Integrand size = 23, antiderivative size = 148 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} b d^3 n x^3-\frac {b e^3 n x^{3 (1+r)}}{9 (1+r)^2}-\frac {3 b d^2 e n x^{3+r}}{(3+r)^2}-\frac {3 b d e^2 n x^{3+2 r}}{(3+2 r)^2}+\frac {1}{3} \left (d^3 x^3+\frac {e^3 x^{3 (1+r)}}{1+r}+\frac {9 d^2 e x^{3+r}}{3+r}+\frac {9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/9*b*d^3*n*x^3-1/9*b*e^3*n*x^(3+3*r)/(1+r)^2-3*b*d^2*e*n*x^(3+r)/(3+r)^2 -3*b*d*e^2*n*x^(3+2*r)/(3+2*r)^2+1/3*(d^3*x^3+e^3*x^(3+3*r)/(1+r)+9*d^2*e* x^(3+r)/(3+r)+9*d*e^2*x^(3+2*r)/(3+2*r))*(a+b*ln(c*x^n))
Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.19 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{9} x^3 \left (b n \left (-d^3-\frac {27 d^2 e x^r}{(3+r)^2}-\frac {27 d e^2 x^{2 r}}{(3+2 r)^2}-\frac {e^3 x^{3 r}}{(1+r)^2}\right )+3 a \left (d^3+\frac {9 d^2 e x^r}{3+r}+\frac {9 d e^2 x^{2 r}}{3+2 r}+\frac {e^3 x^{3 r}}{1+r}\right )+3 b \left (d^3+\frac {9 d^2 e x^r}{3+r}+\frac {9 d e^2 x^{2 r}}{3+2 r}+\frac {e^3 x^{3 r}}{1+r}\right ) \log \left (c x^n\right )\right ) \]
(x^3*(b*n*(-d^3 - (27*d^2*e*x^r)/(3 + r)^2 - (27*d*e^2*x^(2*r))/(3 + 2*r)^ 2 - (e^3*x^(3*r))/(1 + r)^2) + 3*a*(d^3 + (9*d^2*e*x^r)/(3 + r) + (9*d*e^2 *x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r)) + 3*b*(d^3 + (9*d^2*e*x^r)/(3 + r) + (9*d*e^2*x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r))*Log[c*x^n]))/ 9
Time = 0.57 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2771, 27, 2010, 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.
| \(\displaystyle \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2771 |
| \(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{3} x^2 \left (\frac {9 d^2 e x^r}{r+3}+\frac {9 d e^2 x^{2 r}}{2 r+3}+\frac {e^3 x^{3 r}}{r+1}+d^3\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int x^2 \left (\frac {9 d^2 e x^r}{r+3}+\frac {9 d e^2 x^{2 r}}{2 r+3}+\frac {e^3 x^{3 r}}{r+1}+d^3\right )dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int \left (\frac {9 d e^2 x^{2 (r+1)}}{2 r+3}+\frac {9 d^2 e x^{r+2}}{r+3}+\frac {e^3 x^{3 r+2}}{r+1}+d^3 x^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \left (\frac {d^3 x^3}{3}+\frac {9 d^2 e x^{r+3}}{(r+3)^2}+\frac {9 d e^2 x^{2 r+3}}{(2 r+3)^2}+\frac {e^3 x^{3 (r+1)}}{3 (r+1)^2}\right )\) |
-1/3*(b*n*((d^3*x^3)/3 + (e^3*x^(3*(1 + r)))/(3*(1 + r)^2) + (9*d^2*e*x^(3 + r))/(3 + r)^2 + (9*d*e^2*x^(3 + 2*r))/(3 + 2*r)^2)) + ((d^3*x^3 + (e^3* x^(3*(1 + r)))/(1 + r) + (9*d^2*e*x^(3 + r))/(3 + r) + (9*d*e^2*x^(3 + 2*r ))/(3 + 2*r))*(a + b*Log[c*x^n]))/3
3.4.99.3.1 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[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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[u*(a + b*Log[c*x^n]), x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1256\) vs. \(2(142)=284\).
