Integrand size = 24, antiderivative size = 480 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {5 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2}{2 e^6}+\frac {40 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3}{27 e^6}-\frac {5 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4}{8 e^6}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^6}{54 e^6}+\frac {4 b^2 d^5 n^2}{e^5 \sqrt {x}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {4 b d^5 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^6}+\frac {5 b d^4 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{e^6}-\frac {40 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 e^6}+\frac {5 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^6}-\frac {4 b d n \left (d+\frac {e}{\sqrt {x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt {x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{9 e^6}+\frac {2 b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{3 x^3} \]
-1/3*b^2*d^6*n^2*ln(d+e/x^(1/2))^2/e^6+2/3*b*d^6*n*ln(d+e/x^(1/2))*(a+b*ln (c*(d+e/x^(1/2))^n))/e^6-1/3*(a+b*ln(c*(d+e/x^(1/2))^n))^2/x^3-4*b*d^5*n*( a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))/e^6-5/2*b^2*d^4*n^2*(d+e/x^(1/2)) ^2/e^6+5*b*d^4*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^2/e^6+40/27*b^2 *d^3*n^2*(d+e/x^(1/2))^3/e^6-40/9*b*d^3*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e /x^(1/2))^3/e^6-5/8*b^2*d^2*n^2*(d+e/x^(1/2))^4/e^6+5/2*b*d^2*n*(a+b*ln(c* (d+e/x^(1/2))^n))*(d+e/x^(1/2))^4/e^6+4/25*b^2*d*n^2*(d+e/x^(1/2))^5/e^6-4 /5*b*d*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^5/e^6-1/54*b^2*n^2*(d+e /x^(1/2))^6/e^6+1/9*b*n*(a+b*ln(c*(d+e/x^(1/2))^n))*(d+e/x^(1/2))^6/e^6+4* b^2*d^5*n^2/e^5/x^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.37 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=\frac {-1800 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b n \left (600 a e^6-100 b e^6 n-720 a d e^5 \sqrt {x}+264 b d e^5 n \sqrt {x}+900 a d^2 e^4 x-555 b d^2 e^4 n x-1200 a d^3 e^3 x^{3/2}+1140 b d^3 e^3 n x^{3/2}+1800 a d^4 e^2 x^2-2610 b d^4 e^2 n x^2-3600 a d^5 e x^{5/2}+8820 b d^5 e n x^{5/2}-8820 b d^6 n x^3 \log \left (d+\frac {e}{\sqrt {x}}\right )+600 b e^6 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-720 b d e^5 \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+900 b d^2 e^4 x \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-1200 b d^3 e^3 x^{3/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+1800 b d^4 e^2 x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-3600 b d^5 e x^{5/2} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+3600 a d^6 x^3 \log \left (e+d \sqrt {x}\right )+3600 b d^6 x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (e+d \sqrt {x}\right )-1800 b d^6 n x^3 \log ^2\left (e+d \sqrt {x}\right )+3600 b d^6 x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )+3600 b d^6 n x^3 \log \left (e+d \sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )-1800 a d^6 x^3 \log (x)+3600 b d^6 n x^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )+3600 b d^6 n x^3 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )}{e^6}}{5400 x^3} \]
