Integrand size = 24, antiderivative size = 680 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=-\frac {6 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac {56 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}-\frac {21 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}+\frac {84 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}-\frac {14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}+\frac {24 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9}{243 e^9}+\frac {6 b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac {b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{3 e^9}-\frac {6 b d^8 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^9}+\frac {12 b d^7 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^9}-\frac {56 b d^6 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}+\frac {21 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^9}-\frac {84 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 e^9}+\frac {28 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}-\frac {24 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{7 e^9}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^9}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{27 e^9}+\frac {2 b d^9 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \]
-6*b^2*d^7*n^2*(d+e*x^(1/3))^2/e^9+56/9*b^2*d^6*n^2*(d+e*x^(1/3))^3/e^9-21 /4*b^2*d^5*n^2*(d+e*x^(1/3))^4/e^9+84/25*b^2*d^4*n^2*(d+e*x^(1/3))^5/e^9-1 4/9*b^2*d^3*n^2*(d+e*x^(1/3))^6/e^9+24/49*b^2*d^2*n^2*(d+e*x^(1/3))^7/e^9- 3/32*b^2*d*n^2*(d+e*x^(1/3))^8/e^9+2/243*b^2*n^2*(d+e*x^(1/3))^9/e^9+6*b^2 *d^8*n^2*x^(1/3)/e^8-1/3*b^2*d^9*n^2*ln(d+e*x^(1/3))^2/e^9-6*b*d^8*n*(d+e* x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+12*b*d^7*n*(d+e*x^(1/3))^2*(a+b*l n(c*(d+e*x^(1/3))^n))/e^9-56/3*b*d^6*n*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1 /3))^n))/e^9+21*b*d^5*n*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9-84 /5*b*d^4*n*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+28/3*b*d^3*n*(d +e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9-24/7*b*d^2*n*(d+e*x^(1/3))^7 *(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+3/4*b*d*n*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e* x^(1/3))^n))/e^9-2/27*b*n*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+ 2/3*b*d^9*n*ln(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+1/3*x^3*(a+b*l n(c*(d+e*x^(1/3))^n))^2
Time = 0.39 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.63 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {e \sqrt [3]{x} \left (3175200 a^2 e^8 x^{8/3}-2520 a b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+840 d^6 e^2 x^{2/3}-630 d^5 e^3 x+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )+b^2 n^2 \left (17965080 d^8-5807340 d^7 e \sqrt [3]{x}+2813160 d^6 e^2 x^{2/3}-1580670 d^5 e^3 x+947016 d^4 e^4 x^{4/3}-577500 d^3 e^5 x^{5/3}+343800 d^2 e^6 x^2-187425 d e^7 x^{7/3}+78400 e^8 x^{8/3}\right )\right )+2520 b d^9 n (2520 a-7129 b n) \log \left (d+e \sqrt [3]{x}\right )-2520 b e \sqrt [3]{x} \left (-2520 a e^8 x^{8/3}+b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+840 d^6 e^2 x^{2/3}-630 d^5 e^3 x+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 \left (d^9+e^9 x^3\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{9525600 e^9} \]