Time = 9.67 (sec) , antiderivative size = 1257, normalized size of antiderivative = 8.49
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1257\) |
| risch | \(\text {Expression too large to display}\) | \(4027\) |
-1/9*(-243*a*d^3*x^3-108*x^3*x^r*a*d^2*e*r^5-864*x^3*x^r*a*d^2*e*r^4+324*x ^3*b*d^3*n*r-579*x^3*a*d^3*r^4-1296*x^3*a*d^3*r^3-1566*x^3*a*d^3*r^2-972*x ^3*a*d^3*r-729*x^3*d^2*e*x^r*b*ln(c*x^n)-243*x^3*e^3*(x^r)^3*a-243*x^3*b*l n(c*x^n)*d^3+81*x^3*(x^r)^3*b*e^3*n+4*x^3*b*d^3*n*r^6+44*x^3*b*d^3*n*r^5+1 93*x^3*b*d^3*n*r^4+432*x^3*b*d^3*n*r^3+522*x^3*b*d^3*n*r^2-729*x^3*d^2*e*x ^r*a-243*x^3*e^3*(x^r)^3*b*ln(c*x^n)-729*x^3*d*e^2*(x^r)^2*a+540*x^3*x^r*b *d^2*e*n*r^3+999*x^3*x^r*b*d^2*e*n*r^2+810*x^3*x^r*b*d^2*e*n*r-120*x^3*(x^ r)^3*a*e^3*r^4-459*x^3*(x^r)^3*a*e^3*r^3-837*x^3*(x^r)^3*a*e^3*r^2-12*x^3* ln(c*x^n)*b*d^3*r^6-132*x^3*ln(c*x^n)*b*d^3*r^5-579*x^3*ln(c*x^n)*b*d^3*r^ 4-1296*x^3*ln(c*x^n)*b*d^3*r^3-12*x^3*a*d^3*r^6-132*x^3*a*d^3*r^5-729*x^3* d*e^2*(x^r)^2*b*ln(c*x^n)+27*x^3*(x^r)^2*b*d*e^2*n*r^4+216*x^3*(x^r)^2*b*d *e^2*n*r^3+594*x^3*(x^r)^2*b*d*e^2*n*r^2+648*x^3*(x^r)^2*b*d*e^2*n*r-108*x ^3*x^r*ln(c*x^n)*b*d^2*e*r^5-864*x^3*x^r*ln(c*x^n)*b*d^2*e*r^4-2619*x^3*x^ r*ln(c*x^n)*b*d^2*e*r^3-3807*x^3*x^r*ln(c*x^n)*b*d^2*e*r^2-54*x^3*(x^r)^2* ln(c*x^n)*b*d*e^2*r^5+108*x^3*x^r*b*d^2*e*n*r^4-513*x^3*(x^r)^2*ln(c*x^n)* b*d*e^2*r^4-1836*x^3*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3-3078*x^3*(x^r)^2*ln(c*x ^n)*b*d*e^2*r^2-2673*e*d^2*b*ln(c*x^n)*x^r*r*x^3-729*x^3*(x^r)^3*a*e^3*r+1 17*x^3*(x^r)^3*b*e^3*n*r^2+162*x^3*(x^r)^3*b*e^3*n*r+243*x^3*(x^r)^2*b*d*e ^2*n+243*x^3*x^r*b*d^2*e*n+4*x^3*(x^r)^3*b*e^3*n*r^4+81*b*d^3*n*x^3-12*x^3 *(x^r)^3*a*e^3*r^5-1566*x^3*ln(c*x^n)*b*d^3*r^2-972*x^3*ln(c*x^n)*b*d^3...
Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (142) = 284\).
