Integrand size = 24, antiderivative size = 482 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^6}+\frac {10 b^2 d^3 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^4}{32 e^6}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^6}{72 e^6}+\frac {3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{4 e^6}-\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4} \] Output:
-15/8*b^2*d^4*n^2*(d+e/x^(2/3))^2/e^6+10/9*b^2*d^3*n^2*(d+e/x^(2/3))^3/e^6 -15/32*b^2*d^2*n^2*(d+e/x^(2/3))^4/e^6+3/25*b^2*d*n^2*(d+e/x^(2/3))^5/e^6- 1/72*b^2*n^2*(d+e/x^(2/3))^6/e^6+3*b^2*d^5*n^2/e^5/x^(2/3)-1/4*b^2*d^6*n^2 *ln(d+e/x^(2/3))^2/e^6-3*b*d^5*n*(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n)) /e^6+15/4*b*d^4*n*(d+e/x^(2/3))^2*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-10/3*b*d ^3*n*(d+e/x^(2/3))^3*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6+15/8*b*d^2*n*(d+e/x^( 2/3))^4*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-3/5*b*d*n*(d+e/x^(2/3))^5*(a+b*ln( c*(d+e/x^(2/3))^n))/e^6+1/12*b*n*(d+e/x^(2/3))^6*(a+b*ln(c*(d+e/x^(2/3))^n ))/e^6+1/2*b*d^6*n*ln(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-1/4*(a+ b*ln(c*(d+e/x^(2/3))^n))^2/x^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.91 (sec) , antiderivative size = 988, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^5,x]
Output:
-1/7200*(1800*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + (b*n*(-600*a*e^6 + 100* b*e^6*n + 720*a*d*e^5*x^(2/3) - 264*b*d*e^5*n*x^(2/3) - 900*a*d^2*e^4*x^(4 /3) + 555*b*d^2*e^4*n*x^(4/3) + 1200*a*d^3*e^3*x^2 - 1140*b*d^3*e^3*n*x^2 - 1800*a*d^4*e^2*x^(8/3) + 2610*b*d^4*e^2*n*x^(8/3) + 3600*a*d^5*e*x^(10/3 ) - 8820*b*d^5*e*n*x^(10/3) + 8820*b*d^6*n*x^4*Log[d + e/x^(2/3)] - 600*b* e^6*Log[c*(d + e/x^(2/3))^n] + 720*b*d*e^5*x^(2/3)*Log[c*(d + e/x^(2/3))^n ] - 900*b*d^2*e^4*x^(4/3)*Log[c*(d + e/x^(2/3))^n] + 1200*b*d^3*e^3*x^2*Lo g[c*(d + e/x^(2/3))^n] - 1800*b*d^4*e^2*x^(8/3)*Log[c*(d + e/x^(2/3))^n] + 3600*b*d^5*e*x^(10/3)*Log[c*(d + e/x^(2/3))^n] - 3600*a*d^6*x^4*Log[Sqrt[ e] - Sqrt[-d]*x^(1/3)] - 3600*b*d^6*x^4*Log[c*(d + e/x^(2/3))^n]*Log[Sqrt[ e] - Sqrt[-d]*x^(1/3)] + 1800*b*d^6*n*x^4*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^ 2 - 3600*a*d^6*x^4*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] - 3600*b*d^6*x^4*Log[c* (d + e/x^(2/3))^n]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 1800*b*d^6*n*x^4*Log[ Sqrt[e] + Sqrt[-d]*x^(1/3)]^2 + 3600*b*d^6*n*x^4*Log[Sqrt[e] + Sqrt[-d]*x^ (1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 3600*b*d^6*n*x^4*Log[Sq rt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 3600*b *d^6*x^4*Log[c*(d + e/x^(2/3))^n]*Log[-(e/(d*x^(2/3)))] - 7200*b*d^6*n*x^4 *Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])] - 7200 *b*d^6*n*x^4*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e ]] + 2400*a*d^6*x^4*Log[x] - 3600*b*d^6*n*x^4*PolyLog[2, 1 + e/(d*x^(2/...
