Integrand size = 24, antiderivative size = 263 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]
6*b^2*e^2*n^2*ln(-e*x^(1/2)/d)*(a+b*ln(c*(d+e*x^(1/2))^n))/d^2-(a+b*ln(c*( d+e*x^(1/2))^n))^3/x-3*b*e^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*ln(1-d/(d+e*x ^(1/2)))/d^2+6*b^2*e^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*polylog(2,d/(d+e*x^ (1/2)))/d^2+6*b^3*e^2*n^3*polylog(2,1+e*x^(1/2)/d)/d^2+6*b^3*e^2*n^3*polyl og(3,d/(d+e*x^(1/2)))/d^2-3*b*e*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/ 2))/d^2/x^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(536\) vs. \(2(263)=526\).
Time = 0.53 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-3 b d e n \sqrt {x} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-3 b d^2 n \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+3 b e^2 n x \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-d^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-\frac {3}{2} b e^2 n x \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)+3 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\left (d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right ) \left (-2 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-2 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-2 e^2 x \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )\right )+b^3 n^3 \left (\left (d+e \sqrt {x}\right ) \log ^2\left (d+e \sqrt {x}\right ) \left (-3 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-3 e^2 x \left (-2+\log \left (d+e \sqrt {x}\right )\right ) \log \left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-6 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt {x}}{d}\right )+6 e^2 x \operatorname {PolyLog}\left (3,1+\frac {e \sqrt {x}}{d}\right )\right )}{d^2 x} \]
(-3*b*d*e*n*Sqrt[x]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^ n])^2 - 3*b*d^2*n*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c *(d + e*Sqrt[x])^n])^2 + 3*b*e^2*n*x*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e *Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 - d^2*(a - b*n*Log[d + e*Sqrt[x] ] + b*Log[c*(d + e*Sqrt[x])^n])^3 - (3*b*e^2*n*x*(a - b*n*Log[d + e*Sqrt[x ]] + b*Log[c*(d + e*Sqrt[x])^n])^2*Log[x])/2 + 3*b^2*n^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])*((d + e*Sqrt[x])*Log[d + e*Sqrt[x ]]*(-2*e*Sqrt[x] + (-d + e*Sqrt[x])*Log[d + e*Sqrt[x]]) - 2*e^2*x*(-1 + Lo g[d + e*Sqrt[x]])*Log[-((e*Sqrt[x])/d)] - 2*e^2*x*PolyLog[2, 1 + (e*Sqrt[x ])/d]) + b^3*n^3*((d + e*Sqrt[x])*Log[d + e*Sqrt[x]]^2*(-3*e*Sqrt[x] + (-d + e*Sqrt[x])*Log[d + e*Sqrt[x]]) - 3*e^2*x*(-2 + Log[d + e*Sqrt[x]])*Log[ d + e*Sqrt[x]]*Log[-((e*Sqrt[x])/d)] - 6*e^2*x*(-1 + Log[d + e*Sqrt[x]])*P olyLog[2, 1 + (e*Sqrt[x])/d] + 6*e^2*x*PolyLog[3, 1 + (e*Sqrt[x])/d]))/(d^ 2*x)
Time = 1.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2904, 2845, 2858, 27, 2789, 2755, 2754, 2779, 2821, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 2 \left (\frac {3}{2} b e n \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{\left (d+e \sqrt {x}\right ) x}d\sqrt {x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 2 \left (\frac {3}{2} b n \int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{x^{3/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x^{3/2}}d\left (d+e \sqrt {x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {\int \frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e^2 x}d\left (d+e \sqrt {x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \int -\frac {a+b \log \left (c x^{n/2}\right )}{e \sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/2}\right )\right )^2}{e x}d\left (d+e \sqrt {x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\frac {2 b n \int \frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt {x}}{d}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right )}{\sqrt {x}}d\left (d+e \sqrt {x}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 2 \left (\frac {3}{2} b e^2 n \left (\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e \sqrt {x}}{d}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e \sqrt {x}}{d}\right )\right )}{d}-\frac {\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d e \sqrt {x}}}{d}+\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )+b n \operatorname {PolyLog}\left (3,\frac {d}{\sqrt {x}}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )^2}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 x}\right )\) |
2*(-1/2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x + (3*b*e^2*n*((-(((d + e*Sqrt [x])*(a + b*Log[c*x^(n/2)])^2)/(d*e*Sqrt[x])) - (2*b*n*(-(Log[1 - (d + e*S qrt[x])/d]*(a + b*Log[c*x^(n/2)])) - b*n*PolyLog[2, (d + e*Sqrt[x])/d]))/d )/d + (-((Log[1 - d/Sqrt[x]]*(a + b*Log[c*x^(n/2)])^2)/d) + (2*b*n*((a + b *Log[c*x^(n/2)])*PolyLog[2, d/Sqrt[x]] + b*n*PolyLog[3, d/Sqrt[x]]))/d)/d) )/2)
3.5.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[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]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[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])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
integral((b^3*log((e*sqrt(x) + d)^n*c)^3 + 3*a*b^2*log((e*sqrt(x) + d)^n*c )^2 + 3*a^2*b*log((e*sqrt(x) + d)^n*c) + a^3)/x^2, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
-1/2*(2*b^3*d^2*sqrt(x)*log((e*sqrt(x) + d)^n)^3 - 3*(2*b^3*e^2*n*x^(3/2)* log(e*sqrt(x) + d) - 2*b^3*d*e*n*x - (b^3*e^2*n*x*log(x) + 2*b^3*d^2*log(c ) + 2*a*b^2*d^2)*sqrt(x))*log((e*sqrt(x) + d)^n)^2)/(d^2*x^(3/2)) - integr ate(-1/2*(2*(b^3*d^2*e*log(c)^3 + 3*a*b^2*d^2*e*log(c)^2 + 3*a^2*b*d^2*e*l og(c) + a^3*d^2*e)*x^(3/2) + 2*(b^3*d^3*log(c)^3 + 3*a*b^2*d^3*log(c)^2 + 3*a^2*b*d^3*log(c) + a^3*d^3)*x - 3*(2*b^3*e^3*n^2*x^(5/2)*log(e*sqrt(x) + d) - 2*b^3*d*e^2*n^2*x^2 - 2*(b^3*d^2*e*log(c)^2 + 2*a*b^2*d^2*e*log(c) + a^2*b*d^2*e)*x^(3/2) - 2*(b^3*d^3*log(c)^2 + 2*a*b^2*d^3*log(c) + a^2*b*d ^3)*x - (b^3*e^3*n^2*x^2*log(x) + 2*(b^3*d^2*e*n*log(c) + a*b^2*d^2*e*n)*x )*sqrt(x))*log((e*sqrt(x) + d)^n))/(d^2*e*x^(7/2) + d^3*x^3), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^2} \,d x \]