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

Optimal. Leaf size=639 \[ \frac{12 b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{24 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{12 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}-\frac{24 b^3 e^2 n^3 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{PolyLog}\left (4,-\frac{f \sqrt{x}}{e}\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b e^2 n \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{e^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{6 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right )}{f^2}+\frac{12 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{90 b^3 e n^3 \sqrt{x}}{f}+12 b^3 n^3 x \]

[Out]

(-90*b^3*e*n^3*Sqrt[x])/f - 6*a*b^2*n^2*x + 12*b^3*n^3*x + (6*b^3*e^2*n^3*Log[e + f*Sqrt[x]])/f^2 - 6*b^3*n^3*
x*Log[d*(e + f*Sqrt[x])] + (12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 - 6*b^3*n^2*x*Log[c*x
^n] + (42*b^2*e*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/f - 3*b^2*n^2*x*(a + b*Log[c*x^n]) - (6*b^2*e^2*n^2*Log[e + f*
Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + 6*b^2*n^2*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (9*b*e*n*Sqrt[x]*(a
+ b*Log[c*x^n])^2)/f + 3*b*n*x*(a + b*Log[c*x^n])^2 - 3*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (
3*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 + (e*Sqrt[x]*(a + b*Log[c*x^n])^3)/f - (x*(a + b*Lo
g[c*x^n])^3)/2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n]
)^3)/f^2 + (12*b^3*e^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 + (12*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2,
-((f*Sqrt[x])/e)])/f^2 - (6*b*e^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 - (24*b^3*e^2*n^3*P
olyLog[3, -((f*Sqrt[x])/e)])/f^2 + (24*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 - (48*
b^3*e^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^2

________________________________________________________________________________________

Rubi [A]  time = 0.855547, antiderivative size = 639, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {2448, 266, 43, 2370, 2296, 2295, 2305, 2304, 2375, 2337, 2374, 2383, 6589, 2454, 2394, 2315} \[ \frac{12 b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{24 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{12 b^3 e^2 n^3 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}-\frac{24 b^3 e^2 n^3 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{PolyLog}\left (4,-\frac{f \sqrt{x}}{e}\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b e^2 n \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{e^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{6 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right )}{f^2}+\frac{12 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{90 b^3 e n^3 \sqrt{x}}{f}+12 b^3 n^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-90*b^3*e*n^3*Sqrt[x])/f - 6*a*b^2*n^2*x + 12*b^3*n^3*x + (6*b^3*e^2*n^3*Log[e + f*Sqrt[x]])/f^2 - 6*b^3*n^3*
x*Log[d*(e + f*Sqrt[x])] + (12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 - 6*b^3*n^2*x*Log[c*x
^n] + (42*b^2*e*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/f - 3*b^2*n^2*x*(a + b*Log[c*x^n]) - (6*b^2*e^2*n^2*Log[e + f*
Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + 6*b^2*n^2*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (9*b*e*n*Sqrt[x]*(a
+ b*Log[c*x^n])^2)/f + 3*b*n*x*(a + b*Log[c*x^n])^2 - 3*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (
3*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 + (e*Sqrt[x]*(a + b*Log[c*x^n])^3)/f - (x*(a + b*Lo
g[c*x^n])^3)/2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n]
)^3)/f^2 + (12*b^3*e^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 + (12*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2,
-((f*Sqrt[x])/e)])/f^2 - (6*b*e^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 - (24*b^3*e^2*n^3*P
olyLog[3, -((f*Sqrt[x])/e)])/f^2 + (24*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 - (48*
b^3*e^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^2

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2370

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], 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, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 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 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 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]