Time = 0.32 (sec) , antiderivative size = 1022, normalized size of antiderivative = 6.91 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (4 \, b d^{3} r^{6} + 44 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} + 432 \, b d^{3} r^{3} + 522 \, b d^{3} r^{2} + 324 \, b d^{3} r + 81 \, b d^{3}\right )} x^{3} \log \left (c\right ) + 3 \, {\left (4 \, b d^{3} n r^{6} + 44 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} + 432 \, b d^{3} n r^{3} + 522 \, b d^{3} n r^{2} + 324 \, b d^{3} n r + 81 \, b d^{3} n\right )} x^{3} \log \left (x\right ) - {\left (4 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r^{6} + 44 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r^{5} + 81 \, b d^{3} n + 193 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r^{4} - 243 \, a d^{3} + 432 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r^{3} + 522 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r^{2} + 324 \, {\left (b d^{3} n - 3 \, a d^{3}\right )} r\right )} x^{3} + {\left (3 \, {\left (4 \, b e^{3} r^{5} + 40 \, b e^{3} r^{4} + 153 \, b e^{3} r^{3} + 279 \, b e^{3} r^{2} + 243 \, b e^{3} r + 81 \, b e^{3}\right )} x^{3} \log \left (c\right ) + 3 \, {\left (4 \, b e^{3} n r^{5} + 40 \, b e^{3} n r^{4} + 153 \, b e^{3} n r^{3} + 279 \, b e^{3} n r^{2} + 243 \, b e^{3} n r + 81 \, b e^{3} n\right )} x^{3} \log \left (x\right ) + {\left (12 \, a e^{3} r^{5} - 81 \, b e^{3} n - 4 \, {\left (b e^{3} n - 30 \, a e^{3}\right )} r^{4} + 243 \, a e^{3} - 9 \, {\left (4 \, b e^{3} n - 51 \, a e^{3}\right )} r^{3} - 9 \, {\left (13 \, b e^{3} n - 93 \, a e^{3}\right )} r^{2} - 81 \, {\left (2 \, b e^{3} n - 9 \, a e^{3}\right )} r\right )} x^{3}\right )} x^{3 \, r} + 27 \, {\left ({\left (2 \, b d e^{2} r^{5} + 19 \, b d e^{2} r^{4} + 68 \, b d e^{2} r^{3} + 114 \, b d e^{2} r^{2} + 90 \, b d e^{2} r + 27 \, b d e^{2}\right )} x^{3} \log \left (c\right ) + {\left (2 \, b d e^{2} n r^{5} + 19 \, b d e^{2} n r^{4} + 68 \, b d e^{2} n r^{3} + 114 \, b d e^{2} n r^{2} + 90 \, b d e^{2} n r + 27 \, b d e^{2} n\right )} x^{3} \log \left (x\right ) + {\left (2 \, a d e^{2} r^{5} - 9 \, b d e^{2} n - {\left (b d e^{2} n - 19 \, a d e^{2}\right )} r^{4} + 27 \, a d e^{2} - 4 \, {\left (2 \, b d e^{2} n - 17 \, a d e^{2}\right )} r^{3} - 2 \, {\left (11 \, b d e^{2} n - 57 \, a d e^{2}\right )} r^{2} - 6 \, {\left (4 \, b d e^{2} n - 15 \, a d e^{2}\right )} r\right )} x^{3}\right )} x^{2 \, r} + 27 \, {\left ({\left (4 \, b d^{2} e r^{5} + 32 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} + 141 \, b d^{2} e r^{2} + 99 \, b d^{2} e r + 27 \, b d^{2} e\right )} x^{3} \log \left (c\right ) + {\left (4 \, b d^{2} e n r^{5} + 32 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} + 141 \, b d^{2} e n r^{2} + 99 \, b d^{2} e n r + 27 \, b d^{2} e n\right )} x^{3} \log \left (x\right ) + {\left (4 \, a d^{2} e r^{5} - 9 \, b d^{2} e n - 4 \, {\left (b d^{2} e n - 8 \, a d^{2} e\right )} r^{4} + 27 \, a d^{2} e - {\left (20 \, b d^{2} e n - 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n - 141 \, a d^{2} e\right )} r^{2} - 3 \, {\left (10 \, b d^{2} e n - 33 \, a d^{2} e\right )} r\right )} x^{3}\right )} x^{r}}{9 \, {\left (4 \, r^{6} + 44 \, r^{5} + 193 \, r^{4} + 432 \, r^{3} + 522 \, r^{2} + 324 \, r + 81\right )}} \]
