Integrand size = 23, antiderivative size = 105 \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^2 n x^5-\frac {2 b d e n x^{5+r}}{(5+r)^2}-\frac {b e^2 n x^{5+2 r}}{(5+2 r)^2}+\frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{5+r}}{5+r}+\frac {5 e^2 x^{5+2 r}}{5+2 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/25*b*d^2*n*x^5-2*b*d*e*n*x^(5+r)/(5+r)^2-b*e^2*n*x^(5+2*r)/(5+2*r)^2+1/ 5*(d^2*x^5+10*d*e*x^(5+r)/(5+r)+5*e^2*x^(5+2*r)/(5+2*r))*(a+b*ln(c*x^n))
Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18 \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{25} x^5 \left (b n \left (-d^2-\frac {50 d e x^r}{(5+r)^2}-\frac {25 e^2 x^{2 r}}{(5+2 r)^2}\right )+5 a \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right )+5 b \left (d^2+\frac {10 d e x^r}{5+r}+\frac {5 e^2 x^{2 r}}{5+2 r}\right ) \log \left (c x^n\right )\right ) \]
(x^5*(b*n*(-d^2 - (50*d*e*x^r)/(5 + r)^2 - (25*e^2*x^(2*r))/(5 + 2*r)^2) + 5*a*(d^2 + (10*d*e*x^r)/(5 + r) + (5*e^2*x^(2*r))/(5 + 2*r)) + 5*b*(d^2 + (10*d*e*x^r)/(5 + r) + (5*e^2*x^(2*r))/(5 + 2*r))*Log[c*x^n]))/25
Time = 0.38 (sec) , antiderivative size = 106, 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.
| \(\displaystyle \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2771 |
| \(\displaystyle \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{5} x^4 \left (\frac {10 d e x^r}{r+5}+\frac {5 e^2 x^{2 r}}{2 r+5}+d^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int x^4 \left (\frac {10 d e x^r}{r+5}+\frac {5 e^2 x^{2 r}}{2 r+5}+d^2\right )dx\) |
| \(\Big \downarrow \) 1691 |
| \(\displaystyle \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int \left (\frac {5 e^2 x^{2 (r+2)}}{2 r+5}+\frac {10 d e x^{r+4}}{r+5}+d^2 x^4\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (d^2 x^5+\frac {10 d e x^{r+5}}{r+5}+\frac {5 e^2 x^{2 r+5}}{2 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \left (\frac {d^2 x^5}{5}+\frac {10 d e x^{r+5}}{(r+5)^2}+\frac {5 e^2 x^{2 r+5}}{(2 r+5)^2}\right )\) |
-1/5*(b*n*((d^2*x^5)/5 + (10*d*e*x^(5 + r))/(5 + r)^2 + (5*e^2*x^(5 + 2*r) )/(5 + 2*r)^2)) + ((d^2*x^5 + (10*d*e*x^(5 + r))/(5 + r) + (5*e^2*x^(5 + 2 *r))/(5 + 2*r))*(a + b*Log[c*x^n]))/5
3.4.85.3.1 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. \(585\) vs. \(2(101)=202\).
