Integrand size = 23, antiderivative size = 147 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} b d^3 n x^6-\frac {b e^3 n x^{3 (2+r)}}{9 (2+r)^2}-\frac {3 b d e^2 n x^{2 (3+r)}}{4 (3+r)^2}-\frac {3 b d^2 e n x^{6+r}}{(6+r)^2}+\frac {1}{6} \left (d^3 x^6+\frac {2 e^3 x^{3 (2+r)}}{2+r}+\frac {9 d e^2 x^{2 (3+r)}}{3+r}+\frac {18 d^2 e x^{6+r}}{6+r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/36*b*d^3*n*x^6-1/9*b*e^3*n*x^(6+3*r)/(2+r)^2-3/4*b*d*e^2*n*x^(6+2*r)/(3 +r)^2-3*b*d^2*e*n*x^(6+r)/(6+r)^2+1/6*(d^3*x^6+2*e^3*x^(6+3*r)/(2+r)+9*d*e ^2*x^(6+2*r)/(3+r)+18*d^2*e*x^(6+r)/(6+r))*(a+b*ln(c*x^n))
Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.17 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{36} x^6 \left (b n \left (-d^3-\frac {108 d^2 e x^r}{(6+r)^2}-\frac {27 d e^2 x^{2 r}}{(3+r)^2}-\frac {4 e^3 x^{3 r}}{(2+r)^2}\right )+6 a \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right )+6 b \left (d^3+\frac {18 d^2 e x^r}{6+r}+\frac {9 d e^2 x^{2 r}}{3+r}+\frac {2 e^3 x^{3 r}}{2+r}\right ) \log \left (c x^n\right )\right ) \]
(x^6*(b*n*(-d^3 - (108*d^2*e*x^r)/(6 + r)^2 - (27*d*e^2*x^(2*r))/(3 + r)^2 - (4*e^3*x^(3*r))/(2 + r)^2) + 6*a*(d^3 + (18*d^2*e*x^r)/(6 + r) + (9*d*e ^2*x^(2*r))/(3 + r) + (2*e^3*x^(3*r))/(2 + r)) + 6*b*(d^3 + (18*d^2*e*x^r) /(6 + r) + (9*d*e^2*x^(2*r))/(3 + r) + (2*e^3*x^(3*r))/(2 + r))*Log[c*x^n] ))/36
Time = 0.55 (sec) , antiderivative size = 146, 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.
| \(\displaystyle \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2771 |
| \(\displaystyle \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{6} x^5 \left (\frac {18 d^2 e x^r}{r+6}+\frac {9 d e^2 x^{2 r}}{r+3}+\frac {2 e^3 x^{3 r}}{r+2}+d^3\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int x^5 \left (\frac {18 d^2 e x^r}{r+6}+\frac {9 d e^2 x^{2 r}}{r+3}+\frac {2 e^3 x^{3 r}}{r+2}+d^3\right )dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \int \left (\frac {18 d^2 e x^{r+5}}{r+6}+\frac {9 d e^2 x^{2 r+5}}{r+3}+\frac {2 e^3 x^{3 r+5}}{r+2}+d^3 x^5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (d^3 x^6+\frac {18 d^2 e x^{r+6}}{r+6}+\frac {9 d e^2 x^{2 (r+3)}}{r+3}+\frac {2 e^3 x^{3 (r+2)}}{r+2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} b n \left (\frac {d^3 x^6}{6}+\frac {18 d^2 e x^{r+6}}{(r+6)^2}+\frac {9 d e^2 x^{2 (r+3)}}{2 (r+3)^2}+\frac {2 e^3 x^{3 (r+2)}}{3 (r+2)^2}\right )\) |
-1/6*(b*n*((d^3*x^6)/6 + (2*e^3*x^(3*(2 + r)))/(3*(2 + r)^2) + (9*d*e^2*x^ (2*(3 + r)))/(2*(3 + r)^2) + (18*d^2*e*x^(6 + r))/(6 + r)^2)) + ((d^3*x^6 + (2*e^3*x^(3*(2 + r)))/(2 + r) + (9*d*e^2*x^(2*(3 + r)))/(3 + r) + (18*d^ 2*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6
3.4.92.3.1 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. \(1248\) vs. \(2(139)=278\).