1/9*(3*(4*b*d^3*r^6 + 44*b*d^3*r^5 + 193*b*d^3*r^4 + 432*b*d^3*r^3 + 522*b *d^3*r^2 + 324*b*d^3*r + 81*b*d^3)*x^3*log(c) + 3*(4*b*d^3*n*r^6 + 44*b*d^ 3*n*r^5 + 193*b*d^3*n*r^4 + 432*b*d^3*n*r^3 + 522*b*d^3*n*r^2 + 324*b*d^3* n*r + 81*b*d^3*n)*x^3*log(x) - (4*(b*d^3*n - 3*a*d^3)*r^6 + 44*(b*d^3*n - 3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b*d^3*n - 3*a*d^3)*r^4 - 243*a*d^3 + 432* (b*d^3*n - 3*a*d^3)*r^3 + 522*(b*d^3*n - 3*a*d^3)*r^2 + 324*(b*d^3*n - 3*a *d^3)*r)*x^3 + (3*(4*b*e^3*r^5 + 40*b*e^3*r^4 + 153*b*e^3*r^3 + 279*b*e^3* r^2 + 243*b*e^3*r + 81*b*e^3)*x^3*log(c) + 3*(4*b*e^3*n*r^5 + 40*b*e^3*n*r ^4 + 153*b*e^3*n*r^3 + 279*b*e^3*n*r^2 + 243*b*e^3*n*r + 81*b*e^3*n)*x^3*l og(x) + (12*a*e^3*r^5 - 81*b*e^3*n - 4*(b*e^3*n - 30*a*e^3)*r^4 + 243*a*e^ 3 - 9*(4*b*e^3*n - 51*a*e^3)*r^3 - 9*(13*b*e^3*n - 93*a*e^3)*r^2 - 81*(2*b *e^3*n - 9*a*e^3)*r)*x^3)*x^(3*r) + 27*((2*b*d*e^2*r^5 + 19*b*d*e^2*r^4 + 68*b*d*e^2*r^3 + 114*b*d*e^2*r^2 + 90*b*d*e^2*r + 27*b*d*e^2)*x^3*log(c) + (2*b*d*e^2*n*r^5 + 19*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 + 114*b*d*e^2*n*r^ 2 + 90*b*d*e^2*n*r + 27*b*d*e^2*n)*x^3*log(x) + (2*a*d*e^2*r^5 - 9*b*d*e^2 *n - (b*d*e^2*n - 19*a*d*e^2)*r^4 + 27*a*d*e^2 - 4*(2*b*d*e^2*n - 17*a*d*e ^2)*r^3 - 2*(11*b*d*e^2*n - 57*a*d*e^2)*r^2 - 6*(4*b*d*e^2*n - 15*a*d*e^2) *r)*x^3)*x^(2*r) + 27*((4*b*d^2*e*r^5 + 32*b*d^2*e*r^4 + 97*b*d^2*e*r^3 + 141*b*d^2*e*r^2 + 99*b*d^2*e*r + 27*b*d^2*e)*x^3*log(c) + (4*b*d^2*e*n*r^5 + 32*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 + 141*b*d^2*e*n*r^2 + 99*b*d^2*e...
Timed out. \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.51 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{3} \, b d^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{3} x^{3} + \frac {b e^{3} x^{3 \, r + 3} \log \left (c x^{n}\right )}{3 \, {\left (r + 1\right )}} + \frac {3 \, b d e^{2} x^{2 \, r + 3} \log \left (c x^{n}\right )}{2 \, r + 3} + \frac {3 \, b d^{2} e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e^{3} n x^{3 \, r + 3}}{9 \, {\left (r + 1\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 3}}{3 \, {\left (r + 1\right )}} - \frac {3 \, b d e^{2} n x^{2 \, r + 3}}{{\left (2 \, r + 3\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 3}}{2 \, r + 3} - \frac {3 \, b d^{2} e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 3}}{r + 3} \]
-1/9*b*d^3*n*x^3 + 1/3*b*d^3*x^3*log(c*x^n) + 1/3*a*d^3*x^3 + 1/3*b*e^3*x^ (3*r + 3)*log(c*x^n)/(r + 1) + 3*b*d*e^2*x^(2*r + 3)*log(c*x^n)/(2*r + 3) + 3*b*d^2*e*x^(r + 3)*log(c*x^n)/(r + 3) - 1/9*b*e^3*n*x^(3*r + 3)/(r + 1) ^2 + 1/3*a*e^3*x^(3*r + 3)/(r + 1) - 3*b*d*e^2*n*x^(2*r + 3)/(2*r + 3)^2 + 3*a*d*e^2*x^(2*r + 3)/(2*r + 3) - 3*b*d^2*e*n*x^(r + 3)/(r + 3)^2 + 3*a*d ^2*e*x^(r + 3)/(r + 3)
Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (142) = 284\).
