3.124 \(\int \log (d (e+f \sqrt{x})) (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=405 \[ -\frac{4 b e^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{4 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}+\frac{8 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{e^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{2 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right )}{f^2}-\frac{4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{14 b^2 e n^2 \sqrt{x}}{f}-3 b^2 n^2 x \]

[Out]

(14*b^2*e*n^2*Sqrt[x])/f + a*b*n*x - 3*b^2*n^2*x - (2*b^2*e^2*n^2*Log[e + f*Sqrt[x]])/f^2 + 2*b^2*n^2*x*Log[d*
(e + f*Sqrt[x])] - (4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + b^2*n*x*Log[c*x^n] - (6*b*e*
n*Sqrt[x]*(a + b*Log[c*x^n]))/f + b*n*x*(a + b*Log[c*x^n]) + (2*b*e^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))
/f^2 - 2*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (e*Sqrt[x]*(a + b*Log[c*x^n])^2)/f - (x*(a + b*Log[
c*x^n])^2)/2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^
2)/f^2 - (4*b^2*e^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 - (4*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqr
t[x])/e)])/f^2 + (8*b^2*e^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/f^2

________________________________________________________________________________________

Rubi [A]  time = 0.441201, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2448, 266, 43, 2370, 2295, 2304, 2375, 2337, 2374, 6589, 2454, 2394, 2315} \[ -\frac{4 b e^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{4 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^2}+\frac{8 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac{e^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{2 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right )}{f^2}-\frac{4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{14 b^2 e n^2 \sqrt{x}}{f}-3 b^2 n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*b^2*e*n^2*Sqrt[x])/f + a*b*n*x - 3*b^2*n^2*x - (2*b^2*e^2*n^2*Log[e + f*Sqrt[x]])/f^2 + 2*b^2*n^2*x*Log[d*
(e + f*Sqrt[x])] - (4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + b^2*n*x*Log[c*x^n] - (6*b*e*
n*Sqrt[x]*(a + b*Log[c*x^n]))/f + b*n*x*(a + b*Log[c*x^n]) + (2*b*e^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))
/f^2 - 2*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (e*Sqrt[x]*(a + b*Log[c*x^n])^2)/f - (x*(a + b*Log[
c*x^n])^2)/2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^
2)/f^2 - (4*b^2*e^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 - (4*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqr
t[x])/e)])/f^2 + (8*b^2*e^2*n^2*PolyLog[3, -((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 2295

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

Rule 2304

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

Rule 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 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 )^2 \, dx &=\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2-\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-(2 b n) \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{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2-\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-(b n) \int \left (-a-b \log \left (c x^n\right )\right ) \, dx-(2 b n) \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{\left (2 b e^2 n\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{(2 b e n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{f}\\ &=\frac{8 b^2 e n^2 \sqrt{x}}{f}+a b n x-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \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 (b^2 n\right ) \int \log \left (c x^n\right ) \, dx+\left (2 b^2 n^2\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{12 b^2 e n^2 \sqrt{x}}{f}+a b n x-2 b^2 n^2 x+b^2 n x \log \left (c x^n\right )-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac{\left (2 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 )}{x} \, dx}{f^2}+\left (2 b^2 n^2\right ) \int \log \left (d \left (e+f \sqrt{x}\right )\right ) \, dx-\frac{\left (2 b^2 e^2 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{f^2}\\ &=\frac{12 b^2 e n^2 \sqrt{x}}{f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{4 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{\left (4 b^2 e^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{f^2}-\frac{\left (4 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{f^2}-\left (b^2 f n^2\right ) \int \frac{\sqrt{x}}{e+f \sqrt{x}} \, dx\\ &=\frac{12 b^2 e n^2 \sqrt{x}}{f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+b^2 n x \log \left (c x^n\right )-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{4 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{8 b^2 e^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{\left (4 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{f}-\left (2 b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{12 b^2 e n^2 \sqrt{x}}{f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+b^2 n x \log \left (c x^n\right )-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{4 b^2 e^2 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{4 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{8 b^2 e^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}-\left (2 b^2 f n^2\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{14 b^2 e n^2 \sqrt{x}}{f}+a b n x-3 b^2 n^2 x-\frac{2 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right )}{f^2}+2 b^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )-\frac{4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+b^2 n x \log \left (c x^n\right )-\frac{6 b e n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{e \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{1}{2} x \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 )^2-\frac{e^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac{4 b^2 e^2 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^2}-\frac{4 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}+\frac{8 b^2 e^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{f^2}\\ \end{align*}

