Integrand size = 23, antiderivative size = 149 \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d^3 n x^4-\frac {3 b d e^2 n x^{2 (2+r)}}{4 (2+r)^2}-\frac {3 b d^2 e n x^{4+r}}{(4+r)^2}-\frac {b e^3 n x^{4+3 r}}{(4+3 r)^2}+\frac {1}{4} \left (d^3 x^4+\frac {6 d e^2 x^{2 (2+r)}}{2+r}+\frac {12 d^2 e x^{4+r}}{4+r}+\frac {4 e^3 x^{4+3 r}}{4+3 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/16*b*d^3*n*x^4-3/4*b*d*e^2*n*x^(4+2*r)/(2+r)^2-3*b*d^2*e*n*x^(4+r)/(4+r )^2-b*e^3*n*x^(4+3*r)/(4+3*r)^2+1/4*(d^3*x^4+6*d*e^2*x^(4+2*r)/(2+r)+12*d^ 2*e*x^(4+r)/(4+r)+4*e^3*x^(4+3*r)/(4+3*r))*(a+b*ln(c*x^n))
Time = 0.24 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.19 \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{16} x^4 \left (b n \left (-d^3-\frac {48 d^2 e x^r}{(4+r)^2}-\frac {12 d e^2 x^{2 r}}{(2+r)^2}-\frac {16 e^3 x^{3 r}}{(4+3 r)^2}\right )+4 a \left (d^3+\frac {12 d^2 e x^r}{4+r}+\frac {6 d e^2 x^{2 r}}{2+r}+\frac {4 e^3 x^{3 r}}{4+3 r}\right )+4 b \left (d^3+\frac {12 d^2 e x^r}{4+r}+\frac {6 d e^2 x^{2 r}}{2+r}+\frac {4 e^3 x^{3 r}}{4+3 r}\right ) \log \left (c x^n\right )\right ) \]
(x^4*(b*n*(-d^3 - (48*d^2*e*x^r)/(4 + r)^2 - (12*d*e^2*x^(2*r))/(2 + r)^2 - (16*e^3*x^(3*r))/(4 + 3*r)^2) + 4*a*(d^3 + (12*d^2*e*x^r)/(4 + r) + (6*d *e^2*x^(2*r))/(2 + r) + (4*e^3*x^(3*r))/(4 + 3*r)) + 4*b*(d^3 + (12*d^2*e* x^r)/(4 + r) + (6*d*e^2*x^(2*r))/(2 + r) + (4*e^3*x^(3*r))/(4 + 3*r))*Log[ c*x^n]))/16
Time = 0.59 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{4} \left (d^3 x^4+\frac {12 d^2 e x^{r+4}}{r+4}+\frac {6 d e^2 x^{2 (r+2)}}{r+2}+\frac {4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{4} x^3 \left (\frac {12 d^2 e x^r}{r+4}+\frac {6 d e^2 x^{2 r}}{r+2}+\frac {4 e^3 x^{3 r}}{3 r+4}+d^3\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \left (d^3 x^4+\frac {12 d^2 e x^{r+4}}{r+4}+\frac {6 d e^2 x^{2 (r+2)}}{r+2}+\frac {4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \int x^3 \left (\frac {12 d^2 e x^r}{r+4}+\frac {6 d e^2 x^{2 r}}{r+2}+\frac {4 e^3 x^{3 r}}{3 r+4}+d^3\right )dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{4} \left (d^3 x^4+\frac {12 d^2 e x^{r+4}}{r+4}+\frac {6 d e^2 x^{2 (r+2)}}{r+2}+\frac {4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \int \left (\frac {4 e^3 x^{3 (r+1)}}{3 r+4}+\frac {12 d^2 e x^{r+3}}{r+4}+\frac {6 d e^2 x^{2 r+3}}{r+2}+d^3 x^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (d^3 x^4+\frac {12 d^2 e x^{r+4}}{r+4}+\frac {6 d e^2 x^{2 (r+2)}}{r+2}+\frac {4 e^3 x^{3 r+4}}{3 r+4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n \left (\frac {d^3 x^4}{4}+\frac {12 d^2 e x^{r+4}}{(r+4)^2}+\frac {3 d e^2 x^{2 (r+2)}}{(r+2)^2}+\frac {4 e^3 x^{3 r+4}}{(3 r+4)^2}\right )\) |
-1/4*(b*n*((d^3*x^4)/4 + (3*d*e^2*x^(2*(2 + r)))/(2 + r)^2 + (12*d^2*e*x^( 4 + r))/(4 + r)^2 + (4*e^3*x^(4 + 3*r))/(4 + 3*r)^2)) + ((d^3*x^4 + (6*d*e ^2*x^(2*(2 + r)))/(2 + r) + (12*d^2*e*x^(4 + r))/(4 + r) + (4*e^3*x^(4 + 3 *r))/(4 + 3*r))*(a + b*Log[c*x^n]))/4
3.4.93.3.1 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. \(1261\) vs. \(2(143)=286\).