(e*x^(1/3)*(3175200*a^2*e^8*x^(8/3) - 2520*a*b*n*(2520*d^8 - 1260*d^7*e*x^ (1/3) + 840*d^6*e^2*x^(2/3) - 630*d^5*e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^ 3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315*d*e^7*x^(7/3) + 280*e^8*x^(8/3)) + b ^2*n^2*(17965080*d^8 - 5807340*d^7*e*x^(1/3) + 2813160*d^6*e^2*x^(2/3) - 1 580670*d^5*e^3*x + 947016*d^4*e^4*x^(4/3) - 577500*d^3*e^5*x^(5/3) + 34380 0*d^2*e^6*x^2 - 187425*d*e^7*x^(7/3) + 78400*e^8*x^(8/3))) + 2520*b*d^9*n* (2520*a - 7129*b*n)*Log[d + e*x^(1/3)] - 2520*b*e*x^(1/3)*(-2520*a*e^8*x^( 8/3) + b*n*(2520*d^8 - 1260*d^7*e*x^(1/3) + 840*d^6*e^2*x^(2/3) - 630*d^5* e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315* d*e^7*x^(7/3) + 280*e^8*x^(8/3)))*Log[c*(d + e*x^(1/3))^n] + 3175200*b^2*( d^9 + e^9*x^3)*Log[c*(d + e*x^(1/3))^n]^2)/(9525600*e^9)
Time = 0.60 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.59, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2904, 2845, 2858, 25, 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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {2}{9} b e n \int \frac {x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d+e \sqrt [3]{x}}d\sqrt [3]{x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 3 \left (\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {2}{9} b n \int x^{8/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {2}{9} b n \int -x^{8/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {2 b n \int -e^9 x^{8/3} \left (a+b \log \left (c x^{n/3}\right )\right )d\left (d+e \sqrt [3]{x}\right )}{9 e^9}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle 3 \left (\frac {2 b n \left (-b n \int \left (\frac {\log \left (d+e \sqrt [3]{x}\right ) d^9}{\sqrt [3]{x}}-9 d^8+18 \left (d+e \sqrt [3]{x}\right ) d^7-28 x^{2/3} d^6+\frac {63 x d^5}{2}-\frac {126}{5} x^{4/3} d^4+14 x^{5/3} d^3-\frac {36 x^2 d^2}{7}+\frac {9}{8} x^{7/3} d-\frac {x^{8/3}}{9}\right )d\left (d+e \sqrt [3]{x}\right )+d^9 \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )-9 d^8 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )+18 d^7 x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )-28 d^6 x \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {63}{2} d^5 x^{4/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {126}{5} d^4 x^{5/3} \left (a+b \log \left (c x^{n/3}\right )\right )+14 d^3 x^2 \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {36}{7} d^2 x^{7/3} \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {9}{8} d x^{8/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {1}{9} x^3 \left (a+b \log \left (c x^{n/3}\right )\right )\right )}{9 e^9}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {2 b n \left (d^9 \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )-9 d^8 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )+18 d^7 x^{2/3} \left (a+b \log \left (c x^{n/3}\right )\right )-28 d^6 x \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {63}{2} d^5 x^{4/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {126}{5} d^4 x^{5/3} \left (a+b \log \left (c x^{n/3}\right )\right )+14 d^3 x^2 \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {36}{7} d^2 x^{7/3} \left (a+b \log \left (c x^{n/3}\right )\right )+\frac {9}{8} d x^{8/3} \left (a+b \log \left (c x^{n/3}\right )\right )-\frac {1}{9} x^3 \left (a+b \log \left (c x^{n/3}\right )\right )-b n \left (\frac {1}{2} d^9 \log ^2\left (d+e \sqrt [3]{x}\right )-9 d^8 \left (d+e \sqrt [3]{x}\right )+9 d^7 x^{2/3}-\frac {28 d^6 x}{3}+\frac {63}{8} d^5 x^{4/3}-\frac {126}{25} d^4 x^{5/3}+\frac {7 d^3 x^2}{3}-\frac {36}{49} d^2 x^{7/3}+\frac {9}{64} d x^{8/3}-\frac {x^3}{81}\right )\right )}{9 e^9}+\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\right )\) |