Mathematica [A]  time = 0.383844, size = 718, normalized size = 1.77 \[ -\frac{8 b e^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )-16 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )-2 a^2 f^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )+2 a^2 e^2 \log \left (e+f \sqrt{x}\right )-2 a^2 e f \sqrt{x}+a^2 f^2 x-4 a b f^2 x \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )\right )+4 a b e^2 \log \left (c x^n\right ) \log \left (e+f \sqrt{x}\right )-4 a b e f \sqrt{x} \log \left (c x^n\right )+2 a b f^2 x \log \left (c x^n\right )+4 a b f^2 n x \log \left (d \left (e+f \sqrt{x}\right )\right )-4 a b e^2 n \log \left (e+f \sqrt{x}\right )-4 a b e^2 n \log (x) \log \left (e+f \sqrt{x}\right )+4 a b e^2 n \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+12 a b e f n \sqrt{x}-4 a b f^2 n x-2 b^2 f^2 x \log ^2\left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )\right )+4 b^2 f^2 n x \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )\right )+2 b^2 e^2 \log ^2\left (c x^n\right ) \log \left (e+f \sqrt{x}\right )-4 b^2 e^2 n \log \left (c x^n\right ) \log \left (e+f \sqrt{x}\right )-4 b^2 e^2 n \log (x) \log \left (c x^n\right ) \log \left (e+f \sqrt{x}\right )+4 b^2 e^2 n \log (x) \log \left (c x^n\right ) \log \left (\frac{f \sqrt{x}}{e}+1\right )-2 b^2 e f \sqrt{x} \log ^2\left (c x^n\right )+12 b^2 e f n \sqrt{x} \log \left (c x^n\right )+b^2 f^2 x \log ^2\left (c x^n\right )-4 b^2 f^2 n x \log \left (c x^n\right )-4 b^2 f^2 n^2 x \log \left (d \left (e+f \sqrt{x}\right )\right )+2 b^2 e^2 n^2 \log ^2(x) \log \left (e+f \sqrt{x}\right )-2 b^2 e^2 n^2 \log ^2(x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+4 b^2 e^2 n^2 \log \left (e+f \sqrt{x}\right )+4 b^2 e^2 n^2 \log (x) \log \left (e+f \sqrt{x}\right )-4 b^2 e^2 n^2 \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )-28 b^2 e f n^2 \sqrt{x}+6 b^2 f^2 n^2 x}{2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*a^2*e*f*Sqrt[x] + 12*a*b*e*f*n*Sqrt[x] - 28*b^2*e*f*n^2*Sqrt[x] + a^2*f^2*x - 4*a*b*f^2*n*x + 6*b^2*f^2*n
^2*x + 2*a^2*e^2*Log[e + f*Sqrt[x]] - 4*a*b*e^2*n*Log[e + f*Sqrt[x]] + 4*b^2*e^2*n^2*Log[e + f*Sqrt[x]] - 2*a^
2*f^2*x*Log[d*(e + f*Sqrt[x])] + 4*a*b*f^2*n*x*Log[d*(e + f*Sqrt[x])] - 4*b^2*f^2*n^2*x*Log[d*(e + f*Sqrt[x])]
- 4*a*b*e^2*n*Log[e + f*Sqrt[x]]*Log[x] + 4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x] + 4*a*b*e^2*n*Log[1 + (f*Sq
rt[x])/e]*Log[x] - 4*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] + 2*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x]^2 - 2
*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 4*a*b*e*f*Sqrt[x]*Log[c*x^n] + 12*b^2*e*f*n*Sqrt[x]*Log[c*x^n]
+ 2*a*b*f^2*x*Log[c*x^n] - 4*b^2*f^2*n*x*Log[c*x^n] + 4*a*b*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 4*b^2*e^2*n*Lo
g[e + f*Sqrt[x]]*Log[c*x^n] - 4*a*b*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 4*b^2*f^2*n*x*Log[d*(e + f*Sqrt[
x])]*Log[c*x^n] - 4*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] + 4*b^2*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x]
*Log[c*x^n] - 2*b^2*e*f*Sqrt[x]*Log[c*x^n]^2 + b^2*f^2*x*Log[c*x^n]^2 + 2*b^2*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n
]^2 - 2*b^2*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + 8*b*e^2*n*(a - b*n + b*Log[c*x^n])*PolyLog[2, -((f*Sqr
t[x])/e)] - 16*b^2*e^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/(2*f^2)

