Integrand size = 23, antiderivative size = 103 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} b d^2 n x^6-\frac {b e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {2 b d e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^2 x^6+\frac {3 e^2 x^{2 (3+r)}}{3+r}+\frac {12 d e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/36*b*d^2*n*x^6-1/4*b*e^2*n*x^(6+2*r)/(3+r)^2-2*b*d*e*n*x^(6+r)/(6+r)^2+ 1/6*(d^2*x^6+3*e^2*x^(6+2*r)/(3+r)+12*d*e*x^(6+r)/(6+r))*(a+b*ln(c*x^n))
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{36} x^6 \left (b n \left (-d^2-\frac {72 d e x^r}{(6+r)^2}-\frac {9 e^2 x^{2 r}}{(3+r)^2}\right )+6 a \left (d^2+\frac {12 d e x^r}{6+r}+\frac {3 e^2 x^{2 r}}{3+r}\right )+6 b \left (d^2+\frac {12 d e x^r}{6+r}+\frac {3 e^2 x^{2 r}}{3+r}\right ) \log \left (c x^n\right )\right ) \]
(x^6*(b*n*(-d^2 - (72*d*e*x^r)/(6 + r)^2 - (9*e^2*x^(2*r))/(3 + r)^2) + 6* a*(d^2 + (12*d*e*x^r)/(6 + r) + (3*e^2*x^(2*r))/(3 + r)) + 6*b*(d^2 + (12* d*e*x^r)/(6 + r) + (3*e^2*x^(2*r))/(3 + r))*Log[c*x^n]))/36
Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2771, 27, 1691, 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.
| \(\displaystyle \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2771 |
| \(\displaystyle \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{6} x^5 \left (\frac {12 d e x^r}{r+6}+\frac {3 e^2 x^{2 r}}{r+3}+d^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int x^5 \left (\frac {12 d e x^r}{r+6}+\frac {3 e^2 x^{2 r}}{r+3}+d^2\right )dx\) |
| \(\Big \downarrow \) 1691 |
| \(\displaystyle \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int \left (\frac {12 d e x^{r+5}}{r+6}+\frac {3 e^2 x^{2 r+5}}{r+3}+d^2 x^5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (d^2 x^6+\frac {12 d e x^{r+6}}{r+6}+\frac {3 e^2 x^{2 (r+3)}}{r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \left (\frac {d^2 x^6}{6}+\frac {12 d e x^{r+6}}{(r+6)^2}+\frac {3 e^2 x^{2 (r+3)}}{2 (r+3)^2}\right )\) |
-1/6*(b*n*((d^2*x^6)/6 + (3*e^2*x^(2*(3 + r)))/(2*(3 + r)^2) + (12*d*e*x^( 6 + r))/(6 + r)^2)) + ((d^2*x^6 + (3*e^2*x^(2*(3 + r)))/(3 + r) + (12*d*e* x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6
3.4.79.3.1 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[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && !IntegerQ [Simplify[(m + 1)/n]]
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. \(582\) vs. \(2(97)=194\).
Time = 15.65 (sec) , antiderivative size = 583, normalized size of antiderivative = 5.66
| method | result | size |
| parallelrisch | \(-\frac {-1944 a \,d^{2} x^{6}-18 x^{6} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}-270 x^{6} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-1296 x^{6} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +18 x^{6} b \,d^{2} n \,r^{3}+117 x^{6} b \,d^{2} n \,r^{2}+324 x^{6} b \,d^{2} n r +108 x^{6} x^{2 r} b \,e^{2} n r -72 x^{6} x^{r} a d e \,r^{3}-864 x^{6} x^{r} a d e \,r^{2}-3240 x^{6} x^{r} a d e r +648 x^{6} x^{r} b d e n -3888 b d e \ln \left (c \,x^{n}\right ) x^{r} x^{6}+9 x^{6} x^{2 r} b \,e^{2} n \,r^{2}-6 x^{6} a \,d^{2} r^{4}-108 x^{6} a \,d^{2} r^{3}-702 x^{6} a \,d^{2} r^{2}-1944 x^{6} a \,d^{2} r -1944 e^{2} b \ln \left (c \,x^{n}\right ) x^{2 r} x^{6}-1296 x^{6} x^{2 r} a \,e^{2} r +324 x^{6} x^{2 r} b \,e^{2} n -18 x^{6} x^{2 r} a \,e^{2} r^{3}-270 x^{6} x^{2 r} a \,e^{2} r^{2}-1944 x^{6} x^{2 r} a \,e^{2}+72 x^{6} x^{r} b d e n \,r^{2}-6 x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-108 x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}-702 x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-1944 x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2} r -3888 x^{6} x^{r} a d e +x^{6} b \,d^{2} n \,r^{4}-72 x^{6} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}-864 x^{6} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-3240 x^{6} x^{r} \ln \left (c \,x^{n}\right ) b d e r +432 x^{6} x^{r} b d e n r +324 b \,d^{2} n \,x^{6}-1944 x^{6} \ln \left (c \,x^{n}\right ) b \,d^{2}}{36 \left (r^{2}+6 r +9\right ) \left (6+r \right )^{2}}\) | \(583\) |
| risch | \(\text {Expression too large to display}\) | \(1924\) |
-1/36*(-1944*a*d^2*x^6-1944*e^2*b*ln(c*x^n)*(x^r)^2*x^6+18*x^6*b*d^2*n*r^3 +117*x^6*b*d^2*n*r^2+324*x^6*b*d^2*n*r-72*x^6*x^r*a*d*e*r^3-864*x^6*x^r*a* d*e*r^2-3240*x^6*x^r*a*d*e*r+648*x^6*x^r*b*d*e*n-3888*b*d*e*ln(c*x^n)*x^r* x^6+9*x^6*(x^r)^2*b*e^2*n*r^2+108*x^6*(x^r)^2*b*e^2*n*r-18*x^6*(x^r)^2*ln( c*x^n)*b*e^2*r^3-270*x^6*(x^r)^2*ln(c*x^n)*b*e^2*r^2-1296*x^6*(x^r)^2*ln(c *x^n)*b*e^2*r-1944*x^6*(x^r)^2*a*e^2-6*x^6*a*d^2*r^4-108*x^6*a*d^2*r^3-702 *x^6*a*d^2*r^2-1944*x^6*a*d^2*r+72*x^6*x^r*b*d*e*n*r^2-1296*x^6*(x^r)^2*a* e^2*r+324*x^6*(x^r)^2*b*e^2*n-6*x^6*ln(c*x^n)*b*d^2*r^4-108*x^6*ln(c*x^n)* b*d^2*r^3-702*x^6*ln(c*x^n)*b*d^2*r^2-1944*x^6*ln(c*x^n)*b*d^2*r-3888*x^6* x^r*a*d*e+x^6*b*d^2*n*r^4-18*x^6*(x^r)^2*a*e^2*r^3-270*x^6*(x^r)^2*a*e^2*r ^2-72*x^6*x^r*ln(c*x^n)*b*d*e*r^3-864*x^6*x^r*ln(c*x^n)*b*d*e*r^2-3240*x^6 *x^r*ln(c*x^n)*b*d*e*r+432*x^6*x^r*b*d*e*n*r+324*b*d^2*n*x^6-1944*x^6*ln(c *x^n)*b*d^2)/(r^2+6*r+9)/(6+r)^2
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (97) = 194\).
Time = 0.30 (sec) , antiderivative size = 489, normalized size of antiderivative = 4.75 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 \, {\left (b d^{2} r^{4} + 18 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} + 324 \, b d^{2} r + 324 \, b d^{2}\right )} x^{6} \log \left (c\right ) + 6 \, {\left (b d^{2} n r^{4} + 18 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} + 324 \, b d^{2} n r + 324 \, b d^{2} n\right )} x^{6} \log \left (x\right ) - {\left ({\left (b d^{2} n - 6 \, a d^{2}\right )} r^{4} + 324 \, b d^{2} n + 18 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r^{3} - 1944 \, a d^{2} + 117 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r^{2} + 324 \, {\left (b d^{2} n - 6 \, a d^{2}\right )} r\right )} x^{6} + 9 \, {\left (2 \, {\left (b e^{2} r^{3} + 15 \, b e^{2} r^{2} + 72 \, b e^{2} r + 108 \, b e^{2}\right )} x^{6} \log \left (c\right ) + 2 \, {\left (b e^{2} n r^{3} + 15 \, b e^{2} n r^{2} + 72 \, b e^{2} n r + 108 \, b e^{2} n\right )} x^{6} \log \left (x\right ) + {\left (2 \, a e^{2} r^{3} - 36 \, b e^{2} n + 216 \, a e^{2} - {\left (b e^{2} n - 30 \, a e^{2}\right )} r^{2} - 12 \, {\left (b e^{2} n - 12 \, a e^{2}\right )} r\right )} x^{6}\right )} x^{2 \, r} + 72 \, {\left ({\left (b d e r^{3} + 12 \, b d e r^{2} + 45 \, b d e r + 54 \, b d e\right )} x^{6} \log \left (c\right ) + {\left (b d e n r^{3} + 12 \, b d e n r^{2} + 45 \, b d e n r + 54 \, b d e n\right )} x^{6} \log \left (x\right ) + {\left (a d e r^{3} - 9 \, b d e n + 54 \, a d e - {\left (b d e n - 12 \, a d e\right )} r^{2} - 3 \, {\left (2 \, b d e n - 15 \, a d e\right )} r\right )} x^{6}\right )} x^{r}}{36 \, {\left (r^{4} + 18 \, r^{3} + 117 \, r^{2} + 324 \, r + 324\right )}} \]
1/36*(6*(b*d^2*r^4 + 18*b*d^2*r^3 + 117*b*d^2*r^2 + 324*b*d^2*r + 324*b*d^ 2)*x^6*log(c) + 6*(b*d^2*n*r^4 + 18*b*d^2*n*r^3 + 117*b*d^2*n*r^2 + 324*b* d^2*n*r + 324*b*d^2*n)*x^6*log(x) - ((b*d^2*n - 6*a*d^2)*r^4 + 324*b*d^2*n + 18*(b*d^2*n - 6*a*d^2)*r^3 - 1944*a*d^2 + 117*(b*d^2*n - 6*a*d^2)*r^2 + 324*(b*d^2*n - 6*a*d^2)*r)*x^6 + 9*(2*(b*e^2*r^3 + 15*b*e^2*r^2 + 72*b*e^ 2*r + 108*b*e^2)*x^6*log(c) + 2*(b*e^2*n*r^3 + 15*b*e^2*n*r^2 + 72*b*e^2*n *r + 108*b*e^2*n)*x^6*log(x) + (2*a*e^2*r^3 - 36*b*e^2*n + 216*a*e^2 - (b* e^2*n - 30*a*e^2)*r^2 - 12*(b*e^2*n - 12*a*e^2)*r)*x^6)*x^(2*r) + 72*((b*d *e*r^3 + 12*b*d*e*r^2 + 45*b*d*e*r + 54*b*d*e)*x^6*log(c) + (b*d*e*n*r^3 + 12*b*d*e*n*r^2 + 45*b*d*e*n*r + 54*b*d*e*n)*x^6*log(x) + (a*d*e*r^3 - 9*b *d*e*n + 54*a*d*e - (b*d*e*n - 12*a*d*e)*r^2 - 3*(2*b*d*e*n - 15*a*d*e)*r) *x^6)*x^r)/(r^4 + 18*r^3 + 117*r^2 + 324*r + 324)
Leaf count of result is larger than twice the leaf count of optimal. 1634 vs. \(2 (97) = 194\).
Time = 26.36 (sec) , antiderivative size = 1634, normalized size of antiderivative = 15.86 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
Piecewise((a*d**2*x**6/6 + 2*a*d*e*log(c*x**n)/n - a*e**2/(6*x**6) - b*d** 2*n*x**6/36 + b*d**2*x**6*log(c*x**n)/6 + b*d*e*log(c*x**n)**2/n - b*e**2* n/(36*x**6) - b*e**2*log(c*x**n)/(6*x**6), Eq(r, -6)), (a*d**2*x**6/6 + 2* a*d*e*x**3/3 + a*e**2*log(c*x**n)/n - b*d**2*n*x**6/36 + b*d**2*x**6*log(c *x**n)/6 - 2*b*d*e*n*x**3/9 + 2*b*d*e*x**3*log(c*x**n)/3 + b*e**2*log(c*x* *n)**2/(2*n), Eq(r, -3)), (6*a*d**2*r**4*x**6/(36*r**4 + 648*r**3 + 4212*r **2 + 11664*r + 11664) + 108*a*d**2*r**3*x**6/(36*r**4 + 648*r**3 + 4212*r **2 + 11664*r + 11664) + 702*a*d**2*r**2*x**6/(36*r**4 + 648*r**3 + 4212*r **2 + 11664*r + 11664) + 1944*a*d**2*r*x**6/(36*r**4 + 648*r**3 + 4212*r** 2 + 11664*r + 11664) + 1944*a*d**2*x**6/(36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) + 72*a*d*e*r**3*x**6*x**r/(36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) + 864*a*d*e*r**2*x**6*x**r/(36*r**4 + 648*r**3 + 4212* r**2 + 11664*r + 11664) + 3240*a*d*e*r*x**6*x**r/(36*r**4 + 648*r**3 + 421 2*r**2 + 11664*r + 11664) + 3888*a*d*e*x**6*x**r/(36*r**4 + 648*r**3 + 421 2*r**2 + 11664*r + 11664) + 18*a*e**2*r**3*x**6*x**(2*r)/(36*r**4 + 648*r* *3 + 4212*r**2 + 11664*r + 11664) + 270*a*e**2*r**2*x**6*x**(2*r)/(36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) + 1296*a*e**2*r*x**6*x**(2*r)/( 36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) + 1944*a*e**2*x**6*x**(2 *r)/(36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) - b*d**2*n*r**4*x** 6/(36*r**4 + 648*r**3 + 4212*r**2 + 11664*r + 11664) - 18*b*d**2*n*r**3...
Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.44 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, b d^{2} n x^{6} + \frac {1}{6} \, b d^{2} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{2} x^{6} + \frac {b e^{2} x^{2 \, r + 6} \log \left (c x^{n}\right )}{2 \, {\left (r + 3\right )}} + \frac {2 \, b d e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e^{2} n x^{2 \, r + 6}}{4 \, {\left (r + 3\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 6}}{2 \, {\left (r + 3\right )}} - \frac {2 \, b d e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {2 \, a d e x^{r + 6}}{r + 6} \]
-1/36*b*d^2*n*x^6 + 1/6*b*d^2*x^6*log(c*x^n) + 1/6*a*d^2*x^6 + 1/2*b*e^2*x ^(2*r + 6)*log(c*x^n)/(r + 3) + 2*b*d*e*x^(r + 6)*log(c*x^n)/(r + 6) - 1/4 *b*e^2*n*x^(2*r + 6)/(r + 3)^2 + 1/2*a*e^2*x^(2*r + 6)/(r + 3) - 2*b*d*e*n *x^(r + 6)/(r + 6)^2 + 2*a*d*e*x^(r + 6)/(r + 6)
Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (97) = 194\).
Time = 0.37 (sec) , antiderivative size = 744, normalized size of antiderivative = 7.22 \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {18 \, b e^{2} n r^{3} x^{6} x^{2 \, r} \log \left (x\right ) + 72 \, b d e n r^{3} x^{6} x^{r} \log \left (x\right ) + 6 \, b d^{2} n r^{4} x^{6} \log \left (x\right ) - b d^{2} n r^{4} x^{6} + 18 \, b e^{2} r^{3} x^{6} x^{2 \, r} \log \left (c\right ) + 72 \, b d e r^{3} x^{6} x^{r} \log \left (c\right ) + 6 \, b d^{2} r^{4} x^{6} \log \left (c\right ) + 270 \, b e^{2} n r^{2} x^{6} x^{2 \, r} \log \left (x\right ) + 864 \, b d e n r^{2} x^{6} x^{r} \log \left (x\right ) + 108 \, b d^{2} n r^{3} x^{6} \log \left (x\right ) - 9 \, b e^{2} n r^{2} x^{6} x^{2 \, r} + 18 \, a e^{2} r^{3} x^{6} x^{2 \, r} - 72 \, b d e n r^{2} x^{6} x^{r} + 72 \, a d e r^{3} x^{6} x^{r} - 18 \, b d^{2} n r^{3} x^{6} + 6 \, a d^{2} r^{4} x^{6} + 270 \, b e^{2} r^{2} x^{6} x^{2 \, r} \log \left (c\right ) + 864 \, b d e r^{2} x^{6} x^{r} \log \left (c\right ) + 108 \, b d^{2} r^{3} x^{6} \log \left (c\right ) + 1296 \, b e^{2} n r x^{6} x^{2 \, r} \log \left (x\right ) + 3240 \, b d e n r x^{6} x^{r} \log \left (x\right ) + 702 \, b d^{2} n r^{2} x^{6} \log \left (x\right ) - 108 \, b e^{2} n r x^{6} x^{2 \, r} + 270 \, a e^{2} r^{2} x^{6} x^{2 \, r} - 432 \, b d e n r x^{6} x^{r} + 864 \, a d e r^{2} x^{6} x^{r} - 117 \, b d^{2} n r^{2} x^{6} + 108 \, a d^{2} r^{3} x^{6} + 1296 \, b e^{2} r x^{6} x^{2 \, r} \log \left (c\right ) + 3240 \, b