3*((x^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/9 + (2*b*n*(-(b*n*(-9*d^8*(d + e*x^(1/3)) + 9*d^7*x^(2/3) - (28*d^6*x)/3 + (63*d^5*x^(4/3))/8 - (126*d^4 *x^(5/3))/25 + (7*d^3*x^2)/3 - (36*d^2*x^(7/3))/49 + (9*d*x^(8/3))/64 - x^ 3/81 + (d^9*Log[d + e*x^(1/3)]^2)/2)) - 9*d^8*(d + e*x^(1/3))*(a + b*Log[c *x^(n/3)]) + 18*d^7*x^(2/3)*(a + b*Log[c*x^(n/3)]) - 28*d^6*x*(a + b*Log[c *x^(n/3)]) + (63*d^5*x^(4/3)*(a + b*Log[c*x^(n/3)]))/2 - (126*d^4*x^(5/3)* (a + b*Log[c*x^(n/3)]))/5 + 14*d^3*x^2*(a + b*Log[c*x^(n/3)]) - (36*d^2*x^ (7/3)*(a + b*Log[c*x^(n/3)]))/7 + (9*d*x^(8/3)*(a + b*Log[c*x^(n/3)]))/8 - (x^3*(a + b*Log[c*x^(n/3)]))/9 + d^9*Log[d + e*x^(1/3)]*(a + b*Log[c*x^(n /3)])))/(9*e^9))
3.5.50.3.1 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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}d x\]
Time = 0.38 (sec) , antiderivative size = 674, normalized size of antiderivative = 0.99 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {3175200 \, b^{2} e^{9} x^{3} \log \left (c\right )^{2} + 39200 \, {\left (2 \, b^{2} e^{9} n^{2} - 18 \, a b e^{9} n + 81 \, a^{2} e^{9}\right )} x^{3} - 2100 \, {\left (275 \, b^{2} d^{3} e^{6} n^{2} - 504 \, a b d^{3} e^{6} n\right )} x^{2} + 3175200 \, {\left (b^{2} e^{9} n^{2} x^{3} + b^{2} d^{9} n^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} + 840 \, {\left (3349 \, b^{2} d^{6} e^{3} n^{2} - 2520 \, a b d^{6} e^{3} n\right )} x + 2520 \, {\left (420 \, b^{2} d^{3} e^{6} n^{2} x^{2} - 840 \, b^{2} d^{6} e^{3} n^{2} x - 7129 \, b^{2} d^{9} n^{2} + 2520 \, a b d^{9} n - 280 \, {\left (b^{2} e^{9} n^{2} - 9 \, a b e^{9} n\right )} x^{3} + 2520 \, {\left (b^{2} e^{9} n x^{3} + b^{2} d^{9} n\right )} \log \left (c\right ) + 63 \, {\left (5 \, b^{2} d e^{8} n^{2} x^{2} - 8 \, b^{2} d^{4} e^{5} n^{2} x + 20 \, b^{2} d^{7} e^{2} n^{2}\right )} x^{\frac {2}{3}} - 90 \, {\left (4 \, b^{2} d^{2} e^{7} n^{2} x^{2} - 7 \, b^{2} d^{5} e^{4} n^{2} x + 28 \, b^{2} d^{8} e n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 352800 \, {\left (3 \, b^{2} d^{3} e^{6} n x^{2} - 6 \, b^{2} d^{6} e^{3} n x - 2 \, {\left (b^{2} e^{9} n - 9 \, a b e^{9}\right )} x^{3}\right )} \log \left (c\right ) - 63 \, {\left (92180 \, b^{2} d^{7} e^{2} n^{2} - 50400 \, a b d^{7} e^{2} n + 175 \, {\left (17 \, b^{2} d e^{8} n^{2} - 72 \, a b d e^{8} n\right )} x^{2} - 8 \, {\left (1879 \, b^{2} d^{4} e^{5} n^{2} - 2520 \, a b d^{4} e^{5} n\right )} x - 2520 \, {\left (5 \, b^{2} d e^{8} n x^{2} - 8 \, b^{2} d^{4} e^{5} n x + 20 \, b^{2} d^{7} e^{2} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 90 \, {\left (199612 \, b^{2} d^{8} e n^{2} - 70560 \, a b d^{8} e n + 20 \, {\left (191 \, b^{2} d^{2} e^{7} n^{2} - 504 \, a b d^{2} e^{7} n\right )} x^{2} - 7 \, {\left (2509 \, b^{2} d^{5} e^{4} n^{2} - 2520 \, a b d^{5} e^{4} n\right )} x - 2520 \, {\left (4 \, b^{2} d^{2} e^{7} n x^{2} - 7 \, b^{2} d^{5} e^{4} n x + 28 \, b^{2} d^{8} e n\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{9525600 \, e^{9}} \]