Time = 0.33 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.89 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/9*(12*b*e^3*n*r^5*x^3*x^(3*r)*log(x) + 54*b*d*e^2*n*r^5*x^3*x^(2*r)*log( x) + 108*b*d^2*e*n*r^5*x^3*x^r*log(x) + 12*b*d^3*n*r^6*x^3*log(x) - 4*b*d^ 3*n*r^6*x^3 + 12*b*e^3*r^5*x^3*x^(3*r)*log(c) + 54*b*d*e^2*r^5*x^3*x^(2*r) *log(c) + 108*b*d^2*e*r^5*x^3*x^r*log(c) + 12*b*d^3*r^6*x^3*log(c) + 120*b *e^3*n*r^4*x^3*x^(3*r)*log(x) + 513*b*d*e^2*n*r^4*x^3*x^(2*r)*log(x) + 864 *b*d^2*e*n*r^4*x^3*x^r*log(x) + 132*b*d^3*n*r^5*x^3*log(x) - 4*b*e^3*n*r^4 *x^3*x^(3*r) + 12*a*e^3*r^5*x^3*x^(3*r) - 27*b*d*e^2*n*r^4*x^3*x^(2*r) + 5 4*a*d*e^2*r^5*x^3*x^(2*r) - 108*b*d^2*e*n*r^4*x^3*x^r + 108*a*d^2*e*r^5*x^ 3*x^r - 44*b*d^3*n*r^5*x^3 + 12*a*d^3*r^6*x^3 + 120*b*e^3*r^4*x^3*x^(3*r)* log(c) + 513*b*d*e^2*r^4*x^3*x^(2*r)*log(c) + 864*b*d^2*e*r^4*x^3*x^r*log( c) + 132*b*d^3*r^5*x^3*log(c) + 459*b*e^3*n*r^3*x^3*x^(3*r)*log(x) + 1836* b*d*e^2*n*r^3*x^3*x^(2*r)*log(x) + 2619*b*d^2*e*n*r^3*x^3*x^r*log(x) + 579 *b*d^3*n*r^4*x^3*log(x) - 36*b*e^3*n*r^3*x^3*x^(3*r) + 120*a*e^3*r^4*x^3*x ^(3*r) - 216*b*d*e^2*n*r^3*x^3*x^(2*r) + 513*a*d*e^2*r^4*x^3*x^(2*r) - 540 *b*d^2*e*n*r^3*x^3*x^r + 864*a*d^2*e*r^4*x^3*x^r - 193*b*d^3*n*r^4*x^3 + 1 32*a*d^3*r^5*x^3 + 459*b*e^3*r^3*x^3*x^(3*r)*log(c) + 1836*b*d*e^2*r^3*x^3 *x^(2*r)*log(c) + 2619*b*d^2*e*r^3*x^3*x^r*log(c) + 579*b*d^3*r^4*x^3*log( c) + 837*b*e^3*n*r^2*x^3*x^(3*r)*log(x) + 3078*b*d*e^2*n*r^2*x^3*x^(2*r)*l og(x) + 3807*b*d^2*e*n*r^2*x^3*x^r*log(x) + 1296*b*d^3*n*r^3*x^3*log(x) - 117*b*e^3*n*r^2*x^3*x^(3*r) + 459*a*e^3*r^3*x^3*x^(3*r) - 594*b*d*e^2*n...
Timed out. \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]