Time = 40.92 (sec) , antiderivative size = 1249, normalized size of antiderivative = 8.50
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1249\) |
| risch | \(\text {Expression too large to display}\) | \(4021\) |
-1/36*(-1026*x^6*(x^r)^2*a*d*e^2*r^4-7344*x^6*(x^r)^2*a*d*e^2*r^3-11664*x^ 6*(x^r)^3*ln(c*x^n)*b*e^3*r-23328*e^2*d*b*ln(c*x^n)*(x^r)^2*x^6+4*x^6*(x^r )^3*b*e^3*n*r^4+72*x^6*(x^r)^3*b*e^3*n*r^3+468*x^6*(x^r)^3*b*e^3*n*r^2+129 6*x^6*(x^r)^3*b*e^3*n*r-7776*a*d^3*x^6-30456*x^6*x^r*a*d^2*e*r^2-42768*x^6 *x^r*a*d^2*e*r+3888*x^6*x^r*b*d^2*e*n-23328*e*d^2*b*ln(c*x^n)*x^r*x^6-12*x ^6*(x^r)^3*ln(c*x^n)*b*e^3*r^5-108*x^6*x^r*a*d^2*e*r^5-1728*x^6*x^r*a*d^2* e*r^4-10476*x^6*x^r*a*d^2*e*r^3-7776*e^3*b*ln(c*x^n)*(x^r)^3*x^6-12528*x^6 *ln(c*x^n)*b*d^3*r^2-15552*x^6*ln(c*x^n)*b*d^3*r-12*x^6*(x^r)^3*a*e^3*r^5- 240*x^6*(x^r)^3*a*e^3*r^4-1836*x^6*(x^r)^3*a*e^3*r^3-6696*x^6*(x^r)^3*a*e^ 3*r^2-11664*x^6*(x^r)^3*a*e^3*r+1296*x^6*(x^r)^3*b*e^3*n-23328*x^6*x^r*a*d ^2*e-23328*x^6*(x^r)^2*a*d*e^2+x^6*b*d^3*n*r^6+22*x^6*b*d^3*n*r^5+193*x^6* b*d^3*n*r^4+864*x^6*b*d^3*n*r^3+2088*x^6*b*d^3*n*r^2+2592*x^6*b*d^3*n*r-6* x^6*ln(c*x^n)*b*d^3*r^6-132*x^6*ln(c*x^n)*b*d^3*r^5-240*x^6*(x^r)^3*ln(c*x ^n)*b*e^3*r^4-1836*x^6*(x^r)^3*ln(c*x^n)*b*e^3*r^3-6696*x^6*(x^r)^3*ln(c*x ^n)*b*e^3*r^2-38880*x^6*(x^r)^2*a*d*e^2*r+3888*x^6*(x^r)^2*b*d*e^2*n-24624 *x^6*(x^r)^2*a*d*e^2*r^2-54*x^6*(x^r)^2*a*d*e^2*r^5-7776*x^6*(x^r)^3*a*e^3 -6*x^6*a*d^3*r^6-132*x^6*a*d^3*r^5-1158*x^6*a*d^3*r^4-5184*x^6*a*d^3*r^3-1 2528*x^6*a*d^3*r^2-15552*x^6*a*d^3*r-1158*x^6*ln(c*x^n)*b*d^3*r^4-5184*x^6 *ln(c*x^n)*b*d^3*r^3-108*x^6*x^r*ln(c*x^n)*b*d^2*e*r^5-1728*x^6*x^r*ln(c*x ^n)*b*d^2*e*r^4-10476*x^6*x^r*ln(c*x^n)*b*d^2*e*r^3-30456*x^6*x^r*ln(c*...
Leaf count of result is larger than twice the leaf count of optimal. 1011 vs. \(2 (139) = 278\).
