2 x}+\frac{45 b^3 n^3 f^2}{16 e^2 x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 f}{6 e x^{3/2}}-\frac{7 b n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 e x^{3/2}}-\frac{37 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 e x^{3/2}}-\frac{175 b^3 n^3 f}{216 e x^{3/2}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{8 x^2}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-175*b^3*f*n^3)/(216*e*x^(3/2)) + (45*b^3*f^2*n^3)/(16*e^2*x) - (255*b^3*f^3*n^3)/(8*e^3*Sqrt[x]) + (3*b^3*f^
4*n^3*Log[e + f*Sqrt[x]])/(8*e^4) - (3*b^3*n^3*Log[d*(e + f*Sqrt[x])])/(8*x^2) - (3*b^3*f^4*n^3*Log[e + f*Sqrt
[x]]*Log[-((f*Sqrt[x])/e)])/(2*e^4) - (3*b^3*f^4*n^3*Log[x])/(16*e^4) + (3*b^3*f^4*n^3*Log[x]^2)/(16*e^4) - (3
7*b^2*f*n^2*(a + b*Log[c*x^n]))/(36*e*x^(3/2)) + (21*b^2*f^2*n^2*(a + b*Log[c*x^n]))/(8*e^2*x) - (63*b^2*f^3*n
^2*(a + b*Log[c*x^n]))/(4*e^3*Sqrt[x]) + (3*b^2*f^4*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) - (3*b^
2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) - (3*b^2*f^4*n^2*Log[x]*(a + b*Log[c*x^n]))/(8*e^4) -
(7*b*f*n*(a + b*Log[c*x^n])^2)/(12*e*x^(3/2)) + (9*b*f^2*n*(a + b*Log[c*x^n])^2)/(8*e^2*x) - (15*b*f^3*n*(a +
b*Log[c*x^n])^2)/(4*e^3*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) + (3*b*f^4*n*L
og[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) - (f^4*(a + b*Log[c*x^n])^3)/(8*e^4) - (f*(a + b*Log[c*x^n
])^3)/(6*e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4*e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) - (Log
[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/(2*x^2) + (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(2*e^4)
- (f^4*(a + b*Log[c*x^n])^4)/(16*b*e^4*n) - (3*b^3*f^4*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) + (3*b^2*f^4
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 + (3*b*f^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sq
rt[x])/e)])/e^4 - (6*b^3*f^4*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^4 - (12*b^2*f^4*n^2*(a + b*Log[c*x^n])*PolyLo
g[3, -((f*Sqrt[x])/e)])/e^4 + (24*b^3*f^4*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^4

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 2375

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

Rule 2337

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

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 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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 2394