Time = 0.97 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.64, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -\frac {3}{2} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^{10/3}}d\frac {1}{x^{2/3}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -\frac {3}{2} \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{6 x^4}-\frac {1}{3} b e n \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{\left (d+\frac {e}{x^{2/3}}\right ) x^4}d\frac {1}{x^{2/3}}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -\frac {3}{2} \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{6 x^4}-\frac {1}{3} b n \int \frac {a+b \log \left (c x^{-2 n/3}\right )}{x^{10/3}}d\left (d+\frac {e}{x^{2/3}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{6 x^4}-\frac {b n \int \frac {e^6 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{x^{10/3}}d\left (d+\frac {e}{x^{2/3}}\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -\frac {3}{2} \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{6 x^4}-\frac {b n \left (-b n \int \left (x^{2/3} \log \left (d+\frac {e}{x^{2/3}}\right ) d^6-6 d^5+\frac {15}{2} \left (d+\frac {e}{x^{2/3}}\right ) d^4-\frac {20 d^3}{3 x^{4/3}}+\frac {15 d^2}{4 x^2}-\frac {6 d}{5 x^{8/3}}+\frac {1}{6 x^{10/3}}\right )d\left (d+\frac {e}{x^{2/3}}\right )+d^6 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-6 d^5 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )+\frac {15 d^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 x^{4/3}}-\frac {20 d^3 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 x^2}+\frac {15 d^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 x^{8/3}}-\frac {6 d \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{5 x^{10/3}}+\frac {a+b \log \left (c x^{-2 n/3}\right )}{6 x^4}\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{6 x^4}-\frac {b n \left (d^6 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-6 d^5 \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )+\frac {15 d^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 x^{4/3}}-\frac {20 d^3 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 x^2}+\frac {15 d^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 x^{8/3}}-\frac {6 d \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{5 x^{10/3}}+\frac {a+b \log \left (c x^{-2 n/3}\right )}{6 x^4}-b n \left (\frac {1}{2} d^6 \log ^2\left (d+\frac {e}{x^{2/3}}\right )-6 d^5 \left (d+\frac {e}{x^{2/3}}\right )+\frac {15 d^4}{4 x^{4/3}}-\frac {20 d^3}{9 x^2}+\frac {15 d^2}{16 x^{8/3}}-\frac {6 d}{25 x^{10/3}}+\frac {1}{36 x^4}\right )\right )}{3 e^6}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^5,x]
Output:
(-3*((a + b*Log[c*(d + e/x^(2/3))^n])^2/(6*x^4) - (b*n*(-(b*n*(-6*d^5*(d + e/x^(2/3)) + 1/(36*x^4) - (6*d)/(25*x^(10/3)) + (15*d^2)/(16*x^(8/3)) - ( 20*d^3)/(9*x^2) + (15*d^4)/(4*x^(4/3)) + (d^6*Log[d + e/x^(2/3)]^2)/2)) - 6*d^5*(d + e/x^(2/3))*(a + b*Log[c/x^((2*n)/3)]) + (a + b*Log[c/x^((2*n)/3 )])/(6*x^4) - (6*d*(a + b*Log[c/x^((2*n)/3)]))/(5*x^(10/3)) + (15*d^2*(a + b*Log[c/x^((2*n)/3)]))/(4*x^(8/3)) - (20*d^3*(a + b*Log[c/x^((2*n)/3)]))/ (3*x^2) + (15*d^4*(a + b*Log[c/x^((2*n)/3)]))/(2*x^(4/3)) + d^6*Log[d + e/ x^(2/3)]*(a + b*Log[c/x^((2*n)/3)])))/(3*e^6)))/2
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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{2}}{x^{5}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^5,x)