1/9525600*(3175200*b^2*e^9*x^3*log(c)^2 + 39200*(2*b^2*e^9*n^2 - 18*a*b*e^ 9*n + 81*a^2*e^9)*x^3 - 2100*(275*b^2*d^3*e^6*n^2 - 504*a*b*d^3*e^6*n)*x^2 + 3175200*(b^2*e^9*n^2*x^3 + b^2*d^9*n^2)*log(e*x^(1/3) + d)^2 + 840*(334 9*b^2*d^6*e^3*n^2 - 2520*a*b*d^6*e^3*n)*x + 2520*(420*b^2*d^3*e^6*n^2*x^2 - 840*b^2*d^6*e^3*n^2*x - 7129*b^2*d^9*n^2 + 2520*a*b*d^9*n - 280*(b^2*e^9 *n^2 - 9*a*b*e^9*n)*x^3 + 2520*(b^2*e^9*n*x^3 + b^2*d^9*n)*log(c) + 63*(5* b^2*d*e^8*n^2*x^2 - 8*b^2*d^4*e^5*n^2*x + 20*b^2*d^7*e^2*n^2)*x^(2/3) - 90 *(4*b^2*d^2*e^7*n^2*x^2 - 7*b^2*d^5*e^4*n^2*x + 28*b^2*d^8*e*n^2)*x^(1/3)) *log(e*x^(1/3) + d) + 352800*(3*b^2*d^3*e^6*n*x^2 - 6*b^2*d^6*e^3*n*x - 2* (b^2*e^9*n - 9*a*b*e^9)*x^3)*log(c) - 63*(92180*b^2*d^7*e^2*n^2 - 50400*a* b*d^7*e^2*n + 175*(17*b^2*d*e^8*n^2 - 72*a*b*d*e^8*n)*x^2 - 8*(1879*b^2*d^ 4*e^5*n^2 - 2520*a*b*d^4*e^5*n)*x - 2520*(5*b^2*d*e^8*n*x^2 - 8*b^2*d^4*e^ 5*n*x + 20*b^2*d^7*e^2*n)*log(c))*x^(2/3) + 90*(199612*b^2*d^8*e*n^2 - 705 60*a*b*d^8*e*n + 20*(191*b^2*d^2*e^7*n^2 - 504*a*b*d^2*e^7*n)*x^2 - 7*(250 9*b^2*d^5*e^4*n^2 - 2520*a*b*d^5*e^4*n)*x - 2520*(4*b^2*d^2*e^7*n*x^2 - 7* b^2*d^5*e^4*n*x + 28*b^2*d^8*e*n)*log(c))*x^(1/3))/e^9
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \]
Time = 0.21 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.62 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + \frac {2}{3} \, a b x^{3} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3780} \, a b e n {\left (\frac {2520 \, d^{9} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{10}} - \frac {280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac {8}{3}} + 360 \, d^{2} e^{6} x^{\frac {7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac {5}{3}} - 630 \, d^{5} e^{3} x^{\frac {4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac {2}{3}} + 2520 \, d^{8} x^{\frac {1}{3}}}{e^{9}}\right )} + \frac {1}{9525600} \, {\left (2520 \, e n {\left (\frac {2520 \, d^{9} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{10}} - \frac {280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac {8}{3}} + 360 \, d^{2} e^{6} x^{\frac {7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac {5}{3}} - 630 \, d^{5} e^{3} x^{\frac {4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac {2}{3}} + 2520 \, d^{8} x^{\frac {1}{3}}}{e^{9}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {{\left (78400 \, e^{9} x^{3} - 187425 \, d e^{8} x^{\frac {8}{3}} + 343800 \, d^{2} e^{7} x^{\frac {7}{3}} - 577500 \, d^{3} e^{6} x^{2} - 3175200 \, d^{9} \log \left (e x^{\frac {1}{3}} + d\right )^{2} + 947016 \, d^{4} e^{5} x^{\frac {5}{3}} - 1580670 \, d^{5} e^{4} x^{\frac {4}{3}} + 2813160 \, d^{6} e^{3} x - 17965080 \, d^{9} \log \left (e x^{\frac {1}{3}} + d\right ) - 5807340 \, d^{7} e^{2} x^{\frac {2}{3}} + 17965080 \, d^{8} e x^{\frac {1}{3}}\right )} n^{2}}{e^{9}}\right )} b^{2} \]
1/3*b^2*x^3*log((e*x^(1/3) + d)^n*c)^2 + 2/3*a*b*x^3*log((e*x^(1/3) + d)^n *c) + 1/3*a^2*x^3 + 1/3780*a*b*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (28 0*e^8*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 50 4*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1260*d^7*e*x^(2/ 3) + 2520*d^8*x^(1/3))/e^9) + 1/9525600*(2520*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d ^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1 260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9)*log((e*x^(1/3) + d)^n*c) + (784 00*e^9*x^3 - 187425*d*e^8*x^(8/3) + 343800*d^2*e^7*x^(7/3) - 577500*d^3*e^ 6*x^2 - 3175200*d^9*log(e*x^(1/3) + d)^2 + 947016*d^4*e^5*x^(5/3) - 158067 0*d^5*e^4*x^(4/3) + 2813160*d^6*e^3*x - 17965080*d^9*log(e*x^(1/3) + d) - 5807340*d^7*e^2*x^(2/3) + 17965080*d^8*e*x^(1/3))*n^2/e^9)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (586) = 1172\).