Time = 15.46 (sec) , antiderivative size = 1262, normalized size of antiderivative = 8.47
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(1262\) |
risch | \(\text {Expression too large to display}\) | \(4027\) |
-1/16*(-4096*a*d^3*x^4-12288*e^2*d*b*ln(c*x^n)*(x^r)^2*x^4-12288*e*d^2*b*l n(c*x^n)*x^r*x^4-432*x^4*x^r*a*d^2*e*r^5-4608*x^4*x^r*a*d^2*e*r^4-18624*x^ 4*x^r*a*d^2*e*r^3-36096*x^4*x^r*a*d^2*e*r^2-33792*x^4*x^r*a*d^2*e*r+3072*x ^4*x^r*b*d^2*e*n-48*x^4*(x^r)^3*ln(c*x^n)*b*e^3*r^5-640*x^4*(x^r)^3*ln(c*x ^n)*b*e^3*r^4-3264*x^4*(x^r)^3*ln(c*x^n)*b*e^3*r^3-7936*x^4*(x^r)^3*ln(c*x ^n)*b*e^3*r^2-9216*x^4*(x^r)^3*ln(c*x^n)*b*e^3*r-216*x^4*(x^r)^2*a*d*e^2*r ^5-2736*x^4*(x^r)^2*a*d*e^2*r^4-13056*x^4*(x^r)^2*a*d*e^2*r^3-29184*x^4*(x ^r)^2*a*d*e^2*r^2-30720*x^4*(x^r)^2*a*d*e^2*r+3072*x^4*(x^r)^2*b*d*e^2*n-4 096*x^4*ln(c*x^n)*b*d^3-12288*x^4*x^r*a*d^2*e+9*x^4*b*d^3*n*r^6+132*x^4*b* d^3*n*r^5+772*x^4*b*d^3*n*r^4+2304*x^4*b*d^3*n*r^3+3712*x^4*b*d^3*n*r^2+30 72*x^4*b*d^3*n*r-12288*x^4*(x^r)^2*a*d*e^2-48*x^4*(x^r)^3*a*e^3*r^5-640*x^ 4*(x^r)^3*a*e^3*r^4-3264*x^4*(x^r)^3*a*e^3*r^3-7936*x^4*(x^r)^3*a*e^3*r^2+ 432*x^4*x^r*b*d^2*e*n*r^4+2880*x^4*x^r*b*d^2*e*n*r^3+7104*x^4*x^r*b*d^2*e* n*r^2+7680*x^4*x^r*b*d^2*e*n*r-216*x^4*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5-2736* x^4*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-13056*x^4*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3- 29184*x^4*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-30720*x^4*(x^r)^2*ln(c*x^n)*b*d*e^ 2*r-432*x^4*x^r*ln(c*x^n)*b*d^2*e*r^5-4608*x^4*x^r*ln(c*x^n)*b*d^2*e*r^4-1 8624*x^4*x^r*ln(c*x^n)*b*d^2*e*r^3-36096*x^4*x^r*ln(c*x^n)*b*d^2*e*r^2-337 92*x^4*x^r*ln(c*x^n)*b*d^2*e*r+108*x^4*(x^r)^2*b*d*e^2*n*r^4+1152*x^4*(x^r )^2*b*d*e^2*n*r^3+4224*x^4*(x^r)^2*b*d*e^2*n*r^2+6144*x^4*(x^r)^2*b*d*e...
Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (143) = 286\).
Time = 0.34 (sec) , antiderivative size = 1022, normalized size of antiderivative = 6.86 \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/16*(4*(9*b*d^3*r^6 + 132*b*d^3*r^5 + 772*b*d^3*r^4 + 2304*b*d^3*r^3 + 37 12*b*d^3*r^2 + 3072*b*d^3*r + 1024*b*d^3)*x^4*log(c) + 4*(9*b*d^3*n*r^6 + 132*b*d^3*n*r^5 + 772*b*d^3*n*r^4 + 2304*b*d^3*n*r^3 + 3712*b*d^3*n*r^2 + 3072*b*d^3*n*r + 1024*b*d^3*n)*x^4*log(x) - (9*(b*d^3*n - 4*a*d^3)*r^6 + 1 32*(b*d^3*n - 4*a*d^3)*r^5 + 1024*b*d^3*n + 772*(b*d^3*n - 4*a*d^3)*r^4 - 4096*a*d^3 + 2304*(b*d^3*n - 4*a*d^3)*r^3 + 3712*(b*d^3*n - 4*a*d^3)*r^2 + 3072*(b*d^3*n - 4*a*d^3)*r)*x^4 + 16*((3*b*e^3*r^5 + 40*b*e^3*r^4 + 204*b *e^3*r^3 + 496*b*e^3*r^2 + 576*b*e^3*r + 256*b*e^3)*x^4*log(c) + (3*b*e^3* n*r^5 + 40*b*e^3*n*r^4 + 204*b*e^3*n*r^3 + 496*b*e^3*n*r^2 + 576*b*e^3*n*r + 256*b*e^3*n)*x^4*log(x) + (3*a*e^3*r^5 - 64*b*e^3*n - (b*e^3*n - 40*a*e ^3)*r^4 + 256*a*e^3 - 12*(b*e^3*n - 17*a*e^3)*r^3 - 4*(13*b*e^3*n - 124*a* e^3)*r^2 - 96*(b*e^3*n - 6*a*e^3)*r)*x^4)*x^(3*r) + 12*(2*(9*b*d*e^2*r^5 + 114*b*d*e^2*r^4 + 544*b*d*e^2*r^3 + 1216*b*d*e^2*r^2 + 1280*b*d*e^2*r + 5 12*b*d*e^2)*x^4*log(c) + 2*(9*b*d*e^2*n*r^5 + 114*b*d*e^2*n*r^4 + 544*b*d* e^2*n*r^3 + 1216*b*d*e^2*n*r^2 + 1280*b*d*e^2*n*r + 512*b*d*e^2*n)*x^4*log (x) + (18*a*d*e^2*r^5 - 256*b*d*e^2*n - 3*(3*b*d*e^2*n - 76*a*d*e^2)*r^4 + 1024*a*d*e^2 - 32*(3*b*d*e^2*n - 34*a*d*e^2)*r^3 - 32*(11*b*d*e^2*n - 76* a*d*e^2)*r^2 - 512*(b*d*e^2*n - 5*a*d*e^2)*r)*x^4)*x^(2*r) + 48*((9*b*d^2* e*r^5 + 96*b*d^2*e*r^4 + 388*b*d^2*e*r^3 + 752*b*d^2*e*r^2 + 704*b*d^2*e*r + 256*b*d^2*e)*x^4*log(c) + (9*b*d^2*e*n*r^5 + 96*b*d^2*e*n*r^4 + 388*...