Output:
int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^5,x)
Time = 0.10 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {100 \, b^{2} e^{6} n^{2} + 1800 \, b^{2} e^{6} \log \left (c\right )^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{4} - b^{2} e^{6} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{2} + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x^{2} - b^{2} e^{6} n + 6 \, a b e^{6}\right )} \log \left (c\right ) + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x^{2} - 10 \, b^{2} e^{6} n^{2} + 60 \, a b e^{6} n + 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{4} - 60 \, {\left (b^{2} d^{6} n x^{4} - b^{2} e^{6} n\right )} \log \left (c\right ) - 6 \, {\left (5 \, b^{2} d^{4} e^{2} n^{2} x^{2} - 2 \, b^{2} d e^{5} n^{2}\right )} x^{\frac {2}{3}} + 15 \, {\left (4 \, b^{2} d^{5} e n^{2} x^{3} - b^{2} d^{2} e^{4} n^{2} x\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 6 \, {\left (44 \, b^{2} d e^{5} n^{2} - 120 \, a b d e^{5} n - 15 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x^{2} + 60 \, {\left (5 \, b^{2} d^{4} e^{2} n x^{2} - 2 \, b^{2} d e^{5} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 15 \, {\left (12 \, {\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x^{3} - {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x - 60 \, {\left (4 \, b^{2} d^{5} e n x^{3} - b^{2} d^{2} e^{4} n x\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{7200 \, e^{6} x^{4}} \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^5,x, algorithm="fricas")
Output:
-1/7200*(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 - 60*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x^2 - 1800*(b^2*d^6*n^2 *x^4 - b^2*e^6*n^2)*log((d*x + e*x^(1/3))/x)^2 + 600*(2*b^2*d^3*e^3*n*x^2 - b^2*e^6*n + 6*a*b*e^6)*log(c) + 60*(20*b^2*d^3*e^3*n^2*x^2 - 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^4 - 60*(b^2*d^6*n *x^4 - b^2*e^6*n)*log(c) - 6*(5*b^2*d^4*e^2*n^2*x^2 - 2*b^2*d*e^5*n^2)*x^( 2/3) + 15*(4*b^2*d^5*e*n^2*x^3 - b^2*d^2*e^4*n^2*x)*x^(1/3))*log((d*x + e* x^(1/3))/x) - 6*(44*b^2*d*e^5*n^2 - 120*a*b*d*e^5*n - 15*(29*b^2*d^4*e^2*n ^2 - 20*a*b*d^4*e^2*n)*x^2 + 60*(5*b^2*d^4*e^2*n*x^2 - 2*b^2*d*e^5*n)*log( c))*x^(2/3) - 15*(12*(49*b^2*d^5*e*n^2 - 20*a*b*d^5*e*n)*x^3 - (37*b^2*d^2 *e^4*n^2 - 60*a*b*d^2*e^4*n)*x - 60*(4*b^2*d^5*e*n*x^3 - b^2*d^2*e^4*n*x)* log(c))*x^(1/3))/(e^6*x^4)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(2/3))**n))**2/x**5,x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {1}{120} \, a b e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{7}} - \frac {60 \, d^{6} \log \left (x^{\frac {2}{3}}\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {10}{3}} - 30 \, d^{4} e x^{\frac {8}{3}} + 20 \, d^{3} e^{2} x^{2} - 15 \, d^{2} e^{3} x^{\frac {4}{3}} + 12 \, d e^{4} x^{\frac {2}{3}} - 10 \, e^{5}}{e^{6} x^{4}}\right )} + \frac {1}{7200} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{7}} - \frac {60 \, d^{6} \log \left (x^{\frac {2}{3}}\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {10}{3}} - 30 \, d^{4} e x^{\frac {8}{3}} + 20 \, d^{3} e^{2} x^{2} - 15 \, d^{2} e^{3} x^{\frac {4}{3}} + 12 \, d e^{4} x^{\frac {2}{3}} - 10 \, e^{5}}{e^{6} x^{4}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{4} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 800 \, d^{6} x^{4} \log \left (x\right )^{2} - 5880 \, d^{6} x^{4} \log \left (x\right ) - 8820 \, d^{5} e x^{\frac {10}{3}} + 2610 \, d^{4} e^{2} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 264 \, d e^{5} x^{\frac {2}{3}} + 100 \, e^{6} - 60 \, {\left (40 \, d^{6} x^{4} \log \left (x\right ) - 147 \, d^{6} x^{4}\right )} \log \left (d x^{\frac {2}{3}} + e\right )\right )} n^{2}}{e^{6} x^{4}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{4 \, x^{4}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{2 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^5,x, algorithm="maxima")
Output:
1/120*a*b*e*n*(60*d^6*log(d*x^(2/3) + e)/e^7 - 60*d^6*log(x^(2/3))/e^7 - ( 60*d^5*x^(10/3) - 30*d^4*e*x^(8/3) + 20*d^3*e^2*x^2 - 15*d^2*e^3*x^(4/3) + 12*d*e^4*x^(2/3) - 10*e^5)/(e^6*x^4)) + 1/7200*(60*e*n*(60*d^6*log(d*x^(2 /3) + e)/e^7 - 60*d^6*log(x^(2/3))/e^7 - (60*d^5*x^(10/3) - 30*d^4*e*x^(8/ 3) + 20*d^3*e^2*x^2 - 15*d^2*e^3*x^(4/3) + 12*d*e^4*x^(2/3) - 10*e^5)/(e^6 *x^4))*log(c*(d + e/x^(2/3))^n) - (1800*d^6*x^4*log(d*x^(2/3) + e)^2 + 800 *d^6*x^4*log(x)^2 - 5880*d^6*x^4*log(x) - 8820*d^5*e*x^(10/3) + 2610*d^4*e ^2*x^(8/3) - 1140*d^3*e^3*x^2 + 555*d^2*e^4*x^(4/3) - 264*d*e^5*x^(2/3) + 100*e^6 - 60*(40*d^6*x^4*log(x) - 147*d^6*x^4)*log(d*x^(2/3) + e))*n^2/(e^ 6*x^4))*b^2 - 1/4*b^2*log(c*(d + e/x^(2/3))^n)^2/x^4 - 1/2*a*b*log(c*(d + e/x^(2/3))^n)/x^4 - 1/4*a^2/x^4
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{5}} \,d x } \] Input:
integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^5,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(2/3))^n) + a)^2/x^5, x)
Time = 27.02 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2}{4\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2}{4\,x^4}-\frac {b^2\,n^2}{72\,x^4}-\frac {a\,b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{2\,x^4}-\frac {a^2}{4\,x^4}+\frac {a\,b\,n}{12\,x^4}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{12\,x^4}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{x^{2/3}}\right )}{40\,e^6}+\frac {19\,b^2\,d^3\,n^2}{120\,e^3\,x^2}-\frac {37\,b^2\,d^2\,n^2}{480\,e^2\,x^{8/3}}-\frac {29\,b^2\,d^4\,n^2}{80\,e^4\,x^{4/3}}+\frac {49\,b^2\,d^5\,n^2}{40\,e^5\,x^{2/3}}+\frac {11\,b^2\,d\,n^2}{300\,e\,x^{10/3}}-\frac {b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{6\,e^3\,x^2}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{8\,e^2\,x^{8/3}}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{4\,e^4\,x^{4/3}}-\frac {b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{2\,e^5\,x^{2/3}}-\frac {a\,b\,d\,n}{10\,e\,x^{10/3}}+\frac {a\,b\,d^6\,n\,\ln \left (d+\frac {e}{x^{2/3}}\right )}{2\,e^6}-\frac {b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{10\,e\,x^{10/3}}-\frac {a\,b\,d^3\,n}{6\,e^3\,x^2}+\frac {a\,b\,d^2\,n}{8\,e^2\,x^{8/3}}+\frac {a\,b\,d^4\,n}{4\,e^4\,x^{4/3}}-\frac {a\,b\,d^5\,n}{2\,e^5\,x^{2/3}} \] Input:
int((a + b*log(c*(d + e/x^(2/3))^n))^2/x^5,x)
Output:
