Integrand size = 28, antiderivative size = 673 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {90 b^3 f n^3}{e \sqrt {x}}+\frac {6 b^3 f^2 n^3 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {6 b^3 n^3 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {12 b^3 f^2 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {3 b^3 f^2 n^3 \log (x)}{e^2}+\frac {3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac {42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {6 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {6 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \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}{x}+\frac {3 b f^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac {12 b^3 f^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {24 b^3 f^2 n^3 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {48 b^3 f^2 n^3 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{e^2} \]
-3*b^3*f^2*n^3*ln(x)/e^2+3/2*b^3*f^2*n^3*ln(x)^2/e^2-3*b^2*f^2*n^2*ln(x)*( a+b*ln(c*x^n))/e^2-1/2*f^2*(a+b*ln(c*x^n))^3/e^2-1/8*f^2*(a+b*ln(c*x^n))^4 /b/e^2/n+6*b^3*f^2*n^3*ln(e+f*x^(1/2))/e^2+6*b^2*f^2*n^2*(a+b*ln(c*x^n))*l n(e+f*x^(1/2))/e^2-12*b^3*f^2*n^3*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e^2-6*b ^3*n^3*ln(d*(e+f*x^(1/2)))/x-6*b^2*n^2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))) /x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))/x-(a+b*ln(c*x^n))^3*ln(d*(e +f*x^(1/2)))/x+3*b*f^2*n*(a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e)/e^2+f^2*(a+b* ln(c*x^n))^3*ln(1+f*x^(1/2)/e)/e^2+12*b^2*f^2*n^2*(a+b*ln(c*x^n))*polylog( 2,-f*x^(1/2)/e)/e^2+6*b*f^2*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x^(1/2)/e)/e^ 2-12*b^3*f^2*n^3*polylog(2,1+f*x^(1/2)/e)/e^2-24*b^3*f^2*n^3*polylog(3,-f* x^(1/2)/e)/e^2-24*b^2*f^2*n^2*(a+b*ln(c*x^n))*polylog(3,-f*x^(1/2)/e)/e^2+ 48*b^3*f^2*n^3*polylog(4,-f*x^(1/2)/e)/e^2-90*b^3*f*n^3/e/x^(1/2)-42*b^2*f *n^2*(a+b*ln(c*x^n))/e/x^(1/2)-9*b*f*n*(a+b*ln(c*x^n))^2/e/x^(1/2)-f*(a+b* ln(c*x^n))^3/e/x^(1/2)
Time = 0.70 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=-\frac {e^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 b \left (a^2+2 a b n+2 b^2 n^2\right ) \log \left (c x^n\right )+3 b^2 (a+b n) \log ^2\left (c x^n\right )+b^3 \log ^3\left (c x^n\right )\right )+e f \sqrt {x} \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-f^2 x \log \left (e+f \sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+\frac {1}{2} f^2 x \log (x) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+3 b f n \sqrt {x} \left (a^2+2 a b n+2 b^2 n^2+2 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (2 e+\left (e-f \sqrt {x} \log \left (1+\frac {f \sqrt {x}}{e}\right )\right ) \log (x)+\frac {1}{4} f \sqrt {x} \log ^2(x)-2 f \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )\right )+b^2 f n^2 \sqrt {x} \left (a+b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (24 e+12 e \log (x)+3 e \log ^2(x)-3 f \sqrt {x} \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+\frac {1}{2} f \sqrt {x} \log ^3(x)-12 f \sqrt {x} \log (x) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )+24 f \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )\right )+b^3 n^3 \left (6 f^2 x \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e}{f \sqrt {x}}\right )+f \sqrt {x} \left (48 e+24 e \log (x)+6 e \log ^2(x)+e \log ^3(x)-f \sqrt {x} \log \left (1+\frac {e}{f \sqrt {x}}\right ) \log ^3(x)+24 f \sqrt {x} \log (x) \operatorname {PolyLog}\left (3,-\frac {e}{f \sqrt {x}}\right )+48 f \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {e}{f \sqrt {x}}\right )\right )\right )}{e^2 x} \]
-((e^2*Log[d*(e + f*Sqrt[x])]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*b*(a^2 + 2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a + b*n)*Log[c*x^n]^2 + b^3*Log[c*x^n]^3) + e*f*Sqrt[x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3* n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c* x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[ c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[ c*x^n])^3) - f^2*x*Log[e + f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b ^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log [c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + L og[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + L og[c*x^n])^3) + (f^2*x*Log[x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n ])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n ])^3))/2 + 3*b*f*n*Sqrt[x]*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*Log[x]) + L og[c*x^n])^2)*(2*e + (e - f*Sqrt[x]*Log[1 + (f*Sqrt[x])/e])*Log[x] + (f*Sq rt[x]*Log[x]^2)/4 - 2*f*Sqrt[x]*PolyLog[2, -((f*Sqrt[x])/e)]) + b^2*f*n^2* Sqrt[x]*(a + b*n - b*n*Log[x] + b*Log[c*x^n])*(24*e + 12*e*Log[x] + 3*e*Lo g[x]^2 - 3*f*Sqrt[x]*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + (f*Sqrt[x]*Log[x...