Time = 9.10 (sec) , antiderivative size = 586, normalized size of antiderivative = 5.58
| method | result | size |
| parallelrisch | \(-\frac {-3125 x^{5} a \,d^{2}-6250 b d e \ln \left (c \,x^{n}\right ) x^{r} r \,x^{5}-200 x^{5} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}-2000 x^{5} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-625 x^{5} x^{2 r} a \,e^{2} r^{2}+625 x^{5} x^{2 r} b \,e^{2} n -3125 x^{5} e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )-2500 x^{5} x^{2 r} a \,e^{2} r -50 x^{5} x^{2 r} a \,e^{2} r^{3}-3125 x^{5} e^{2} x^{2 r} a -20 x^{5} a \,d^{2} r^{4}-300 x^{5} a \,d^{2} r^{3}-1625 x^{5} a \,d^{2} r^{2}-3750 x^{5} a \,d^{2} r +4 x^{5} b \,d^{2} n \,r^{4}+60 x^{5} b \,d^{2} n \,r^{3}+325 x^{5} b \,d^{2} n \,r^{2}+750 x^{5} b \,d^{2} n r -6250 x^{5} d e \,x^{r} b \ln \left (c \,x^{n}\right )-3125 x^{5} b \ln \left (c \,x^{n}\right ) d^{2}-3750 x^{5} \ln \left (c \,x^{n}\right ) b \,d^{2} r -6250 x^{5} d e \,x^{r} a -2500 e^{2} b \ln \left (c \,x^{n}\right ) x^{2 r} x^{5} r -50 x^{5} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}-625 x^{5} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}+25 x^{5} x^{2 r} b \,e^{2} n \,r^{2}+250 x^{5} x^{2 r} b \,e^{2} n r -20 x^{5} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-300 x^{5} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}-1625 x^{5} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}+1250 x^{5} x^{r} b d e n +200 x^{5} x^{r} b d e n \,r^{2}+1000 x^{5} x^{r} b d e n r +625 b \,d^{2} n \,x^{5}-200 x^{5} x^{r} a d e \,r^{3}-2000 x^{5} x^{r} a d e \,r^{2}-6250 x^{5} x^{r} a d e r}{25 \left (r^{2}+10 r +25\right ) \left (5+2 r \right )^{2}}\) | \(586\) |
| risch | \(\text {Expression too large to display}\) | \(1930\) |
-1/25*(-3125*x^5*a*d^2-2500*e^2*b*ln(c*x^n)*(x^r)^2*x^5*r-6250*b*d*e*ln(c* x^n)*x^r*r*x^5-625*x^5*(x^r)^2*a*e^2*r^2-200*x^5*x^r*ln(c*x^n)*b*d*e*r^3-2 000*x^5*x^r*ln(c*x^n)*b*d*e*r^2-20*x^5*a*d^2*r^4-300*x^5*a*d^2*r^3-1625*x^ 5*a*d^2*r^2-3750*x^5*a*d^2*r+625*x^5*(x^r)^2*b*e^2*n+4*x^5*b*d^2*n*r^4+60* x^5*b*d^2*n*r^3+325*x^5*b*d^2*n*r^2+750*x^5*b*d^2*n*r-6250*x^5*d*e*x^r*b*l n(c*x^n)-3125*x^5*e^2*(x^r)^2*a-3125*x^5*b*ln(c*x^n)*d^2-3750*x^5*ln(c*x^n )*b*d^2*r-6250*x^5*d*e*x^r*a-3125*x^5*e^2*(x^r)^2*b*ln(c*x^n)-2500*x^5*(x^ r)^2*a*e^2*r-50*x^5*(x^r)^2*ln(c*x^n)*b*e^2*r^3-625*x^5*(x^r)^2*ln(c*x^n)* b*e^2*r^2-50*x^5*(x^r)^2*a*e^2*r^3-20*x^5*ln(c*x^n)*b*d^2*r^4-300*x^5*ln(c *x^n)*b*d^2*r^3-1625*x^5*ln(c*x^n)*b*d^2*r^2+1250*x^5*x^r*b*d*e*n+25*x^5*( x^r)^2*b*e^2*n*r^2+250*x^5*(x^r)^2*b*e^2*n*r+200*x^5*x^r*b*d*e*n*r^2+1000* x^5*x^r*b*d*e*n*r+625*b*d^2*n*x^5-200*x^5*x^r*a*d*e*r^3-2000*x^5*x^r*a*d*e *r^2-6250*x^5*x^r*a*d*e*r)/(r^2+10*r+25)/(5+2*r)^2
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (101) = 202\).
