Integrand size = 23, antiderivative size = 105 \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} b d^2 n x^3-\frac {2 b d e n x^{3+r}}{(3+r)^2}-\frac {b e^2 n x^{3+2 r}}{(3+2 r)^2}+\frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{3+r}}{3+r}+\frac {3 e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/9*b*d^2*n*x^3-2*b*d*e*n*x^(3+r)/(3+r)^2-b*e^2*n*x^(3+2*r)/(3+2*r)^2+1/3 *(d^2*x^3+6*d*e*x^(3+r)/(3+r)+3*e^2*x^(3+2*r)/(3+2*r))*(a+b*ln(c*x^n))
Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.18 \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{9} x^3 \left (b n \left (-d^2-\frac {18 d e x^r}{(3+r)^2}-\frac {9 e^2 x^{2 r}}{(3+2 r)^2}\right )+3 a \left (d^2+\frac {6 d e x^r}{3+r}+\frac {3 e^2 x^{2 r}}{3+2 r}\right )+3 b \left (d^2+\frac {6 d e x^r}{3+r}+\frac {3 e^2 x^{2 r}}{3+2 r}\right ) \log \left (c x^n\right )\right ) \]
(x^3*(b*n*(-d^2 - (18*d*e*x^r)/(3 + r)^2 - (9*e^2*x^(2*r))/(3 + 2*r)^2) + 3*a*(d^2 + (6*d*e*x^r)/(3 + r) + (3*e^2*x^(2*r))/(3 + 2*r)) + 3*b*(d^2 + ( 6*d*e*x^r)/(3 + r) + (3*e^2*x^(2*r))/(3 + 2*r))*Log[c*x^n]))/9
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.
| \(\displaystyle \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2771 |
| \(\displaystyle \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{3} x^2 \left (\frac {6 d e x^r}{r+3}+\frac {3 e^2 x^{2 r}}{2 r+3}+d^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int x^2 \left (\frac {6 d e x^r}{r+3}+\frac {3 e^2 x^{2 r}}{2 r+3}+d^2\right )dx\) |
| \(\Big \downarrow \) 1691 |
| \(\displaystyle \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int \left (\frac {3 e^2 x^{2 (r+1)}}{2 r+3}+\frac {6 d e x^{r+2}}{r+3}+d^2 x^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (d^2 x^3+\frac {6 d e x^{r+3}}{r+3}+\frac {3 e^2 x^{2 r+3}}{2 r+3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \left (\frac {d^2 x^3}{3}+\frac {6 d e x^{r+3}}{(r+3)^2}+\frac {3 e^2 x^{2 r+3}}{(2 r+3)^2}\right )\) |
-1/3*(b*n*((d^2*x^3)/3 + (6*d*e*x^(3 + r))/(3 + r)^2 + (3*e^2*x^(3 + 2*r)) /(3 + 2*r)^2)) + ((d^2*x^3 + (6*d*e*x^(3 + r))/(3 + r) + (3*e^2*x^(3 + 2*r ))/(3 + 2*r))*(a + b*Log[c*x^n]))/3
3.4.86.3.1 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. \(580\) vs. \(2(101)=202\).
Time = 3.03 (sec) , antiderivative size = 581, normalized size of antiderivative = 5.53
| method | result | size |