Rubi steps

\begin{align*} \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )^2+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f \sqrt{x}}-\frac{e^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2 x}+\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{2} (3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(3 b n) \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac{\left (3 b e^2 n\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{f^2}-\frac{(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{x}} \, dx}{f}\\ &=-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{2 f}-\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (6 b^2 n^2\right ) \int \left (\frac{1}{2} \left (-a-b \log \left (c x^n\right )\right )+\frac{e \left (a+b \log \left (c x^n\right )\right )}{f \sqrt{x}}-\frac{e^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2 x}+\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx+\frac{\left (12 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{f}\\ &=-\frac{48 b^3 e n^3 \sqrt{x}}{f}-3 a b^2 n^2 x+\frac{24 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{\left (3 b e^2 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}{f^2}+\left (3 b^2 n^2\right ) \int \left (-a-b \log \left (c x^n\right )\right ) \, dx+\left (6 b^2 n^2\right ) \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (6 b^2 e^2 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{f^2}+\frac{\left (6 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{f}\\ &=-\frac{72 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+3 b^3 n^3 x-3 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{\left (3 b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{2 f}-\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx+\frac{\left (12 b^2 e^2 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}{f^2}-\left (6 b^3 n^3\right ) \int \left (-\frac{1}{2}+\frac{e}{f \sqrt{x}}-\frac{e^2 \log \left (e+f \sqrt{x}\right )}{f^2 x}+\log \left (d \left (e+f \sqrt{x}\right )\right )\right ) \, dx\\ &=-\frac{84 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{\left (6 b^2 e^2 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}{f^2}-\left (6 b^3 n^3\right ) \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \, dx+\frac{\left (6 b^3 e^2 n^3\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{f^2}-\frac{\left (24 b^3 e^2 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{f^2}\\ &=-\frac{84 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )-6 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{\left (12 b^3 e^2 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{f^2}+\frac{\left (12 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{f^2}+\left (3 b^3 f n^3\right ) \int \frac{\sqrt{x}}{e+f \sqrt{x}} \, dx\\ &=-\frac{84 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{12 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{24 b^3 e^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{\left (12 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{f}+\left (6 b^3 f n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{84 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+9 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{12 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{12 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{24 b^3 e^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\left (6 b^3 f n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{90 b^3 e n^3 \sqrt{x}}{f}-6 a b^2 n^2 x+12 b^3 n^3 x+\frac{6 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right )}{f^2}-6 b^3 n^3 x \log \left (d \left (e+f \sqrt{x}\right )\right )+\frac{12 b^3 e^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac{42 b^2 e n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{9 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 b e^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac{12 b^3 e^2 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{24 b^3 e^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{48 b^3 e^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}\\ \end{align*}

Mathematica [B]  time = 0.653998, size = 1522, normalized size = 2.38 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*a^3*e*f*Sqrt[x] + 18*a^2*b*e*f*n*Sqrt[x] - 84*a*b^2*e*f*n^2*Sqrt[x] + 180*b^3*e*f*n^3*Sqrt[x] + a^3*f^2*x
- 6*a^2*b*f^2*n*x + 18*a*b^2*f^2*n^2*x - 24*b^3*f^2*n^3*x + 2*a^3*e^2*Log[e + f*Sqrt[x]] - 6*a^2*b*e^2*n*Log[
e + f*Sqrt[x]] + 12*a*b^2*e^2*n^2*Log[e + f*Sqrt[x]] - 12*b^3*e^2*n^3*Log[e + f*Sqrt[x]] - 2*a^3*f^2*x*Log[d*(
e + f*Sqrt[x])] + 6*a^2*b*f^2*n*x*Log[d*(e + f*Sqrt[x])] - 12*a*b^2*f^2*n^2*x*Log[d*(e + f*Sqrt[x])] + 12*b^3*
f^2*n^3*x*Log[d*(e + f*Sqrt[x])] - 6*a^2*b*e^2*n*Log[e + f*Sqrt[x]]*Log[x] + 12*a*b^2*e^2*n^2*Log[e + f*Sqrt[x
]]*Log[x] - 12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x] + 6*a^2*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 12*a*b^2*
e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] + 12*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x] + 6*a*b^2*e^2*n^2*Log[e +
f*Sqrt[x]]*Log[x]^2 - 6*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x]^2 - 6*a*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[
x]^2 + 6*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 2*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x]^3 + 2*b^3*e^2*n
^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^3 - 6*a^2*b*e*f*Sqrt[x]*Log[c*x^n] + 36*a*b^2*e*f*n*Sqrt[x]*Log[c*x^n] - 84*b
^3*e*f*n^2*Sqrt[x]*Log[c*x^n] + 3*a^2*b*f^2*x*Log[c*x^n] - 12*a*b^2*f^2*n*x*Log[c*x^n] + 18*b^3*f^2*n^2*x*Log[
c*x^n] + 6*a^2*b*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 12*a*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n] + 12*b^3*e^2
*n^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 6*a^2*b*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 12*a*b^2*f^2*n*x*Log[d*
(e + f*Sqrt[x])]*Log[c*x^n] - 12*b^3*f^2*n^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] - 12*a*b^2*e^2*n*Log[e + f*Sq
rt[x]]*Log[x]*Log[c*x^n] + 12*b^3*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] + 12*a*b^2*e^2*n*Log[1 + (f*Sqr
t[x])/e]*Log[x]*Log[c*x^n] - 12*b^3*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] + 6*b^3*e^2*n^2*Log[e + f
*Sqrt[x]]*Log[x]^2*Log[c*x^n] - 6*b^3*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2*Log[c*x^n] - 6*a*b^2*e*f*Sqrt[x]
*Log[c*x^n]^2 + 18*b^3*e*f*n*Sqrt[x]*Log[c*x^n]^2 + 3*a*b^2*f^2*x*Log[c*x^n]^2 - 6*b^3*f^2*n*x*Log[c*x^n]^2 +
6*a*b^2*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 - 6*b^3*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 - 6*a*b^2*f^2*x*Log[
d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + 6*b^3*f^2*n*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 - 6*b^3*e^2*n*Log[e + f*Sq
rt[x]]*Log[x]*Log[c*x^n]^2 + 6*b^3*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n]^2 - 2*b^3*e*f*Sqrt[x]*Log[c*
x^n]^3 + b^3*f^2*x*Log[c*x^n]^3 + 2*b^3*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n]^3 - 2*b^3*f^2*x*Log[d*(e + f*Sqrt[x]
)]*Log[c*x^n]^3 + 12*b*e^2*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b*(a - b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog
[2, -((f*Sqrt[x])/e)] - 48*b^2*e^2*n^2*(a - b*n + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)] + 96*b^3*e^2*n^3*
PolyLog[4, -((f*Sqrt[x])/e)])/(2*f^2)