Time = 0.30 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.73 \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {5 \, {\left (4 \, b d^{2} r^{4} + 60 \, b d^{2} r^{3} + 325 \, b d^{2} r^{2} + 750 \, b d^{2} r + 625 \, b d^{2}\right )} x^{5} \log \left (c\right ) + 5 \, {\left (4 \, b d^{2} n r^{4} + 60 \, b d^{2} n r^{3} + 325 \, b d^{2} n r^{2} + 750 \, b d^{2} n r + 625 \, b d^{2} n\right )} x^{5} \log \left (x\right ) - {\left (4 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{4} + 625 \, b d^{2} n + 60 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{3} - 3125 \, a d^{2} + 325 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r^{2} + 750 \, {\left (b d^{2} n - 5 \, a d^{2}\right )} r\right )} x^{5} + 25 \, {\left ({\left (2 \, b e^{2} r^{3} + 25 \, b e^{2} r^{2} + 100 \, b e^{2} r + 125 \, b e^{2}\right )} x^{5} \log \left (c\right ) + {\left (2 \, b e^{2} n r^{3} + 25 \, b e^{2} n r^{2} + 100 \, b e^{2} n r + 125 \, b e^{2} n\right )} x^{5} \log \left (x\right ) + {\left (2 \, a e^{2} r^{3} - 25 \, b e^{2} n + 125 \, a e^{2} - {\left (b e^{2} n - 25 \, a e^{2}\right )} r^{2} - 10 \, {\left (b e^{2} n - 10 \, a e^{2}\right )} r\right )} x^{5}\right )} x^{2 \, r} + 50 \, {\left ({\left (4 \, b d e r^{3} + 40 \, b d e r^{2} + 125 \, b d e r + 125 \, b d e\right )} x^{5} \log \left (c\right ) + {\left (4 \, b d e n r^{3} + 40 \, b d e n r^{2} + 125 \, b d e n r + 125 \, b d e n\right )} x^{5} \log \left (x\right ) + {\left (4 \, a d e r^{3} - 25 \, b d e n + 125 \, a d e - 4 \, {\left (b d e n - 10 \, a d e\right )} r^{2} - 5 \, {\left (4 \, b d e n - 25 \, a d e\right )} r\right )} x^{5}\right )} x^{r}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \]
1/25*(5*(4*b*d^2*r^4 + 60*b*d^2*r^3 + 325*b*d^2*r^2 + 750*b*d^2*r + 625*b* d^2)*x^5*log(c) + 5*(4*b*d^2*n*r^4 + 60*b*d^2*n*r^3 + 325*b*d^2*n*r^2 + 75 0*b*d^2*n*r + 625*b*d^2*n)*x^5*log(x) - (4*(b*d^2*n - 5*a*d^2)*r^4 + 625*b *d^2*n + 60*(b*d^2*n - 5*a*d^2)*r^3 - 3125*a*d^2 + 325*(b*d^2*n - 5*a*d^2) *r^2 + 750*(b*d^2*n - 5*a*d^2)*r)*x^5 + 25*((2*b*e^2*r^3 + 25*b*e^2*r^2 + 100*b*e^2*r + 125*b*e^2)*x^5*log(c) + (2*b*e^2*n*r^3 + 25*b*e^2*n*r^2 + 10 0*b*e^2*n*r + 125*b*e^2*n)*x^5*log(x) + (2*a*e^2*r^3 - 25*b*e^2*n + 125*a* e^2 - (b*e^2*n - 25*a*e^2)*r^2 - 10*(b*e^2*n - 10*a*e^2)*r)*x^5)*x^(2*r) + 50*((4*b*d*e*r^3 + 40*b*d*e*r^2 + 125*b*d*e*r + 125*b*d*e)*x^5*log(c) + ( 4*b*d*e*n*r^3 + 40*b*d*e*n*r^2 + 125*b*d*e*n*r + 125*b*d*e*n)*x^5*log(x) + (4*a*d*e*r^3 - 25*b*d*e*n + 125*a*d*e - 4*(b*d*e*n - 10*a*d*e)*r^2 - 5*(4 *b*d*e*n - 25*a*d*e)*r)*x^5)*x^r)/(4*r^4 + 60*r^3 + 325*r^2 + 750*r + 625)
Timed out. \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.45 \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, b d^{2} n x^{5} + \frac {1}{5} \, b d^{2} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{2} x^{5} + \frac {b e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {2 \, b d e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {2 \, b d e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {2 \, a d e x^{r + 5}}{r + 5} \]
-1/25*b*d^2*n*x^5 + 1/5*b*d^2*x^5*log(c*x^n) + 1/5*a*d^2*x^5 + b*e^2*x^(2* r + 5)*log(c*x^n)/(2*r + 5) + 2*b*d*e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^2 *n*x^(2*r + 5)/(2*r + 5)^2 + a*e^2*x^(2*r + 5)/(2*r + 5) - 2*b*d*e*n*x^(r + 5)/(r + 5)^2 + 2*a*d*e*x^(r + 5)/(r + 5)
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (101) = 202\).