(-1800*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*n*(600*a*e^6 - 100*b*e^6*n - 720*a*d*e^5*Sqrt[x] + 264*b*d*e^5*n*Sqrt[x] + 900*a*d^2*e^4*x - 555*b*d^ 2*e^4*n*x - 1200*a*d^3*e^3*x^(3/2) + 1140*b*d^3*e^3*n*x^(3/2) + 1800*a*d^4 *e^2*x^2 - 2610*b*d^4*e^2*n*x^2 - 3600*a*d^5*e*x^(5/2) + 8820*b*d^5*e*n*x^ (5/2) - 8820*b*d^6*n*x^3*Log[d + e/Sqrt[x]] + 600*b*e^6*Log[c*(d + e/Sqrt[ x])^n] - 720*b*d*e^5*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] + 900*b*d^2*e^4*x*Lo g[c*(d + e/Sqrt[x])^n] - 1200*b*d^3*e^3*x^(3/2)*Log[c*(d + e/Sqrt[x])^n] + 1800*b*d^4*e^2*x^2*Log[c*(d + e/Sqrt[x])^n] - 3600*b*d^5*e*x^(5/2)*Log[c* (d + e/Sqrt[x])^n] + 3600*a*d^6*x^3*Log[e + d*Sqrt[x]] + 3600*b*d^6*x^3*Lo g[c*(d + e/Sqrt[x])^n]*Log[e + d*Sqrt[x]] - 1800*b*d^6*n*x^3*Log[e + d*Sqr t[x]]^2 + 3600*b*d^6*x^3*Log[c*(d + e/Sqrt[x])^n]*Log[-(e/(d*Sqrt[x]))] + 3600*b*d^6*n*x^3*Log[e + d*Sqrt[x]]*Log[-((d*Sqrt[x])/e)] - 1800*a*d^6*x^3 *Log[x] + 3600*b*d^6*n*x^3*PolyLog[2, 1 + e/(d*Sqrt[x])] + 3600*b*d^6*n*x^ 3*PolyLog[2, 1 + (d*Sqrt[x])/e]))/e^6)/(5400*x^3)
Time = 0.53 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.63, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2904, 2845, 2858, 27, 2772, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -2 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^{5/2}}d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 x^3}-\frac {1}{3} b e n \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{\left (d+\frac {e}{\sqrt {x}}\right ) x^3}d\frac {1}{\sqrt {x}}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 x^3}-\frac {1}{3} b n \int \frac {a+b \log \left (c x^{-n/2}\right )}{x^{5/2}}d\left (d+\frac {e}{\sqrt {x}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 x^3}-\frac {b n \int \frac {e^6 \left (a+b \log \left (c x^{-n/2}\right )\right )}{x^{5/2}}d\left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 x^3}-\frac {b n \left (-b n \int \left (\sqrt {x} \log \left (d+\frac {e}{\sqrt {x}}\right ) d^6-6 d^5+\frac {15}{2} \left (d+\frac {e}{\sqrt {x}}\right ) d^4-\frac {20 d^3}{3 x}+\frac {15 d^2}{4 x^{3/2}}-\frac {6 d}{5 x^2}+\frac {1}{6 x^{5/2}}\right )d\left (d+\frac {e}{\sqrt {x}}\right )+d^6 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )-6 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )+\frac {15 d^4 \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 x}-\frac {20 d^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 x^{3/2}}+\frac {15 d^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 x^2}-\frac {6 d \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 x^{5/2}}+\frac {a+b \log \left (c x^{-n/2}\right )}{6 x^3}\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{6 x^3}-\frac {b n \left (d^6 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )-6 d^5 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c x^{-n/2}\right )\right )+\frac {15 d^4 \left (a+b \log \left (c x^{-n/2}\right )\right )}{2 x}-\frac {20 d^3 \left (a+b \log \left (c x^{-n/2}\right )\right )}{3 x^{3/2}}+\frac {15 d^2 \left (a+b \log \left (c x^{-n/2}\right )\right )}{4 x^2}-\frac {6 d \left (a+b \log \left (c x^{-n/2}\right )\right )}{5 x^{5/2}}+\frac {a+b \log \left (c x^{-n/2}\right )}{6 x^3}-b n \left (\frac {1}{2} d^6 \log ^2\left (d+\frac {e}{\sqrt {x}}\right )-6 d^5 \left (d+\frac {e}{\sqrt {x}}\right )+\frac {15 d^4}{4 x}-\frac {20 d^3}{9 x^{3/2}}+\frac {15 d^2}{16 x^2}-\frac {6 d}{25 x^{5/2}}+\frac {1}{36 x^3}\right )\right )}{3 e^6}\right )\) |