Time = 0.32 (sec) , antiderivative size = 1389, normalized size of antiderivative = 2.04 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]
1/9525600*(3175200*b^2*e*x^3*log(c)^2 + 6350400*a*b*e*x^3*log(c) + 3175200 *a^2*e*x^3 + (3175200*(e*x^(1/3) + d)^9*log(e*x^(1/3) + d)^2/e^8 - 2857680 0*(e*x^(1/3) + d)^8*d*log(e*x^(1/3) + d)^2/e^8 + 114307200*(e*x^(1/3) + d) ^7*d^2*log(e*x^(1/3) + d)^2/e^8 - 266716800*(e*x^(1/3) + d)^6*d^3*log(e*x^ (1/3) + d)^2/e^8 + 400075200*(e*x^(1/3) + d)^5*d^4*log(e*x^(1/3) + d)^2/e^ 8 - 400075200*(e*x^(1/3) + d)^4*d^5*log(e*x^(1/3) + d)^2/e^8 + 266716800*( e*x^(1/3) + d)^3*d^6*log(e*x^(1/3) + d)^2/e^8 - 114307200*(e*x^(1/3) + d)^ 2*d^7*log(e*x^(1/3) + d)^2/e^8 + 28576800*(e*x^(1/3) + d)*d^8*log(e*x^(1/3 ) + d)^2/e^8 - 705600*(e*x^(1/3) + d)^9*log(e*x^(1/3) + d)/e^8 + 7144200*( e*x^(1/3) + d)^8*d*log(e*x^(1/3) + d)/e^8 - 32659200*(e*x^(1/3) + d)^7*d^2 *log(e*x^(1/3) + d)/e^8 + 88905600*(e*x^(1/3) + d)^6*d^3*log(e*x^(1/3) + d )/e^8 - 160030080*(e*x^(1/3) + d)^5*d^4*log(e*x^(1/3) + d)/e^8 + 200037600 *(e*x^(1/3) + d)^4*d^5*log(e*x^(1/3) + d)/e^8 - 177811200*(e*x^(1/3) + d)^ 3*d^6*log(e*x^(1/3) + d)/e^8 + 114307200*(e*x^(1/3) + d)^2*d^7*log(e*x^(1/ 3) + d)/e^8 - 57153600*(e*x^(1/3) + d)*d^8*log(e*x^(1/3) + d)/e^8 + 78400* (e*x^(1/3) + d)^9/e^8 - 893025*(e*x^(1/3) + d)^8*d/e^8 + 4665600*(e*x^(1/3 ) + d)^7*d^2/e^8 - 14817600*(e*x^(1/3) + d)^6*d^3/e^8 + 32006016*(e*x^(1/3 ) + d)^5*d^4/e^8 - 50009400*(e*x^(1/3) + d)^4*d^5/e^8 + 59270400*(e*x^(1/3 ) + d)^3*d^6/e^8 - 57153600*(e*x^(1/3) + d)^2*d^7/e^8 + 57153600*(e*x^(1/3 ) + d)*d^8/e^8)*b^2*n^2 + 2520*(2520*(e*x^(1/3) + d)^9*log(e*x^(1/3) + ...