| parallelrisch | \(-\frac {-72 x^{3} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{3}-432 x^{3} x^{r} \ln \left (c \,x^{n}\right ) b d e \,r^{2}-810 x^{3} x^{r} \ln \left (c \,x^{n}\right ) b d e r -72 x^{3} x^{r} a d e \,r^{3}-432 x^{3} x^{r} a d e \,r^{2}-810 x^{3} x^{r} a d e r +72 x^{3} x^{r} b d e n \,r^{2}+216 x^{3} x^{r} b d e n r -486 x^{3} d e \,x^{r} b \ln \left (c \,x^{n}\right )-243 x^{3} b \ln \left (c \,x^{n}\right ) d^{2}-108 x^{3} a \,d^{2} r^{3}-351 x^{3} a \,d^{2} r^{2}-486 x^{3} a \,d^{2} r +162 x^{3} x^{r} b d e n -12 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{4}-108 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{3}-243 a \,d^{2} x^{3}-486 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{2} r -486 x^{3} d e \,x^{r} a -18 x^{3} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{3}-135 x^{3} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r^{2}-324 x^{3} x^{2 r} \ln \left (c \,x^{n}\right ) b \,e^{2} r +9 x^{3} x^{2 r} b \,e^{2} n \,r^{2}+54 x^{3} x^{2 r} b \,e^{2} n r -351 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{2} r^{2}-12 x^{3} a \,d^{2} r^{4}-243 x^{3} e^{2} x^{2 r} a +81 b \,d^{2} n \,x^{3}+4 x^{3} b \,d^{2} n \,r^{4}+36 x^{3} b \,d^{2} n \,r^{3}+117 x^{3} b \,d^{2} n \,r^{2}+162 x^{3} b \,d^{2} n r -324 x^{3} x^{2 r} a \,e^{2} r -135 x^{3} x^{2 r} a \,e^{2} r^{2}-243 x^{3} e^{2} x^{2 r} b \ln \left (c \,x^{n}\right )-18 x^{3} x^{2 r} a \,e^{2} r^{3}+81 x^{3} x^{2 r} b \,e^{2} n}{9 \left (3+2 r \right )^{2} \left (3+r \right )^{2}}\) | \(581\) |
| risch | \(\text {Expression too large to display}\) | \(1930\) |
-1/9*(-135*x^3*(x^r)^2*ln(c*x^n)*b*e^2*r^2-72*x^3*x^r*ln(c*x^n)*b*d*e*r^3- 432*x^3*x^r*ln(c*x^n)*b*d*e*r^2-810*x^3*x^r*ln(c*x^n)*b*d*e*r-324*x^3*(x^r )^2*a*e^2*r-324*x^3*(x^r)^2*ln(c*x^n)*b*e^2*r-72*x^3*x^r*a*d*e*r^3-432*x^3 *x^r*a*d*e*r^2-810*x^3*x^r*a*d*e*r+72*x^3*x^r*b*d*e*n*r^2+216*x^3*x^r*b*d* e*n*r-486*x^3*d*e*x^r*b*ln(c*x^n)-243*x^3*e^2*(x^r)^2*a-243*x^3*b*ln(c*x^n )*d^2-135*x^3*(x^r)^2*a*e^2*r^2-108*x^3*a*d^2*r^3-351*x^3*a*d^2*r^2-486*x^ 3*a*d^2*r+162*x^3*x^r*b*d*e*n+9*x^3*(x^r)^2*b*e^2*n*r^2+54*x^3*(x^r)^2*b*e ^2*n*r-12*x^3*ln(c*x^n)*b*d^2*r^4-108*x^3*ln(c*x^n)*b*d^2*r^3-243*x^3*e^2* (x^r)^2*b*ln(c*x^n)-243*a*d^2*x^3-486*x^3*ln(c*x^n)*b*d^2*r-18*x^3*(x^r)^2 *a*e^2*r^3-486*x^3*d*e*x^r*a-18*x^3*(x^r)^2*ln(c*x^n)*b*e^2*r^3-351*x^3*ln (c*x^n)*b*d^2*r^2-12*x^3*a*d^2*r^4+81*b*d^2*n*x^3+81*x^3*(x^r)^2*b*e^2*n+4 *x^3*b*d^2*n*r^4+36*x^3*b*d^2*n*r^3+117*x^3*b*d^2*n*r^2+162*x^3*b*d^2*n*r) /(3+2*r)^2/(3+r)^2
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (101) = 202\).