Time = 0.32 (sec) , antiderivative size = 1011, normalized size of antiderivative = 6.88 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 \, {\left (b d^{3} r^{6} + 22 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} + 864 \, b d^{3} r^{3} + 2088 \, b d^{3} r^{2} + 2592 \, b d^{3} r + 1296 \, b d^{3}\right )} x^{6} \log \left (c\right ) + 6 \, {\left (b d^{3} n r^{6} + 22 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} + 864 \, b d^{3} n r^{3} + 2088 \, b d^{3} n r^{2} + 2592 \, b d^{3} n r + 1296 \, b d^{3} n\right )} x^{6} \log \left (x\right ) - {\left ({\left (b d^{3} n - 6 \, a d^{3}\right )} r^{6} + 22 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{5} + 1296 \, b d^{3} n + 193 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{4} - 7776 \, a d^{3} + 864 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{3} + 2088 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r^{2} + 2592 \, {\left (b d^{3} n - 6 \, a d^{3}\right )} r\right )} x^{6} + 4 \, {\left (3 \, {\left (b e^{3} r^{5} + 20 \, b e^{3} r^{4} + 153 \, b e^{3} r^{3} + 558 \, b e^{3} r^{2} + 972 \, b e^{3} r + 648 \, b e^{3}\right )} x^{6} \log \left (c\right ) + 3 \, {\left (b e^{3} n r^{5} + 20 \, b e^{3} n r^{4} + 153 \, b e^{3} n r^{3} + 558 \, b e^{3} n r^{2} + 972 \, b e^{3} n r + 648 \, b e^{3} n\right )} x^{6} \log \left (x\right ) + {\left (3 \, a e^{3} r^{5} - 324 \, b e^{3} n - {\left (b e^{3} n - 60 \, a e^{3}\right )} r^{4} + 1944 \, a e^{3} - 9 \, {\left (2 \, b e^{3} n - 51 \, a e^{3}\right )} r^{3} - 9 \, {\left (13 \, b e^{3} n - 186 \, a e^{3}\right )} r^{2} - 324 \, {\left (b e^{3} n - 9 \, a e^{3}\right )} r\right )} x^{6}\right )} x^{3 \, r} + 27 \, {\left (2 \, {\left (b d e^{2} r^{5} + 19 \, b d e^{2} r^{4} + 136 \, b d e^{2} r^{3} + 456 \, b d e^{2} r^{2} + 720 \, b d e^{2} r + 432 \, b d e^{2}\right )} x^{6} \log \left (c\right ) + 2 \, {\left (b d e^{2} n r^{5} + 19 \, b d e^{2} n r^{4} + 136 \, b d e^{2} n r^{3} + 456 \, b d e^{2} n r^{2} + 720 \, b d e^{2} n r + 432 \, b d e^{2} n\right )} x^{6} \log \left (x\right ) + {\left (2 \, a d e^{2} r^{5} - 144 \, b d e^{2} n - {\left (b d e^{2} n - 38 \, a d e^{2}\right )} r^{4} + 864 \, a d e^{2} - 16 \, {\left (b d e^{2} n - 17 \, a d e^{2}\right )} r^{3} - 8 \, {\left (11 \, b d e^{2} n - 114 \, a d e^{2}\right )} r^{2} - 96 \, {\left (2 \, b d e^{2} n - 15 \, a d e^{2}\right )} r\right )} x^{6}\right )} x^{2 \, r} + 108 \, {\left ({\left (b d^{2} e r^{5} + 16 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} + 282 \, b d^{2} e r^{2} + 396 \, b d^{2} e r + 216 \, b d^{2} e\right )} x^{6} \log \left (c\right ) + {\left (b d^{2} e n r^{5} + 16 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} + 282 \, b d^{2} e n r^{2} + 396 \, b d^{2} e n r + 216 \, b d^{2} e n\right )} x^{6} \log \left (x\right ) + {\left (a d^{2} e r^{5} - 36 \, b d^{2} e n - {\left (b d^{2} e n - 16 \, a d^{2} e\right )} r^{4} + 216 \, a d^{2} e - {\left (10 \, b d^{2} e n - 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n - 282 \, a d^{2} e\right )} r^{2} - 12 \, {\left (5 \, b d^{2} e n - 33 \, a d^{2} e\right )} r\right )} x^{6}\right )} x^{r}}{36 \, {\left (r^{6} + 22 \, r^{5} + 193 \, r^{4} + 864 \, r^{3} + 2088 \, r^{2} + 2592 \, r + 1296\right )}} \]
1/36*(6*(b*d^3*r^6 + 22*b*d^3*r^5 + 193*b*d^3*r^4 + 864*b*d^3*r^3 + 2088*b *d^3*r^2 + 2592*b*d^3*r + 1296*b*d^3)*x^6*log(c) + 6*(b*d^3*n*r^6 + 22*b*d ^3*n*r^5 + 193*b*d^3*n*r^4 + 864*b*d^3*n*r^3 + 2088*b*d^3*n*r^2 + 2592*b*d ^3*n*r + 1296*b*d^3*n)*x^6*log(x) - ((b*d^3*n - 6*a*d^3)*r^6 + 22*(b*d^3*n - 6*a*d^3)*r^5 + 1296*b*d^3*n + 193*(b*d^3*n - 6*a*d^3)*r^4 - 7776*a*d^3 + 864*(b*d^3*n - 6*a*d^3)*r^3 + 2088*(b*d^3*n - 6*a*d^3)*r^2 + 2592*(b*d^3 *n - 6*a*d^3)*r)*x^6 + 4*(3*(b*e^3*r^5 + 20*b*e^3*r^4 + 153*b*e^3*r^3 + 55 8*b*e^3*r^2 + 972*b*e^3*r + 648*b*e^3)*x^6*log(c) + 3*(b*e^3*n*r^5 + 20*b* e^3*n*r^4 + 153*b*e^3*n*r^3 + 558*b*e^3*n*r^2 + 972*b*e^3*n*r + 648*b*e^3* n)*x^6*log(x) + (3*a*e^3*r^5 - 324*b*e^3*n - (b*e^3*n - 60*a*e^3)*r^4 + 19 44*a*e^3 - 9*(2*b*e^3*n - 51*a*e^3)*r^3 - 9*(13*b*e^3*n - 186*a*e^3)*r^2 - 324*(b*e^3*n - 9*a*e^3)*r)*x^6)*x^(3*r) + 27*(2*(b*d*e^2*r^5 + 19*b*d*e^2 *r^4 + 136*b*d*e^2*r^3 + 456*b*d*e^2*r^2 + 720*b*d*e^2*r + 432*b*d*e^2)*x^ 6*log(c) + 2*(b*d*e^2*n*r^5 + 19*b*d*e^2*n*r^4 + 136*b*d*e^2*n*r^3 + 456*b *d*e^2*n*r^2 + 720*b*d*e^2*n*r + 432*b*d*e^2*n)*x^6*log(x) + (2*a*d*e^2*r^ 5 - 144*b*d*e^2*n - (b*d*e^2*n - 38*a*d*e^2)*r^4 + 864*a*d*e^2 - 16*(b*d*e ^2*n - 17*a*d*e^2)*r^3 - 8*(11*b*d*e^2*n - 114*a*d*e^2)*r^2 - 96*(2*b*d*e^ 2*n - 15*a*d*e^2)*r)*x^6)*x^(2*r) + 108*((b*d^2*e*r^5 + 16*b*d^2*e*r^4 + 9 7*b*d^2*e*r^3 + 282*b*d^2*e*r^2 + 396*b*d^2*e*r + 216*b*d^2*e)*x^6*log(c) + (b*d^2*e*n*r^5 + 16*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 + 282*b*d^2*e*n*...
Leaf count of result is larger than twice the leaf count of optimal. 4100 vs. \(2 (143) = 286\).