Time = 6.03 (sec) , antiderivative size = 608, normalized size of antiderivative = 0.89 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{3}+\frac {2\,b^2\,n^2\,x^3}{243}+\frac {2\,a\,b\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b^2\,d^9\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{3\,e^9}-\frac {2\,a\,b\,n\,x^3}{27}-\frac {2\,b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{27}-\frac {7129\,b^2\,d^9\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{3780\,e^9}-\frac {275\,b^2\,d^3\,n^2\,x^2}{4536\,e^3}+\frac {191\,b^2\,d^2\,n^2\,x^{7/3}}{5292\,e^2}+\frac {1879\,b^2\,d^4\,n^2\,x^{5/3}}{18900\,e^4}-\frac {2509\,b^2\,d^5\,n^2\,x^{4/3}}{15120\,e^5}-\frac {4609\,b^2\,d^7\,n^2\,x^{2/3}}{7560\,e^7}+\frac {7129\,b^2\,d^8\,n^2\,x^{1/3}}{3780\,e^8}-\frac {17\,b^2\,d\,n^2\,x^{8/3}}{864\,e}+\frac {3349\,b^2\,d^6\,n^2\,x}{11340\,e^6}+\frac {b^2\,d^3\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{9\,e^3}-\frac {2\,b^2\,d^2\,n\,x^{7/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{21\,e^2}-\frac {2\,b^2\,d^4\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{15\,e^4}+\frac {b^2\,d^5\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6\,e^5}+\frac {b^2\,d^7\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^7}-\frac {2\,b^2\,d^8\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^8}+\frac {a\,b\,d\,n\,x^{8/3}}{12\,e}-\frac {2\,a\,b\,d^6\,n\,x}{9\,e^6}+\frac {2\,a\,b\,d^9\,n\,\ln \left (d+e\,x^{1/3}\right )}{3\,e^9}+\frac {b^2\,d\,n\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{12\,e}-\frac {2\,b^2\,d^6\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{9\,e^6}+\frac {a\,b\,d^3\,n\,x^2}{9\,e^3}-\frac {2\,a\,b\,d^2\,n\,x^{7/3}}{21\,e^2}-\frac {2\,a\,b\,d^4\,n\,x^{5/3}}{15\,e^4}+\frac {a\,b\,d^5\,n\,x^{4/3}}{6\,e^5}+\frac {a\,b\,d^7\,n\,x^{2/3}}{3\,e^7}-\frac {2\,a\,b\,d^8\,n\,x^{1/3}}{3\,e^8} \]
(a^2*x^3)/3 + (b^2*x^3*log(c*(d + e*x^(1/3))^n)^2)/3 + (2*b^2*n^2*x^3)/243 + (2*a*b*x^3*log(c*(d + e*x^(1/3))^n))/3 + (b^2*d^9*log(c*(d + e*x^(1/3)) ^n)^2)/(3*e^9) - (2*a*b*n*x^3)/27 - (2*b^2*n*x^3*log(c*(d + e*x^(1/3))^n)) /27 - (7129*b^2*d^9*n^2*log(d + e*x^(1/3)))/(3780*e^9) - (275*b^2*d^3*n^2* x^2)/(4536*e^3) + (191*b^2*d^2*n^2*x^(7/3))/(5292*e^2) + (1879*b^2*d^4*n^2 *x^(5/3))/(18900*e^4) - (2509*b^2*d^5*n^2*x^(4/3))/(15120*e^5) - (4609*b^2 *d^7*n^2*x^(2/3))/(7560*e^7) + (7129*b^2*d^8*n^2*x^(1/3))/(3780*e^8) - (17 *b^2*d*n^2*x^(8/3))/(864*e) + (3349*b^2*d^6*n^2*x)/(11340*e^6) + (b^2*d^3* n*x^2*log(c*(d + e*x^(1/3))^n))/(9*e^3) - (2*b^2*d^2*n*x^(7/3)*log(c*(d + e*x^(1/3))^n))/(21*e^2) - (2*b^2*d^4*n*x^(5/3)*log(c*(d + e*x^(1/3))^n))/( 15*e^4) + (b^2*d^5*n*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(6*e^5) + (b^2*d^7* n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(3*e^7) - (2*b^2*d^8*n*x^(1/3)*log(c*( d + e*x^(1/3))^n))/(3*e^8) + (a*b*d*n*x^(8/3))/(12*e) - (2*a*b*d^6*n*x)/(9 *e^6) + (2*a*b*d^9*n*log(d + e*x^(1/3)))/(3*e^9) + (b^2*d*n*x^(8/3)*log(c* (d + e*x^(1/3))^n))/(12*e) - (2*b^2*d^6*n*x*log(c*(d + e*x^(1/3))^n))/(9*e ^6) + (a*b*d^3*n*x^2)/(9*e^3) - (2*a*b*d^2*n*x^(7/3))/(21*e^2) - (2*a*b*d^ 4*n*x^(5/3))/(15*e^4) + (a*b*d^5*n*x^(4/3))/(6*e^5) + (a*b*d^7*n*x^(2/3))/ (3*e^7) - (2*a*b*d^8*n*x^(1/3))/(3*e^8)