Time = 1.42 (sec) , antiderivative size = 779, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 2009}
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.
\(\displaystyle \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx\) |
\(\Big \downarrow \) 2824 |
\(\displaystyle -3 b n \int \left (\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e x^{3/2}}\right )dx-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt {x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 b n \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^4}{24 b^2 e^2 n^2}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {4 b f^2 n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {8 b f^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b e^2 n}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {3 f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}+\frac {14 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}+\frac {4 b^2 f^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{e^2}+\frac {8 b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {16 b^2 f^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}+\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {30 b^2 f n^2}{e \sqrt {x}}\right )-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt {x}}\) |
-((f*(a + b*Log[c*x^n])^3)/(e*Sqrt[x])) + (f^2*Log[e + f*Sqrt[x]]*(a + b*L og[c*x^n])^3)/e^2 - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x - (f^2 *Log[x]*(a + b*Log[c*x^n])^3)/(2*e^2) - 3*b*n*((30*b^2*f*n^2)/(e*Sqrt[x]) - (2*b^2*f^2*n^2*Log[e + f*Sqrt[x]])/e^2 + (2*b^2*n^2*Log[d*(e + f*Sqrt[x] )])/x + (4*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 + (b^ 2*f^2*n^2*Log[x])/e^2 - (b^2*f^2*n^2*Log[x]^2)/(2*e^2) + (14*b*f*n*(a + b* Log[c*x^n]))/(e*Sqrt[x]) - (2*b*f^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n] ))/e^2 + (2*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x + (b*f^2*n*Lo g[x]*(a + b*Log[c*x^n]))/e^2 + (3*f*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) + (L og[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x - (f^2*Log[1 + (f*Sqrt[x])/e ]*(a + b*Log[c*x^n])^2)/e^2 + (f^2*(a + b*Log[c*x^n])^3)/(6*b*e^2*n) + (f^ 2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(3*b*e^2*n) - (f^2*Log[1 + (f*S qrt[x])/e]*(a + b*Log[c*x^n])^3)/(3*b*e^2*n) - (f^2*Log[x]*(a + b*Log[c*x^ n])^3)/(6*b*e^2*n) + (f^2*(a + b*Log[c*x^n])^4)/(24*b^2*e^2*n^2) + (4*b^2* f^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^2 - (4*b*f^2*n*(a + b*Log[c*x^n]) *PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (2*f^2*(a + b*Log[c*x^n])^2*PolyLog[2 , -((f*Sqrt[x])/e)])/e^2 + (8*b^2*f^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/e^ 2 + (8*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 - (16* b^2*f^2*n^2*PolyLog[4, -((f*Sqrt[x])/e)])/e^2)
3.2.31.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ .))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)], x]}, Simp[(a + b*Log[c*x^n])^p u, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1)/x u, 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]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{2}}d x\]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
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^2, x)
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx=\int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]