(b^2*d^6*log(c*(d + e/x^(2/3))^n)^2)/(4*e^6) - (b^2*log(c*(d + e/x^(2/3))^ n)^2)/(4*x^4) - (b^2*n^2)/(72*x^4) - (a*b*log(c*(d + e/x^(2/3))^n))/(2*x^4 ) - a^2/(4*x^4) + (a*b*n)/(12*x^4) + (b^2*n*log(c*(d + e/x^(2/3))^n))/(12* x^4) - (49*b^2*d^6*n^2*log(d + e/x^(2/3)))/(40*e^6) + (19*b^2*d^3*n^2)/(12 0*e^3*x^2) - (37*b^2*d^2*n^2)/(480*e^2*x^(8/3)) - (29*b^2*d^4*n^2)/(80*e^4 *x^(4/3)) + (49*b^2*d^5*n^2)/(40*e^5*x^(2/3)) + (11*b^2*d*n^2)/(300*e*x^(1 0/3)) - (b^2*d^3*n*log(c*(d + e/x^(2/3))^n))/(6*e^3*x^2) + (b^2*d^2*n*log( c*(d + e/x^(2/3))^n))/(8*e^2*x^(8/3)) + (b^2*d^4*n*log(c*(d + e/x^(2/3))^n ))/(4*e^4*x^(4/3)) - (b^2*d^5*n*log(c*(d + e/x^(2/3))^n))/(2*e^5*x^(2/3)) - (a*b*d*n)/(10*e*x^(10/3)) + (a*b*d^6*n*log(d + e/x^(2/3)))/(2*e^6) - (b^ 2*d*n*log(c*(d + e/x^(2/3))^n))/(10*e*x^(10/3)) - (a*b*d^3*n)/(6*e^3*x^2) + (a*b*d^2*n)/(8*e^2*x^(8/3)) + (a*b*d^4*n)/(4*e^4*x^(4/3)) - (a*b*d^5*n)/ (2*e^5*x^(2/3))
Time = 0.16 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {1800 x^{\frac {8}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{4} e^{2} n -720 x^{\frac {2}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d \,e^{5} n +1800 x^{\frac {8}{3}} a b \,d^{4} e^{2} n -720 x^{\frac {2}{3}} a b d \,e^{5} n -2610 x^{\frac {8}{3}} b^{2} d^{4} e^{2} n^{2}+264 x^{\frac {2}{3}} b^{2} d \,e^{5} n^{2}-3600 x^{\frac {10}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{5} e n +900 x^{\frac {4}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{2} e^{4} n -3600 x^{\frac {10}{3}} a b \,d^{5} e n +900 x^{\frac {4}{3}} a b \,d^{2} e^{4} n +8820 x^{\frac {10}{3}} b^{2} d^{5} e \,n^{2}-555 x^{\frac {4}{3}} b^{2} d^{2} e^{4} n^{2}+1800 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} d^{6} x^{4}-1800 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{2} e^{6}+3600 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,d^{6} x^{4}-3600 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a b \,e^{6}-8820 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{6} n \,x^{4}-1200 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} d^{3} e^{3} n \,x^{2}+600 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) b^{2} e^{6} n -1800 a^{2} e^{6}-1200 a b \,d^{3} e^{3} n \,x^{2}+600 a b \,e^{6} n +1140 b^{2} d^{3} e^{3} n^{2} x^{2}-100 b^{2} e^{6} n^{2}}{7200 e^{6} x^{4}} \] Input:
int((a+b*log(c*(d+e/x^(2/3))^n))^2/x^5,x)
Output:
(1800*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**4*e**2*n* x**2 - 720*x**(2/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d*e**5* n + 1800*x**(2/3)*a*b*d**4*e**2*n*x**2 - 720*x**(2/3)*a*b*d*e**5*n - 2610* x**(2/3)*b**2*d**4*e**2*n**2*x**2 + 264*x**(2/3)*b**2*d*e**5*n**2 - 3600*x **(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**5*e*n*x**3 + 900 *x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**2*e**4*n*x - 3 600*x**(1/3)*a*b*d**5*e*n*x**3 + 900*x**(1/3)*a*b*d**2*e**4*n*x + 8820*x** (1/3)*b**2*d**5*e*n**2*x**3 - 555*x**(1/3)*b**2*d**2*e**4*n**2*x + 1800*lo g(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*d**6*x**4 - 1800*log(((x** (2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**2*e**6 + 3600*log(((x**(2/3)*d + e) **n*c)/x**((2*n)/3))*a*b*d**6*x**4 - 3600*log(((x**(2/3)*d + e)**n*c)/x**( (2*n)/3))*a*b*e**6 - 8820*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d **6*n*x**4 - 1200*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*d**3*e**3 *n*x**2 + 600*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*b**2*e**6*n - 1800 *a**2*e**6 - 1200*a*b*d**3*e**3*n*x**2 + 600*a*b*e**6*n + 1140*b**2*d**3*e **3*n**2*x**2 - 100*b**2*e**6*n**2)/(7200*e**6*x**4)