Time = 0.35 (sec) , antiderivative size = 746, normalized size of antiderivative = 7.10 \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {50 \, b e^{2} n r^{3} x^{5} x^{2 \, r} \log \left (x\right ) + 200 \, b d e n r^{3} x^{5} x^{r} \log \left (x\right ) + 20 \, b d^{2} n r^{4} x^{5} \log \left (x\right ) - 4 \, b d^{2} n r^{4} x^{5} + 50 \, b e^{2} r^{3} x^{5} x^{2 \, r} \log \left (c\right ) + 200 \, b d e r^{3} x^{5} x^{r} \log \left (c\right ) + 20 \, b d^{2} r^{4} x^{5} \log \left (c\right ) + 625 \, b e^{2} n r^{2} x^{5} x^{2 \, r} \log \left (x\right ) + 2000 \, b d e n r^{2} x^{5} x^{r} \log \left (x\right ) + 300 \, b d^{2} n r^{3} x^{5} \log \left (x\right ) - 25 \, b e^{2} n r^{2} x^{5} x^{2 \, r} + 50 \, a e^{2} r^{3} x^{5} x^{2 \, r} - 200 \, b d e n r^{2} x^{5} x^{r} + 200 \, a d e r^{3} x^{5} x^{r} - 60 \, b d^{2} n r^{3} x^{5} + 20 \, a d^{2} r^{4} x^{5} + 625 \, b e^{2} r^{2} x^{5} x^{2 \, r} \log \left (c\right ) + 2000 \, b d e r^{2} x^{5} x^{r} \log \left (c\right ) + 300 \, b d^{2} r^{3} x^{5} \log \left (c\right ) + 2500 \, b e^{2} n r x^{5} x^{2 \, r} \log \left (x\right ) + 6250 \, b d e n r x^{5} x^{r} \log \left (x\right ) + 1625 \, b d^{2} n r^{2} x^{5} \log \left (x\right ) - 250 \, b e^{2} n r x^{5} x^{2 \, r} + 625 \, a e^{2} r^{2} x^{5} x^{2 \, r} - 1000 \, b d e n r x^{5} x^{r} + 2000 \, a d e r^{2} x^{5} x^{r} - 325 \, b d^{2} n r^{2} x^{5} + 300 \, a d^{2} r^{3} x^{5} + 2500 \, b e^{2} r x^{5} x^{2 \, r} \log \left (c\right ) + 6250 \, b d e r x^{5} x^{r} \log \left (c\right ) + 1625 \, b d^{2} r^{2} x^{5} \log \left (c\right ) + 3125 \, b e^{2} n x^{5} x^{2 \, r} \log \left (x\right ) + 6250 \, b d e n x^{5} x^{r} \log \left (x\right ) + 3750 \, b d^{2} n r x^{5} \log \left (x\right ) - 625 \, b e^{2} n x^{5} x^{2 \, r} + 2500 \, a e^{2} r x^{5} x^{2 \, r} - 1250 \, b d e n x^{5} x^{r} + 6250 \, a d e r x^{5} x^{r} - 750 \, b d^{2} n r x^{5} + 1625 \, a d^{2} r^{2} x^{5} + 3125 \, b e^{2} x^{5} x^{2 \, r} \log \left (c\right ) + 6250 \, b d e x^{5} x^{r} \log \left (c\right ) + 3750 \, b d^{2} r x^{5} \log \left (c\right ) + 3125 \, b d^{2} n x^{5} \log \left (x\right ) + 3125 \, a e^{2} x^{5} x^{2 \, r} + 6250 \, a d e x^{5} x^{r} - 625 \, b d^{2} n x^{5} + 3750 \, a d^{2} r x^{5} + 3125 \, b d^{2} x^{5} \log \left (c\right ) + 3125 \, a d^{2} x^{5}}{25 \, {\left (4 \, r^{4} + 60 \, r^{3} + 325 \, r^{2} + 750 \, r + 625\right )}} \]
1/25*(50*b*e^2*n*r^3*x^5*x^(2*r)*log(x) + 200*b*d*e*n*r^3*x^5*x^r*log(x) + 20*b*d^2*n*r^4*x^5*log(x) - 4*b*d^2*n*r^4*x^5 + 50*b*e^2*r^3*x^5*x^(2*r)* log(c) + 200*b*d*e*r^3*x^5*x^r*log(c) + 20*b*d^2*r^4*x^5*log(c) + 625*b*e^ 2*n*r^2*x^5*x^(2*r)*log(x) + 2000*b*d*e*n*r^2*x^5*x^r*log(x) + 300*b*d^2*n *r^3*x^5*log(x) - 25*b*e^2*n*r^2*x^5*x^(2*r) + 50*a*e^2*r^3*x^5*x^(2*r) - 200*b*d*e*n*r^2*x^5*x^r + 200*a*d*e*r^3*x^5*x^r - 60*b*d^2*n*r^3*x^5 + 20* a*d^2*r^4*x^5 + 625*b*e^2*r^2*x^5*x^(2*r)*log(c) + 2000*b*d*e*r^2*x^5*x^r* log(c) + 300*b*d^2*r^3*x^5*log(c) + 2500*b*e^2*n*r*x^5*x^(2*r)*log(x) + 62 50*b*d*e*n*r*x^5*x^r*log(x) + 1625*b*d^2*n*r^2*x^5*log(x) - 250*b*e^2*n*r* x^5*x^(2*r) + 625*a*e^2*r^2*x^5*x^(2*r) - 1000*b*d*e*n*r*x^5*x^r + 2000*a* d*e*r^2*x^5*x^r - 325*b*d^2*n*r^2*x^5 + 300*a*d^2*r^3*x^5 + 2500*b*e^2*r*x ^5*x^(2*r)*log(c) + 6250*b*d*e*r*x^5*x^r*log(c) + 1625*b*d^2*r^2*x^5*log(c ) + 3125*b*e^2*n*x^5*x^(2*r)*log(x) + 6250*b*d*e*n*x^5*x^r*log(x) + 3750*b *d^2*n*r*x^5*log(x) - 625*b*e^2*n*x^5*x^(2*r) + 2500*a*e^2*r*x^5*x^(2*r) - 1250*b*d*e*n*x^5*x^r + 6250*a*d*e*r*x^5*x^r - 750*b*d^2*n*r*x^5 + 1625*a* d^2*r^2*x^5 + 3125*b*e^2*x^5*x^(2*r)*log(c) + 6250*b*d*e*x^5*x^r*log(c) + 3750*b*d^2*r*x^5*log(c) + 3125*b*d^2*n*x^5*log(x) + 3125*a*e^2*x^5*x^(2*r) + 6250*a*d*e*x^5*x^r - 625*b*d^2*n*x^5 + 3750*a*d^2*r*x^5 + 3125*b*d^2*x^ 5*log(c) + 3125*a*d^2*x^5)/(4*r^4 + 60*r^3 + 325*r^2 + 750*r + 625)
Timed out. \[ \int x^4 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^4\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]