Time = 92.42 (sec) , antiderivative size = 4100, normalized size of antiderivative = 27.89 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
Piecewise((a*d**3*x**6/6 + 3*a*d**2*e*log(c*x**n)/n - a*d*e**2/(2*x**6) - a*e**3/(12*x**12) - b*d**3*n*x**6/36 + b*d**3*x**6*log(c*x**n)/6 + 3*b*d** 2*e*log(c*x**n)**2/(2*n) - b*d*e**2*n/(12*x**6) - b*d*e**2*log(c*x**n)/(2* x**6) - b*e**3*n/(144*x**12) - b*e**3*log(c*x**n)/(12*x**12), Eq(r, -6)), (a*d**3*x**6/6 + a*d**2*e*x**3 + 3*a*d*e**2*log(c*x**n)/n - a*e**3/(3*x**3 ) - b*d**3*n*x**6/36 + b*d**3*x**6*log(c*x**n)/6 - b*d**2*e*n*x**3/3 + b*d **2*e*x**3*log(c*x**n) + 3*b*d*e**2*log(c*x**n)**2/(2*n) - b*e**3*n/(9*x** 3) - b*e**3*log(c*x**n)/(3*x**3), Eq(r, -3)), (a*d**3*x**6/6 + 3*a*d**2*e* x**4/4 + 3*a*d*e**2*x**2/2 + a*e**3*log(c*x**n)/n - b*d**3*n*x**6/36 + b*d **3*x**6*log(c*x**n)/6 - 3*b*d**2*e*n*x**4/16 + 3*b*d**2*e*x**4*log(c*x**n )/4 - 3*b*d*e**2*n*x**2/4 + 3*b*d*e**2*x**2*log(c*x**n)/2 + b*e**3*log(c*x **n)**2/(2*n), Eq(r, -2)), (6*a*d**3*r**6*x**6/(36*r**6 + 792*r**5 + 6948* r**4 + 31104*r**3 + 75168*r**2 + 93312*r + 46656) + 132*a*d**3*r**5*x**6/( 36*r**6 + 792*r**5 + 6948*r**4 + 31104*r**3 + 75168*r**2 + 93312*r + 46656 ) + 1158*a*d**3*r**4*x**6/(36*r**6 + 792*r**5 + 6948*r**4 + 31104*r**3 + 7 5168*r**2 + 93312*r + 46656) + 5184*a*d**3*r**3*x**6/(36*r**6 + 792*r**5 + 6948*r**4 + 31104*r**3 + 75168*r**2 + 93312*r + 46656) + 12528*a*d**3*r** 2*x**6/(36*r**6 + 792*r**5 + 6948*r**4 + 31104*r**3 + 75168*r**2 + 93312*r + 46656) + 15552*a*d**3*r*x**6/(36*r**6 + 792*r**5 + 6948*r**4 + 31104*r* *3 + 75168*r**2 + 93312*r + 46656) + 7776*a*d**3*x**6/(36*r**6 + 792*r*...
Time = 0.19 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.48 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{36} \, b d^{3} n x^{6} + \frac {1}{6} \, b d^{3} x^{6} \log \left (c x^{n}\right ) + \frac {1}{6} \, a d^{3} x^{6} + \frac {b e^{3} x^{3 \, r + 6} \log \left (c x^{n}\right )}{3 \, {\left (r + 2\right )}} + \frac {3 \, b d e^{2} x^{2 \, r + 6} \log \left (c x^{n}\right )}{2 \, {\left (r + 3\right )}} + \frac {3 \, b d^{2} e x^{r + 6} \log \left (c x^{n}\right )}{r + 6} - \frac {b e^{3} n x^{3 \, r + 6}}{9 \, {\left (r + 2\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 6}}{3 \, {\left (r + 2\right )}} - \frac {3 \, b d e^{2} n x^{2 \, r + 6}}{4 \, {\left (r + 3\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 6}}{2 \, {\left (r + 3\right )}} - \frac {3 \, b d^{2} e n x^{r + 6}}{{\left (r + 6\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 6}}{r + 6} \]
-1/36*b*d^3*n*x^6 + 1/6*b*d^3*x^6*log(c*x^n) + 1/6*a*d^3*x^6 + 1/3*b*e^3*x ^(3*r + 6)*log(c*x^n)/(r + 2) + 3/2*b*d*e^2*x^(2*r + 6)*log(c*x^n)/(r + 3) + 3*b*d^2*e*x^(r + 6)*log(c*x^n)/(r + 6) - 1/9*b*e^3*n*x^(3*r + 6)/(r + 2 )^2 + 1/3*a*e^3*x^(3*r + 6)/(r + 2) - 3/4*b*d*e^2*n*x^(2*r + 6)/(r + 3)^2 + 3/2*a*d*e^2*x^(2*r + 6)/(r + 3) - 3*b*d^2*e*n*x^(r + 6)/(r + 6)^2 + 3*a* d^2*e*x^(r + 6)/(r + 6)
Leaf count of result is larger than twice the leaf count of optimal. 1609 vs. \(2 (139) = 278\).