Time = 0.28 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.73 \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (4 \, b d^{2} r^{4} + 36 \, b d^{2} r^{3} + 117 \, b d^{2} r^{2} + 162 \, b d^{2} r + 81 \, b d^{2}\right )} x^{3} \log \left (c\right ) + 3 \, {\left (4 \, b d^{2} n r^{4} + 36 \, b d^{2} n r^{3} + 117 \, b d^{2} n r^{2} + 162 \, b d^{2} n r + 81 \, b d^{2} n\right )} x^{3} \log \left (x\right ) - {\left (4 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{4} + 81 \, b d^{2} n + 36 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{3} - 243 \, a d^{2} + 117 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r^{2} + 162 \, {\left (b d^{2} n - 3 \, a d^{2}\right )} r\right )} x^{3} + 9 \, {\left ({\left (2 \, b e^{2} r^{3} + 15 \, b e^{2} r^{2} + 36 \, b e^{2} r + 27 \, b e^{2}\right )} x^{3} \log \left (c\right ) + {\left (2 \, b e^{2} n r^{3} + 15 \, b e^{2} n r^{2} + 36 \, b e^{2} n r + 27 \, b e^{2} n\right )} x^{3} \log \left (x\right ) + {\left (2 \, a e^{2} r^{3} - 9 \, b e^{2} n + 27 \, a e^{2} - {\left (b e^{2} n - 15 \, a e^{2}\right )} r^{2} - 6 \, {\left (b e^{2} n - 6 \, a e^{2}\right )} r\right )} x^{3}\right )} x^{2 \, r} + 18 \, {\left ({\left (4 \, b d e r^{3} + 24 \, b d e r^{2} + 45 \, b d e r + 27 \, b d e\right )} x^{3} \log \left (c\right ) + {\left (4 \, b d e n r^{3} + 24 \, b d e n r^{2} + 45 \, b d e n r + 27 \, b d e n\right )} x^{3} \log \left (x\right ) + {\left (4 \, a d e r^{3} - 9 \, b d e n + 27 \, a d e - 4 \, {\left (b d e n - 6 \, a d e\right )} r^{2} - 3 \, {\left (4 \, b d e n - 15 \, a d e\right )} r\right )} x^{3}\right )} x^{r}}{9 \, {\left (4 \, r^{4} + 36 \, r^{3} + 117 \, r^{2} + 162 \, r + 81\right )}} \]
1/9*(3*(4*b*d^2*r^4 + 36*b*d^2*r^3 + 117*b*d^2*r^2 + 162*b*d^2*r + 81*b*d^ 2)*x^3*log(c) + 3*(4*b*d^2*n*r^4 + 36*b*d^2*n*r^3 + 117*b*d^2*n*r^2 + 162* b*d^2*n*r + 81*b*d^2*n)*x^3*log(x) - (4*(b*d^2*n - 3*a*d^2)*r^4 + 81*b*d^2 *n + 36*(b*d^2*n - 3*a*d^2)*r^3 - 243*a*d^2 + 117*(b*d^2*n - 3*a*d^2)*r^2 + 162*(b*d^2*n - 3*a*d^2)*r)*x^3 + 9*((2*b*e^2*r^3 + 15*b*e^2*r^2 + 36*b*e ^2*r + 27*b*e^2)*x^3*log(c) + (2*b*e^2*n*r^3 + 15*b*e^2*n*r^2 + 36*b*e^2*n *r + 27*b*e^2*n)*x^3*log(x) + (2*a*e^2*r^3 - 9*b*e^2*n + 27*a*e^2 - (b*e^2 *n - 15*a*e^2)*r^2 - 6*(b*e^2*n - 6*a*e^2)*r)*x^3)*x^(2*r) + 18*((4*b*d*e* r^3 + 24*b*d*e*r^2 + 45*b*d*e*r + 27*b*d*e)*x^3*log(c) + (4*b*d*e*n*r^3 + 24*b*d*e*n*r^2 + 45*b*d*e*n*r + 27*b*d*e*n)*x^3*log(x) + (4*a*d*e*r^3 - 9* b*d*e*n + 27*a*d*e - 4*(b*d*e*n - 6*a*d*e)*r^2 - 3*(4*b*d*e*n - 15*a*d*e)* r)*x^3)*x^r)/(4*r^4 + 36*r^3 + 117*r^2 + 162*r + 81)
Time = 83.39 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.29 \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{2} x^{3}}{3} + 2 a d e \left (\begin {cases} \frac {x^{3} x^{r}}{r + 3} & \text {for}\: r \neq -3 \\x^{3} x^{r} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) + a e^{2} \left (\begin {cases} \frac {x^{3} x^{2 r}}{2 r + 3} & \text {for}\: r \neq - \frac {3}{2} \\x^{3} x^{2 r} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{2} n x^{3}}{9} + \frac {b d^{2} x^{3} \log {\left (c x^{n} \right )}}{3} - 2 b d e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r + 3}}{r + 3} & \text {for}\: r \neq -3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r + 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq -3 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 b d e \left (\begin {cases} \frac {x^{r + 3}}{r + 3} & \text {for}\: r \neq -3 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r + 3}}{2 r + 3} & \text {for}\: r \neq - \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r + 3} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq - \frac {3}{2} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{2} \left (\begin {cases} \frac {x^{2 r + 3}}{2 r + 3} & \text {for}\: r \neq - \frac {3}{2} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
a*d**2*x**3/3 + 2*a*d*e*Piecewise((x**3*x**r/(r + 3), Ne(r, -3)), (x**3*x* *r*log(x), True)) + a*e**2*Piecewise((x**3*x**(2*r)/(2*r + 3), Ne(r, -3/2) ), (x**3*x**(2*r)*log(x), True)) - b*d**2*n*x**3/9 + b*d**2*x**3*log(c*x** n)/3 - 2*b*d*e*n*Piecewise((Piecewise((x**(r + 3)/(r + 3), Ne(r, -3)), (lo g(x), True))/(r + 3), (r > -oo) & (r < oo) & Ne(r, -3)), (log(x)**2/2, Tru e)) + 2*b*d*e*Piecewise((x**(r + 3)/(r + 3), Ne(r, -3)), (log(x), True))*l og(c*x**n) - b*e**2*n*Piecewise((Piecewise((x**(2*r + 3)/(2*r + 3), Ne(r, -3/2)), (log(x), True))/(2*r + 3), (r > -oo) & (r < oo) & Ne(r, -3/2)), (l og(x)**2/2, True)) + b*e**2*Piecewise((x**(2*r + 3)/(2*r + 3), Ne(r, -3/2) ), (log(x), True))*log(c*x**n)
Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.45 \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, b d^{2} n x^{3} + \frac {1}{3} \, b d^{2} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{2} x^{3} + \frac {b e^{2} x^{2 \, r + 3} \log \left (c x^{n}\right )}{2 \, r + 3} + \frac {2 \, b d e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e^{2} n x^{2 \, r + 3}}{{\left (2 \, r + 3\right )}^{2}} + \frac {a e^{2} x^{2 \, r + 3}}{2 \, r + 3} - \frac {2 \, b d e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {2 \, a d e x^{r + 3}}{r + 3} \]
-1/9*b*d^2*n*x^3 + 1/3*b*d^2*x^3*log(c*x^n) + 1/3*a*d^2*x^3 + b*e^2*x^(2*r + 3)*log(c*x^n)/(2*r + 3) + 2*b*d*e*x^(r + 3)*log(c*x^n)/(r + 3) - b*e^2* n*x^(2*r + 3)/(2*r + 3)^2 + a*e^2*x^(2*r + 3)/(2*r + 3) - 2*b*d*e*n*x^(r + 3)/(r + 3)^2 + 2*a*d*e*x^(r + 3)/(r + 3)
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (101) = 202\).
Time = 0.37 (sec) , antiderivative size = 746, normalized size of antiderivative = 7.10 \[ \int x^2 \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^{3} x^{2 \, r} \log \left (x\right ) + 72 \, b d e n r^{3} x^{3} x^{r} \log \left (x\right ) + 12 \, b d^{2} n r^{4} x^{3} \log \left (x\right ) - 4 \, b d^{2} n r^{4} x^{3} + 18 \, b e^{2} r^{3} x^{3} x^{2 \, r} \log \left (c\right ) + 72 \, b d e r^{3} x^{3} x^{r} \log \left (c\right ) + 12 \, b d^{2} r^{4} x^{3} \log \left (c\right ) + 135 \, b e^{2} n r^{2} x^{3} x^{2 \, r} \log \left (x\right ) + 432 \, b d e n r^{2} x^{3} x^{r} \log \left (x\right ) + 108 \, b d^{2} n r^{3} x^{3} \log \left (x\right ) - 9 \, b e^{2} n r^{2} x^{3} x^{2 \, r} + 18 \, a