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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, algorithm="maxima")

[Out]

1/27*(27*b^3*e*x*log(d)*log(x^n)^3 + 81*(a*b^2*e*log(d) - (e*n*log(d) - e*log(c)*log(d))*b^3)*x*log(x^n)^2 + 8
1*(a^2*b*e*log(d) - 2*(e*n*log(d) - e*log(c)*log(d))*a*b^2 + (2*e*n^2*log(d) - 2*e*n*log(c)*log(d) + e*log(c)^
2*log(d))*b^3)*x*log(x^n) + 27*(a^3*e*log(d) - 3*(e*n*log(d) - e*log(c)*log(d))*a^2*b + 3*(2*e*n^2*log(d) - 2*
e*n*log(c)*log(d) + e*log(c)^2*log(d))*a*b^2 - (6*e*n^3*log(d) - 6*e*n^2*log(c)*log(d) + 3*e*n*log(c)^2*log(d)
- e*log(c)^3*log(d))*b^3)*x + 27*(b^3*e*x*log(x^n)^3 - 3*((e*n - e*log(c))*b^3 - a*b^2*e)*x*log(x^n)^2 - 3*(2
*(e*n - e*log(c))*a*b^2 - (2*e*n^2 - 2*e*n*log(c) + e*log(c)^2)*b^3 - a^2*b*e)*x*log(x^n) - (3*(e*n - e*log(c)
)*a^2*b - 3*(2*e*n^2 - 2*e*n*log(c) + e*log(c)^2)*a*b^2 + (6*e*n^3 - 6*e*n^2*log(c) + 3*e*n*log(c)^2 - e*log(c
)^3)*b^3 - a^3*e)*x)*log(f*sqrt(x) + e) - (9*b^3*f*x^2*log(x^n)^3 - 9*((5*f*n - 3*f*log(c))*b^3 - 3*a*b^2*f)*x
^2*log(x^n)^2 - 3*(6*(5*f*n - 3*f*log(c))*a*b^2 - (38*f*n^2 - 30*f*n*log(c) + 9*f*log(c)^2)*b^3 - 9*a^2*b*f)*x
^2*log(x^n) - (9*(5*f*n - 3*f*log(c))*a^2*b - 3*(38*f*n^2 - 30*f*n*log(c) + 9*f*log(c)^2)*a*b^2 + (130*f*n^3 -
114*f*n^2*log(c) + 45*f*n*log(c)^2 - 9*f*log(c)^3)*b^3 - 9*a^3*f)*x^2)/sqrt(x))/e + integrate(1/2*(b^3*f^2*x*
log(x^n)^3 + 3*(a*b^2*f^2 - (f^2*n - f^2*log(c))*b^3)*x*log(x^n)^2 + 3*(a^2*b*f^2 - 2*(f^2*n - f^2*log(c))*a*b
^2 + (2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*b^3)*x*log(x^n) + (a^3*f^2 - 3*(f^2*n - f^2*log(c))*a^2*b + 3
*(2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*a*b^2 - (6*f^2*n^3 - 6*f^2*n^2*log(c) + 3*f^2*n*log(c)^2 - f^2*lo
g(c)^3)*b^3)*x)/(e*f*sqrt(x) + e^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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\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, 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)

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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)

[Out]

Timed out

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

[Out]

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