Time = 0.37 (sec) , antiderivative size = 1609, normalized size of antiderivative = 10.95 \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/36*(12*b*e^3*n*r^5*x^6*x^(3*r)*log(x) + 54*b*d*e^2*n*r^5*x^6*x^(2*r)*log (x) + 108*b*d^2*e*n*r^5*x^6*x^r*log(x) + 6*b*d^3*n*r^6*x^6*log(x) - b*d^3* n*r^6*x^6 + 12*b*e^3*r^5*x^6*x^(3*r)*log(c) + 54*b*d*e^2*r^5*x^6*x^(2*r)*l og(c) + 108*b*d^2*e*r^5*x^6*x^r*log(c) + 6*b*d^3*r^6*x^6*log(c) + 240*b*e^ 3*n*r^4*x^6*x^(3*r)*log(x) + 1026*b*d*e^2*n*r^4*x^6*x^(2*r)*log(x) + 1728* b*d^2*e*n*r^4*x^6*x^r*log(x) + 132*b*d^3*n*r^5*x^6*log(x) - 4*b*e^3*n*r^4* x^6*x^(3*r) + 12*a*e^3*r^5*x^6*x^(3*r) - 27*b*d*e^2*n*r^4*x^6*x^(2*r) + 54 *a*d*e^2*r^5*x^6*x^(2*r) - 108*b*d^2*e*n*r^4*x^6*x^r + 108*a*d^2*e*r^5*x^6 *x^r - 22*b*d^3*n*r^5*x^6 + 6*a*d^3*r^6*x^6 + 240*b*e^3*r^4*x^6*x^(3*r)*lo g(c) + 1026*b*d*e^2*r^4*x^6*x^(2*r)*log(c) + 1728*b*d^2*e*r^4*x^6*x^r*log( c) + 132*b*d^3*r^5*x^6*log(c) + 1836*b*e^3*n*r^3*x^6*x^(3*r)*log(x) + 7344 *b*d*e^2*n*r^3*x^6*x^(2*r)*log(x) + 10476*b*d^2*e*n*r^3*x^6*x^r*log(x) + 1 158*b*d^3*n*r^4*x^6*log(x) - 72*b*e^3*n*r^3*x^6*x^(3*r) + 240*a*e^3*r^4*x^ 6*x^(3*r) - 432*b*d*e^2*n*r^3*x^6*x^(2*r) + 1026*a*d*e^2*r^4*x^6*x^(2*r) - 1080*b*d^2*e*n*r^3*x^6*x^r + 1728*a*d^2*e*r^4*x^6*x^r - 193*b*d^3*n*r^4*x ^6 + 132*a*d^3*r^5*x^6 + 1836*b*e^3*r^3*x^6*x^(3*r)*log(c) + 7344*b*d*e^2* r^3*x^6*x^(2*r)*log(c) + 10476*b*d^2*e*r^3*x^6*x^r*log(c) + 1158*b*d^3*r^4 *x^6*log(c) + 6696*b*e^3*n*r^2*x^6*x^(3*r)*log(x) + 24624*b*d*e^2*n*r^2*x^ 6*x^(2*r)*log(x) + 30456*b*d^2*e*n*r^2*x^6*x^r*log(x) + 5184*b*d^3*n*r^3*x ^6*log(x) - 468*b*e^3*n*r^2*x^6*x^(3*r) + 1836*a*e^3*r^3*x^6*x^(3*r) - ...
Timed out. \[ \int x^5 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^5\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]