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx &=-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}+\frac{f^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}-(3 b n) \int \left (-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{5/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x^2}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 x^{3/2}}+\frac{f^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^3}-\frac{f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4 x}\right ) \, dx\\ &=-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}+\frac{f^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}+\frac{1}{2} (3 b n) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx+\frac{(b f n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{5/2}} \, dx}{2 e}-\frac{\left (3 b f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{4 e^2}+\frac{\left (3 b f^3 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx}{2 e^3}+\frac{\left (3 b f^4 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{4 e^4}-\frac{\left (3 b f^4 n\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^4}\\ &=-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}+\frac{3 b f^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3 b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{8 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^5 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{4 e^4}-\frac{\left (3 b f^4 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx}{4 e^4}-\left (3 b^2 n^2\right ) \int \left (-\frac{f \left (a+b \log \left (c x^n\right )\right )}{6 e x^{5/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x^2}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^3 x^{3/2}}+\frac{f^4 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}-\frac{f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4 x}\right ) \, dx+\frac{\left (2 b^2 f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{5/2}} \, dx}{3 e}-\frac{\left (3 b^2 f^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{2 e^2}+\frac{\left (6 b^2 f^3 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e^3}\\ &=-\frac{8 b^3 f n^3}{27 e x^{3/2}}+\frac{3 b^3 f^2 n^3}{2 e^2 x}-\frac{24 b^3 f^3 n^3}{e^3 \sqrt{x}}-\frac{4 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{9 e x^{3/2}}+\frac{3 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2 x}-\frac{12 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt{x}}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}+\frac{3 b f^4 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{3 b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{8 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{4 e^4}-\frac{\left (3 b f^4 n\right ) \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^4}+\frac{1}{2} \left (3 b^2 n^2\right ) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac{\left (b^2 f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{5/2}} \, dx}{2 e}-\frac{\left (3 b^2 f^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{4 e^2}+\frac{\left (3 b^2 f^3 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{2 e^3}+\frac{\left (3 b^2 f^4 n^2\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{4 e^4}-\frac{\left (3 b^2 f^4 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e^4}\\ &=-\frac{14 b^3 f n^3}{27 e x^{3/2}}+\frac{9 b^3 f^2 n^3}{4 e^2 x}-\frac{30 b^3 f^3 n^3}{e^3 \sqrt{x}}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{f^4 \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b e^4 n}+\frac{\left (3 b f^5 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{8 e^4}-\frac{\left (3 b^2 f^4 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx}{4 e^4}-\frac{\left (6 b^2 f^4 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^4}-\frac{1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac{f}{6 e x^{5/2}}+\frac{f^2}{4 e^2 x^2}-\frac{f^3}{2 e^3 x^{3/2}}+\frac{f^4 \log \left (e+f \sqrt{x}\right )}{2 e^4 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right )}{2 x^3}-\frac{f^4 \log (x)}{4 e^4 x}\right ) \, dx\\ &=-\frac{37 b^3 f n^3}{54 e x^{3/2}}+\frac{21 b^3 f^2 n^3}{8 e^2 x}-\frac{63 b^3 f^3 n^3}{2 e^3 \sqrt{x}}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac{3 b f^4 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{\left (3 b f^4 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{8 e^4}-\frac{\left (3 b^2 f^4 n^2\right ) \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{2 e^4}+\frac{1}{4} \left (3 b^3 n^3\right ) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right )}{x^3} \, dx+\frac{\left (3 b^3 f^4 n^3\right ) \int \frac{\log (x)}{x} \, dx}{8 e^4}-\frac{\left (3 b^3 f^4 n^3\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{4 e^4}+\frac{\left (12 b^3 f^4 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^4}\\ &=-\frac{37 b^3 f n^3}{54 e x^{3/2}}+\frac{21 b^3 f^2 n^3}{8 e^2 x}-\frac{63 b^3 f^3 n^3}{2 e^3 \sqrt{x}}+\frac{3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac{3 b f^4 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}+\frac{3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{24 b^3 f^4 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{\left (3 f^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac{1}{2} \left (3 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (d (e+f x))}{x^5} \, dx,x,\sqrt{x}\right )-\frac{\left (3 b^3 f^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{2 e^4}-\frac{\left (3 b^3 f^4 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^4}\\ &=-\frac{37 b^3 f n^3}{54 e x^{3/2}}+\frac{21 b^3 f^2 n^3}{8 e^2 x}-\frac{63 b^3 f^3 n^3}{2 e^3 \sqrt{x}}-\frac{3 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{8 x^2}-\frac{3 b^3 f^4 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{2 e^4}+\frac{3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac{3 b f^4 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}+\frac{3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{6 b^3 f^4 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{24 b^3 f^4 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{1}{8} \left (3 b^3 f n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 (e+f x)} \, dx,x,\sqrt{x}\right )+\frac{\left (3 b^3 f^5 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{2 e^4}\\ &=-\frac{37 b^3 f n^3}{54 e x^{3/2}}+\frac{21 b^3 f^2 n^3}{8 e^2 x}-\frac{63 b^3 f^3 n^3}{2 e^3 \sqrt{x}}-\frac{3 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{8 x^2}-\frac{3 b^3 f^4 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{2 e^4}+\frac{3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac{3 b f^4 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}-\frac{3 b^3 f^4 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{2 e^4}+\frac{3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{6 b^3 f^4 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{24 b^3 f^4 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{1}{8} \left (3 b^3 f n^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{e x^4}-\frac{f}{e^2 x^3}+\frac{f^2}{e^3 x^2}-\frac{f^3}{e^4 x}+\frac{f^4}{e^4 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{175 b^3 f n^3}{216 e x^{3/2}}+\frac{45 b^3 f^2 n^3}{16 e^2 x}-\frac{255 b^3 f^3 n^3}{8 e^3 \sqrt{x}}+\frac{3 b^3 f^4 n^3 \log \left (e+f \sqrt{x}\right )}{8 e^4}-\frac{3 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{8 x^2}-\frac{3 b^3 f^4 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{2 e^4}-\frac{3 b^3 f^4 n^3 \log (x)}{16 e^4}+\frac{3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac{37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac{21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac{63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt{x}}+\frac{3 b^2 f^4 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{3 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac{7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac{9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac{15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac{3 b f^4 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^4}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac{f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac{f^4 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac{f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}-\frac{3 b^3 f^4 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{2 e^4}+\frac{3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{6 b^3 f^4 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}+\frac{24 b^3 f^4 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^4}\\ \end{align*}