e^{2} r^{3} x^{3} x^{2 \, r} - 72 \, b d e n r^{2} x^{3} x^{r} + 72 \, a d e r^{3} x^{3} x^{r} - 36 \, b d^{2} n r^{3} x^{3} + 12 \, a d^{2} r^{4} x^{3} + 135 \, b e^{2} r^{2} x^{3} x^{2 \, r} \log \left (c\right ) + 432 \, b d e r^{2} x^{3} x^{r} \log \left (c\right ) + 108 \, b d^{2} r^{3} x^{3} \log \left (c\right ) + 324 \, b e^{2} n r x^{3} x^{2 \, r} \log \left (x\right ) + 810 \, b d e n r x^{3} x^{r} \log \left (x\right ) + 351 \, b d^{2} n r^{2} x^{3} \log \left (x\right ) - 54 \, b e^{2} n r x^{3} x^{2 \, r} + 135 \, a e^{2} r^{2} x^{3} x^{2 \, r} - 216 \, b d e n r x^{3} x^{r} + 432 \, a d e r^{2} x^{3} x^{r} - 117 \, b d^{2} n r^{2} x^{3} + 108 \, a d^{2} r^{3} x^{3} + 324 \, b e^{2} r x^{3} x^{2 \, r} \log \left (c\right ) + 810 \, b d e r x^{3} x^{r} \log \left (c\right ) + 351 \, b d^{2} r^{2} x^{3} \log \left (c\right ) + 243 \, b e^{2} n x^{3} x^{2 \, r} \log \left (x\right ) + 486 \, b d e n x^{3} x^{r} \log \left (x\right ) + 486 \, b d^{2} n r x^{3} \log \left (x\right ) - 81 \, b e^{2} n x^{3} x^{2 \, r} + 324 \, a e^{2} r x^{3} x^{2 \, r} - 162 \, b d e n x^{3} x^{r} + 810 \, a d e r x^{3} x^{r} - 162 \, b d^{2} n r x^{3} + 351 \, a d^{2} r^{2} x^{3} + 243 \, b e^{2} x^{3} x^{2 \, r} \log \left (c\right ) + 486 \, b d e x^{3} x^{r} \log \left (c\right ) + 486 \, b d^{2} r x^{3} \log \left (c\right ) + 243 \, b d^{2} n x^{3} \log \left (x\right ) + 243 \, a e^{2} x^{3} x^{2 \, r} + 486 \, a d e x^{3} x^{r} - 81 \, b d^{2} n x^{3} + 486 \, a d^{2} r x^{3} + 243 \, b d^{2} x^{3} \log \left (c\right ) + 243 \, a d^{2} x^{3}}{9 \, {\left (4 \, r^{4} + 36 \, r^{3} + 117 \, r^{2} + 162 \, r + 81\right )}} \]
1/9*(18*b*e^2*n*r^3*x^3*x^(2*r)*log(x) + 72*b*d*e*n*r^3*x^3*x^r*log(x) + 1 2*b*d^2*n*r^4*x^3*log(x) - 4*b*d^2*n*r^4*x^3 + 18*b*e^2*r^3*x^3*x^(2*r)*lo g(c) + 72*b*d*e*r^3*x^3*x^r*log(c) + 12*b*d^2*r^4*x^3*log(c) + 135*b*e^2*n *r^2*x^3*x^(2*r)*log(x) + 432*b*d*e*n*r^2*x^3*x^r*log(x) + 108*b*d^2*n*r^3 *x^3*log(x) - 9*b*e^2*n*r^2*x^3*x^(2*r) + 18*a*e^2*r^3*x^3*x^(2*r) - 72*b* d*e*n*r^2*x^3*x^r + 72*a*d*e*r^3*x^3*x^r - 36*b*d^2*n*r^3*x^3 + 12*a*d^2*r ^4*x^3 + 135*b*e^2*r^2*x^3*x^(2*r)*log(c) + 432*b*d*e*r^2*x^3*x^r*log(c) + 108*b*d^2*r^3*x^3*log(c) + 324*b*e^2*n*r*x^3*x^(2*r)*log(x) + 810*b*d*e*n *r*x^3*x^r*log(x) + 351*b*d^2*n*r^2*x^3*log(x) - 54*b*e^2*n*r*x^3*x^(2*r) + 135*a*e^2*r^2*x^3*x^(2*r) - 216*b*d*e*n*r*x^3*x^r + 432*a*d*e*r^2*x^3*x^ r - 117*b*d^2*n*r^2*x^3 + 108*a*d^2*r^3*x^3 + 324*b*e^2*r*x^3*x^(2*r)*log( c) + 810*b*d*e*r*x^3*x^r*log(c) + 351*b*d^2*r^2*x^3*log(c) + 243*b*e^2*n*x ^3*x^(2*r)*log(x) + 486*b*d*e*n*x^3*x^r*log(x) + 486*b*d^2*n*r*x^3*log(x) - 81*b*e^2*n*x^3*x^(2*r) + 324*a*e^2*r*x^3*x^(2*r) - 162*b*d*e*n*x^3*x^r + 810*a*d*e*r*x^3*x^r - 162*b*d^2*n*r*x^3 + 351*a*d^2*r^2*x^3 + 243*b*e^2*x ^3*x^(2*r)*log(c) + 486*b*d*e*x^3*x^r*log(c) + 486*b*d^2*r*x^3*log(c) + 24 3*b*d^2*n*x^3*log(x) + 243*a*e^2*x^3*x^(2*r) + 486*a*d*e*x^3*x^r - 81*b*d^ 2*n*x^3 + 486*a*d^2*r*x^3 + 243*b*d^2*x^3*log(c) + 243*a*d^2*x^3)/(4*r^4 + 36*r^3 + 117*r^2 + 162*r + 81)
Timed out. \[ \int x^2 \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]