d e r x^{6} x^{r} \log \left (c\right ) + 702 \, b d^{2} r^{2} x^{6} \log \left (c\right ) + 1944 \, b e^{2} n x^{6} x^{2 \, r} \log \left (x\right ) + 3888 \, b d e n x^{6} x^{r} \log \left (x\right ) + 1944 \, b d^{2} n r x^{6} \log \left (x\right ) - 324 \, b e^{2} n x^{6} x^{2 \, r} + 1296 \, a e^{2} r x^{6} x^{2 \, r} - 648 \, b d e n x^{6} x^{r} + 3240 \, a d e r x^{6} x^{r} - 324 \, b d^{2} n r x^{6} + 702 \, a d^{2} r^{2} x^{6} + 1944 \, b e^{2} x^{6} x^{2 \, r} \log \left (c\right ) + 3888 \, b d e x^{6} x^{r} \log \left (c\right ) + 1944 \, b d^{2} r x^{6} \log \left (c\right ) + 1944 \, b d^{2} n x^{6} \log \left (x\right ) + 1944 \, a e^{2} x^{6} x^{2 \, r} + 3888 \, a d e x^{6} x^{r} - 324 \, b d^{2} n x^{6} + 1944 \, a d^{2} r x^{6} + 1944 \, b d^{2} x^{6} \log \left (c\right ) + 1944 \, a d^{2} x^{6}}{36 \, {\left (r^{4} + 18 \, r^{3} + 117 \, r^{2} + 324 \, r + 324\right )}} \]
1/36*(18*b*e^2*n*r^3*x^6*x^(2*r)*log(x) + 72*b*d*e*n*r^3*x^6*x^r*log(x) + 6*b*d^2*n*r^4*x^6*log(x) - b*d^2*n*r^4*x^6 + 18*b*e^2*r^3*x^6*x^(2*r)*log( c) + 72*b*d*e*r^3*x^6*x^r*log(c) + 6*b*d^2*r^4*x^6*log(c) + 270*b*e^2*n*r^ 2*x^6*x^(2*r)*log(x) + 864*b*d*e*n*r^2*x^6*x^r*log(x) + 108*b*d^2*n*r^3*x^ 6*log(x) - 9*b*e^2*n*r^2*x^6*x^(2*r) + 18*a*e^2*r^3*x^6*x^(2*r) - 72*b*d*e *n*r^2*x^6*x^r + 72*a*d*e*r^3*x^6*x^r - 18*b*d^2*n*r^3*x^6 + 6*a*d^2*r^4*x ^6 + 270*b*e^2*r^2*x^6*x^(2*r)*log(c) + 864*b*d*e*r^2*x^6*x^r*log(c) + 108 *b*d^2*r^3*x^6*log(c) + 1296*b*e^2*n*r*x^6*x^(2*r)*log(x) + 3240*b*d*e*n*r *x^6*x^r*log(x) + 702*b*d^2*n*r^2*x^6*log(x) - 108*b*e^2*n*r*x^6*x^(2*r) + 270*a*e^2*r^2*x^6*x^(2*r) - 432*b*d*e*n*r*x^6*x^r + 864*a*d*e*r^2*x^6*x^r - 117*b*d^2*n*r^2*x^6 + 108*a*d^2*r^3*x^6 + 1296*b*e^2*r*x^6*x^(2*r)*log( c) + 3240*b*d*e*r*x^6*x^r*log(c) + 702*b*d^2*r^2*x^6*log(c) + 1944*b*e^2*n *x^6*x^(2*r)*log(x) + 3888*b*d*e*n*x^6*x^r*log(x) + 1944*b*d^2*n*r*x^6*log (x) - 324*b*e^2*n*x^6*x^(2*r) + 1296*a*e^2*r*x^6*x^(2*r) - 648*b*d*e*n*x^6 *x^r + 3240*a*d*e*r*x^6*x^r - 324*b*d^2*n*r*x^6 + 702*a*d^2*r^2*x^6 + 1944 *b*e^2*x^6*x^(2*r)*log(c) + 3888*b*d*e*x^6*x^r*log(c) + 1944*b*d^2*r*x^6*l og(c) + 1944*b*d^2*n*x^6*log(x) + 1944*a*e^2*x^6*x^(2*r) + 3888*a*d*e*x^6* x^r - 324*b*d^2*n*x^6 + 1944*a*d^2*r*x^6 + 1944*b*d^2*x^6*log(c) + 1944*a* d^2*x^6)/(r^4 + 18*r^3 + 117*r^2 + 324*r + 324)
Timed out. \[ \int x^5 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^5\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]