Mathematica [A]  time = 2.22607, size = 1549, normalized size = 1.69 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(54*e^4*Log[d*(e + f*Sqrt[x])]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^2*n^2)
*Log[c*x^n] + 6*b^2*(2*a + b*n)*Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3) + 18*e^3*f*Sqrt[x]*(4*a^3 + 6*a^2*b*n + 6*a
*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2
*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4
*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 27*e^2*f^2*x*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n
*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b
^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) +
54*e*f^3*x^(3/2)*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^
2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2
+ 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 54*f^4*x^2*Log[e + f*Sqrt[x]]*(
4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) +
Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[
x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 27*f^4*x^2*Log[x]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2
+ 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Lo
g[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(
n*Log[x]) + Log[c*x^n])^3) + 18*b*f*n*Sqrt[x]*(2*a^2 + 2*a*b*n + b^2*n^2 + 4*a*b*(-(n*Log[x]) + Log[c*x^n]) +
2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(e*(4*e^2 - 9*e*f*Sqrt[x] + 36*f^2*x)
+ 3*(2*e^3 - 3*e^2*f*Sqrt[x] + 6*e*f^2*x - 6*f^3*x^(3/2)*Log[1 + (f*Sqrt[x])/e])*Log[x] + (9*f^3*x^(3/2)*Log[
x]^2)/2 - 36*f^3*x^(3/2)*PolyLog[2, -((f*Sqrt[x])/e)]) - 6*b^2*f*n^2*Sqrt[x]*(-2*a - b*n + 2*b*n*Log[x] - 2*b*
Log[c*x^n])*(16*e^3 - 54*e^2*f*Sqrt[x] + 432*e*f^2*x + 24*e^3*Log[x] - 54*e^2*f*Sqrt[x]*Log[x] + 216*e*f^2*x*L
og[x] + 18*e^3*Log[x]^2 - 27*e^2*f*Sqrt[x]*Log[x]^2 + 54*e*f^2*x*Log[x]^2 - 54*f^3*x^(3/2)*Log[1 + (f*Sqrt[x])
/e]*Log[x]^2 + 9*f^3*x^(3/2)*Log[x]^3 - 216*f^3*x^(3/2)*Log[x]*PolyLog[2, -((f*Sqrt[x])/e)] + 432*f^3*x^(3/2)*
PolyLog[3, -((f*Sqrt[x])/e)]) + 4*b^3*n^3*(2*e*f*Sqrt[x]*(16*e^2 - 81*e*f*Sqrt[x] + 1296*f^2*x) + 9*(e*f*Sqrt[
x]*(2*e^2 - 3*e*f*Sqrt[x] + 6*f^2*x) - 6*f^4*x^2*Log[1 + e/(f*Sqrt[x])])*Log[x]^3 + 9*f*Sqrt[x]*Log[x]^2*(e*(4
*e^2 - 9*e*f*Sqrt[x] + 36*f^2*x) + 36*f^3*x^(3/2)*PolyLog[2, -(e/(f*Sqrt[x]))]) + 6*f*Sqrt[x]*Log[x]*(e*(8*e^2
- 27*e*f*Sqrt[x] + 216*f^2*x) + 216*f^3*x^(3/2)*PolyLog[3, -(e/(f*Sqrt[x]))]) + 2592*f^4*x^2*PolyLog[4, -(e/(
f*Sqrt[x]))]))/(432*e^4*x^2)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{{x}^{3}}\ln \left ( d \left ( e+f\sqrt{x} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt{x} + d e\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3*log(d*(e+f*x^(1/2)))/x^3,x, algorithm="fricas")

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log(d*f*sqrt(x) + d*e)/x^3, x)

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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((a+b*ln(c*x**n))**3*ln(d*(e+f*x**(1/2)))/x**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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