-2*((a + b*Log[c*(d + e/Sqrt[x])^n])^2/(6*x^3) - (b*n*(-(b*n*(-6*d^5*(d + e/Sqrt[x]) + 1/(36*x^3) - (6*d)/(25*x^(5/2)) + (15*d^2)/(16*x^2) - (20*d^3 )/(9*x^(3/2)) + (15*d^4)/(4*x) + (d^6*Log[d + e/Sqrt[x]]^2)/2)) - 6*d^5*(d + e/Sqrt[x])*(a + b*Log[c/x^(n/2)]) + (a + b*Log[c/x^(n/2)])/(6*x^3) - (6 *d*(a + b*Log[c/x^(n/2)]))/(5*x^(5/2)) + (15*d^2*(a + b*Log[c/x^(n/2)]))/( 4*x^2) - (20*d^3*(a + b*Log[c/x^(n/2)]))/(3*x^(3/2)) + (15*d^4*(a + b*Log[ c/x^(n/2)]))/(2*x) + d^6*Log[d + e/Sqrt[x]]*(a + b*Log[c/x^(n/2)])))/(3*e^ 6))
3.5.35.3.1 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_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}}{x^{4}}d x\]
Time = 0.34 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {100 \, b^{2} e^{6} n^{2} + 1800 \, b^{2} e^{6} \log \left (c\right )^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} + 90 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{3} - b^{2} e^{6} n^{2}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} + 15 \, {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x - 300 \, {\left (6 \, b^{2} d^{4} e^{2} n x^{2} + 3 \, b^{2} d^{2} e^{4} n x + 2 \, b^{2} e^{6} n - 12 \, a b e^{6}\right )} \log \left (c\right ) - 60 \, {\left (30 \, b^{2} d^{4} e^{2} n^{2} x^{2} + 15 \, b^{2} d^{2} e^{4} n^{2} x + 10 \, b^{2} e^{6} n^{2} - 60 \, a b e^{6} n - 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{3} + 60 \, {\left (b^{2} d^{6} n x^{3} - b^{2} e^{6} n\right )} \log \left (c\right ) - 4 \, {\left (15 \, b^{2} d^{5} e n^{2} x^{2} + 5 \, b^{2} d^{3} e^{3} n^{2} x + 3 \, b^{2} d e^{5} n^{2}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 12 \, {\left (22 \, b^{2} d e^{5} n^{2} - 60 \, a b d e^{5} n + 15 \, {\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x^{2} + 5 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x - 20 \, {\left (15 \, b^{2} d^{5} e n x^{2} + 5 \, b^{2} d^{3} e^{3} n x + 3 \, b^{2} d e^{5} n\right )} \log \left (c\right )\right )} \sqrt {x}}{5400 \, e^{6} x^{3}} \]
-1/5400*(100*b^2*e^6*n^2 + 1800*b^2*e^6*log(c)^2 - 600*a*b*e^6*n + 1800*a^ 2*e^6 + 90*(29*b^2*d^4*e^2*n^2 - 20*a*b*d^4*e^2*n)*x^2 - 1800*(b^2*d^6*n^2 *x^3 - b^2*e^6*n^2)*log((d*x + e*sqrt(x))/x)^2 + 15*(37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n)*x - 300*(6*b^2*d^4*e^2*n*x^2 + 3*b^2*d^2*e^4*n*x + 2*b^2 *e^6*n - 12*a*b*e^6)*log(c) - 60*(30*b^2*d^4*e^2*n^2*x^2 + 15*b^2*d^2*e^4* n^2*x + 10*b^2*e^6*n^2 - 60*a*b*e^6*n - 3*(49*b^2*d^6*n^2 - 20*a*b*d^6*n)* x^3 + 60*(b^2*d^6*n*x^3 - b^2*e^6*n)*log(c) - 4*(15*b^2*d^5*e*n^2*x^2 + 5* b^2*d^3*e^3*n^2*x + 3*b^2*d*e^5*n^2)*sqrt(x))*log((d*x + e*sqrt(x))/x) - 1 2*(22*b^2*d*e^5*n^2 - 60*a*b*d*e^5*n + 15*(49*b^2*d^5*e*n^2 - 20*a*b*d^5*e *n)*x^2 + 5*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x - 20*(15*b^2*d^5*e*n *x^2 + 5*b^2*d^3*e^3*n*x + 3*b^2*d*e^5*n)*log(c))*sqrt(x))/(e^6*x^3)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=\frac {1}{90} \, a b e n {\left (\frac {60 \, d^{6} \log \left (d \sqrt {x} + e\right )}{e^{7}} - \frac {30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac {3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt {x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} + \frac {1}{5400} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (d \sqrt {x} + e\right )}{e^{7}} - \frac {30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac {3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt {x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{3} \log \left (d \sqrt {x} + e\right )^{2} + 450 \, d^{6} x^{3} \log \left (x\right )^{2} - 4410 \, d^{6} x^{3} \log \left (x\right ) - 8820 \, d^{5} e x^{\frac {5}{2}} + 2610 \, d^{4} e^{2} x^{2} - 1140 \, d^{3} e^{3} x^{\frac {3}{2}} + 555 \, d^{2} e^{4} x - 264 \, d e^{5} \sqrt {x} + 100 \, e^{6} - 180 \, {\left (10 \, d^{6} x^{3} \log \left (x\right ) - 49 \, d^{6} x^{3}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{6} x^{3}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a^{2}}{3 \, x^{3}} \]
1/90*a*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5* x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt (x) - 10*e^5)/(e^6*x^3)) + 1/5400*(60*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x) )^n) - (1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4410*d^ 6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2 ) + 555*d^2*e^4*x - 264*d*e^5*sqrt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^6*x^3))*b^2 - 1/3*b^2*log(c*(d + e /sqrt(x))^n)^2/x^3 - 2/3*a*b*log(c*(d + e/sqrt(x))^n)/x^3 - 1/3*a^2/x^3
Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (412) = 824\).
Time = 0.41 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=\text {Too large to display} \]
1/5400*(1800*(6*(d*sqrt(x) + e)*b^2*d^5*n^2/(e^5*sqrt(x)) - 15*(d*sqrt(x) + e)^2*b^2*d^4*n^2/(e^5*x) + 20*(d*sqrt(x) + e)^3*b^2*d^3*n^2/(e^5*x^(3/2) ) - 15*(d*sqrt(x) + e)^4*b^2*d^2*n^2/(e^5*x^2) + 6*(d*sqrt(x) + e)^5*b^2*d *n^2/(e^5*x^(5/2)) - (d*sqrt(x) + e)^6*b^2*n^2/(e^5*x^3))*log((d*sqrt(x) + e)/sqrt(x))^2 + 60*(10*(b^2*n^2 - 6*b^2*n*log(c) - 6*a*b*n)*(d*sqrt(x) + e)^6/(e^5*x^3) - 72*(b^2*d*n^2 - 5*b^2*d*n*log(c) - 5*a*b*d*n)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 225*(b^2*d^2*n^2 - 4*b^2*d^2*n*log(c) - 4*a*b*d^2*n )*(d*sqrt(x) + e)^4/(e^5*x^2) - 400*(b^2*d^3*n^2 - 3*b^2*d^3*n*log(c) - 3* a*b*d^3*n)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 450*(b^2*d^4*n^2 - 2*b^2*d^4* n*log(c) - 2*a*b*d^4*n)*(d*sqrt(x) + e)^2/(e^5*x) - 360*(b^2*d^5*n^2 - b^2 *d^5*n*log(c) - a*b*d^5*n)*(d*sqrt(x) + e)/(e^5*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x)) - 100*(b^2*n^2 - 6*b^2*n*log(c) + 18*b^2*log(c)^2 - 6*a*b*n + 36*a*b*log(c) + 18*a^2)*(d*sqrt(x) + e)^6/(e^5*x^3) + 432*(2*b^2*d*n^2 - 10*b^2*d*n*log(c) + 25*b^2*d*log(c)^2 - 10*a*b*d*n + 50*a*b*d*log(c) + 25* a^2*d)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) - 3375*(b^2*d^2*n^2 - 4*b^2*d^2*n*l og(c) + 8*b^2*d^2*log(c)^2 - 4*a*b*d^2*n + 16*a*b*d^2*log(c) + 8*a^2*d^2)* (d*sqrt(x) + e)^4/(e^5*x^2) + 4000*(2*b^2*d^3*n^2 - 6*b^2*d^3*n*log(c) + 9 *b^2*d^3*log(c)^2 - 6*a*b*d^3*n + 18*a*b*d^3*log(c) + 9*a^2*d^3)*(d*sqrt(x ) + e)^3/(e^5*x^(3/2)) - 13500*(b^2*d^4*n^2 - 2*b^2*d^4*n*log(c) + 2*b^2*d ^4*log(c)^2 - 2*a*b*d^4*n + 4*a*b*d^4*log(c) + 2*a^2*d^4)*(d*sqrt(x) + ...