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\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))^2*ln(d*(e+f*x^(1/2))),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{27 \, b^{2} e x \log \left (d\right ) \log \left (x^{n}\right )^{2} + 54 \,{\left (a b e \log \left (d\right ) -{\left (e n \log \left (d\right ) - e \log \left (c\right ) \log \left (d\right )\right )} b^{2}\right )} x \log \left (x^{n}\right ) + 27 \,{\left (a^{2} e \log \left (d\right ) - 2 \,{\left (e n \log \left (d\right ) - e \log \left (c\right ) \log \left (d\right )\right )} a b +{\left (2 \, e n^{2} \log \left (d\right ) - 2 \, e n \log \left (c\right ) \log \left (d\right ) + e \log \left (c\right )^{2} \log \left (d\right )\right )} b^{2}\right )} x + 27 \,{\left (b^{2} e x \log \left (x^{n}\right )^{2} - 2 \,{\left ({\left (e n - e \log \left (c\right )\right )} b^{2} - a b e\right )} x \log \left (x^{n}\right ) -{\left (2 \,{\left (e n - e \log \left (c\right )\right )} a b -{\left (2 \, e n^{2} - 2 \, e n \log \left (c\right ) + e \log \left (c\right )^{2}\right )} b^{2} - a^{2} e\right )} x\right )} \log \left (f \sqrt{x} + e\right ) - \frac{9 \, b^{2} f x^{2} \log \left (x^{n}\right )^{2} - 6 \,{\left ({\left (5 \, f n - 3 \, f \log \left (c\right )\right )} b^{2} - 3 \, a b f\right )} x^{2} \log \left (x^{n}\right ) -{\left (6 \,{\left (5 \, f n - 3 \, f \log \left (c\right )\right )} a b -{\left (38 \, f n^{2} - 30 \, f n \log \left (c\right ) + 9 \, f \log \left (c\right )^{2}\right )} b^{2} - 9 \, a^{2} f\right )} x^{2}}{\sqrt{x}}}{27 \, e} + \int \frac{b^{2} f^{2} x \log \left (x^{n}\right )^{2} + 2 \,{\left (a b f^{2} -{\left (f^{2} n - f^{2} \log \left (c\right )\right )} b^{2}\right )} x \log \left (x^{n}\right ) +{\left (a^{2} f^{2} - 2 \,{\left (f^{2} n - f^{2} \log \left (c\right )\right )} a b +{\left (2 \, f^{2} n^{2} - 2 \, f^{2} n \log \left (c\right ) + f^{2} \log \left (c\right )^{2}\right )} b^{2}\right )} x}{2 \,{\left (e f \sqrt{x} + e^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(27*b^2*e*x*log(d)*log(x^n)^2 + 54*(a*b*e*log(d) - (e*n*log(d) - e*log(c)*log(d))*b^2)*x*log(x^n) + 27*(a
^2*e*log(d) - 2*(e*n*log(d) - e*log(c)*log(d))*a*b + (2*e*n^2*log(d) - 2*e*n*log(c)*log(d) + e*log(c)^2*log(d)
)*b^2)*x + 27*(b^2*e*x*log(x^n)^2 - 2*((e*n - e*log(c))*b^2 - a*b*e)*x*log(x^n) - (2*(e*n - e*log(c))*a*b - (2
*e*n^2 - 2*e*n*log(c) + e*log(c)^2)*b^2 - a^2*e)*x)*log(f*sqrt(x) + e) - (9*b^2*f*x^2*log(x^n)^2 - 6*((5*f*n -
3*f*log(c))*b^2 - 3*a*b*f)*x^2*log(x^n) - (6*(5*f*n - 3*f*log(c))*a*b - (38*f*n^2 - 30*f*n*log(c) + 9*f*log(c
)^2)*b^2 - 9*a^2*f)*x^2)/sqrt(x))/e + integrate(1/2*(b^2*f^2*x*log(x^n)^2 + 2*(a*b*f^2 - (f^2*n - f^2*log(c))*
b^2)*x*log(x^n) + (a^2*f^2 - 2*(f^2*n - f^2*log(c))*a*b + (2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*b^2)*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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\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))^2*log(d*(e+f*x^(1/2))),x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*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))**2*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 )}^{2} \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))^2*log(d*(e+f*x^(1/2))),x, algorithm="giac")

[Out]

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