Time = 3.00 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{x^4} \, dx=\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2}{3\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2}{3\,x^3}-\frac {b^2\,n^2}{54\,x^3}-\frac {2\,a\,b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,x^3}-\frac {a^2}{3\,x^3}+\frac {a\,b\,n}{9\,x^3}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{9\,x^3}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{30\,e^6}-\frac {37\,b^2\,d^2\,n^2}{360\,e^2\,x^2}-\frac {29\,b^2\,d^4\,n^2}{60\,e^4\,x}+\frac {19\,b^2\,d^3\,n^2}{90\,e^3\,x^{3/2}}+\frac {49\,b^2\,d^5\,n^2}{30\,e^5\,\sqrt {x}}+\frac {11\,b^2\,d\,n^2}{225\,e\,x^{5/2}}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{6\,e^2\,x^2}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,e^4\,x}-\frac {2\,b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{9\,e^3\,x^{3/2}}-\frac {2\,b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,e^5\,\sqrt {x}}-\frac {2\,a\,b\,d\,n}{15\,e\,x^{5/2}}+\frac {2\,a\,b\,d^6\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{3\,e^6}-\frac {2\,b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{15\,e\,x^{5/2}}+\frac {a\,b\,d^2\,n}{6\,e^2\,x^2}+\frac {a\,b\,d^4\,n}{3\,e^4\,x}-\frac {2\,a\,b\,d^3\,n}{9\,e^3\,x^{3/2}}-\frac {2\,a\,b\,d^5\,n}{3\,e^5\,\sqrt {x}} \]
(b^2*d^6*log(c*(d + e/x^(1/2))^n)^2)/(3*e^6) - (b^2*log(c*(d + e/x^(1/2))^ n)^2)/(3*x^3) - (b^2*n^2)/(54*x^3) - (2*a*b*log(c*(d + e/x^(1/2))^n))/(3*x ^3) - a^2/(3*x^3) + (a*b*n)/(9*x^3) + (b^2*n*log(c*(d + e/x^(1/2))^n))/(9* x^3) - (49*b^2*d^6*n^2*log(d + e/x^(1/2)))/(30*e^6) - (37*b^2*d^2*n^2)/(36 0*e^2*x^2) - (29*b^2*d^4*n^2)/(60*e^4*x) + (19*b^2*d^3*n^2)/(90*e^3*x^(3/2 )) + (49*b^2*d^5*n^2)/(30*e^5*x^(1/2)) + (11*b^2*d*n^2)/(225*e*x^(5/2)) + (b^2*d^2*n*log(c*(d + e/x^(1/2))^n))/(6*e^2*x^2) + (b^2*d^4*n*log(c*(d + e /x^(1/2))^n))/(3*e^4*x) - (2*b^2*d^3*n*log(c*(d + e/x^(1/2))^n))/(9*e^3*x^ (3/2)) - (2*b^2*d^5*n*log(c*(d + e/x^(1/2))^n))/(3*e^5*x^(1/2)) - (2*a*b*d *n)/(15*e*x^(5/2)) + (2*a*b*d^6*n*log(d + e/x^(1/2)))/(3*e^6) - (2*b^2*d*n *log(c*(d + e/x^(1/2))^n))/(15*e*x^(5/2)) + (a*b*d^2*n)/(6*e^2*x^2) + (a*b *d^4*n)/(3*e^4*x) - (2*a*b*d^3*n)/(9*e^3*x^(3/2)) - (2*a*b*d^5*n)/(3*e^5*x ^(1/2))