Timed out. \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.49 \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, b d^{3} n x^{4} + \frac {1}{4} \, b d^{3} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{3} x^{4} + \frac {b e^{3} x^{3 \, r + 4} \log \left (c x^{n}\right )}{3 \, r + 4} + \frac {3 \, b d e^{2} x^{2 \, r + 4} \log \left (c x^{n}\right )}{2 \, {\left (r + 2\right )}} + \frac {3 \, b d^{2} e x^{r + 4} \log \left (c x^{n}\right )}{r + 4} - \frac {b e^{3} n x^{3 \, r + 4}}{{\left (3 \, r + 4\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 4}}{3 \, r + 4} - \frac {3 \, b d e^{2} n x^{2 \, r + 4}}{4 \, {\left (r + 2\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 4}}{2 \, {\left (r + 2\right )}} - \frac {3 \, b d^{2} e n x^{r + 4}}{{\left (r + 4\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 4}}{r + 4} \]
-1/16*b*d^3*n*x^4 + 1/4*b*d^3*x^4*log(c*x^n) + 1/4*a*d^3*x^4 + b*e^3*x^(3* r + 4)*log(c*x^n)/(3*r + 4) + 3/2*b*d*e^2*x^(2*r + 4)*log(c*x^n)/(r + 2) + 3*b*d^2*e*x^(r + 4)*log(c*x^n)/(r + 4) - b*e^3*n*x^(3*r + 4)/(3*r + 4)^2 + a*e^3*x^(3*r + 4)/(3*r + 4) - 3/4*b*d*e^2*n*x^(2*r + 4)/(r + 2)^2 + 3/2* a*d*e^2*x^(2*r + 4)/(r + 2) - 3*b*d^2*e*n*x^(r + 4)/(r + 4)^2 + 3*a*d^2*e* x^(r + 4)/(r + 4)
Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (143) = 286\).
Time = 0.48 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.81 \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/16*(48*b*e^3*n*r^5*x^4*x^(3*r)*log(x) + 216*b*d*e^2*n*r^5*x^4*x^(2*r)*lo g(x) + 432*b*d^2*e*n*r^5*x^4*x^r*log(x) + 36*b*d^3*n*r^6*x^4*log(x) - 9*b* d^3*n*r^6*x^4 + 48*b*e^3*r^5*x^4*x^(3*r)*log(c) + 216*b*d*e^2*r^5*x^4*x^(2 *r)*log(c) + 432*b*d^2*e*r^5*x^4*x^r*log(c) + 36*b*d^3*r^6*x^4*log(c) + 64 0*b*e^3*n*r^4*x^4*x^(3*r)*log(x) + 2736*b*d*e^2*n*r^4*x^4*x^(2*r)*log(x) + 4608*b*d^2*e*n*r^4*x^4*x^r*log(x) + 528*b*d^3*n*r^5*x^4*log(x) - 16*b*e^3 *n*r^4*x^4*x^(3*r) + 48*a*e^3*r^5*x^4*x^(3*r) - 108*b*d*e^2*n*r^4*x^4*x^(2 *r) + 216*a*d*e^2*r^5*x^4*x^(2*r) - 432*b*d^2*e*n*r^4*x^4*x^r + 432*a*d^2* e*r^5*x^4*x^r - 132*b*d^3*n*r^5*x^4 + 36*a*d^3*r^6*x^4 + 640*b*e^3*r^4*x^4 *x^(3*r)*log(c) + 2736*b*d*e^2*r^4*x^4*x^(2*r)*log(c) + 4608*b*d^2*e*r^4*x ^4*x^r*log(c) + 528*b*d^3*r^5*x^4*log(c) + 3264*b*e^3*n*r^3*x^4*x^(3*r)*lo g(x) + 13056*b*d*e^2*n*r^3*x^4*x^(2*r)*log(x) + 18624*b*d^2*e*n*r^3*x^4*x^ r*log(x) + 3088*b*d^3*n*r^4*x^4*log(x) - 192*b*e^3*n*r^3*x^4*x^(3*r) + 640 *a*e^3*r^4*x^4*x^(3*r) - 1152*b*d*e^2*n*r^3*x^4*x^(2*r) + 2736*a*d*e^2*r^4 *x^4*x^(2*r) - 2880*b*d^2*e*n*r^3*x^4*x^r + 4608*a*d^2*e*r^4*x^4*x^r - 772 *b*d^3*n*r^4*x^4 + 528*a*d^3*r^5*x^4 + 3264*b*e^3*r^3*x^4*x^(3*r)*log(c) + 13056*b*d*e^2*r^3*x^4*x^(2*r)*log(c) + 18624*b*d^2*e*r^3*x^4*x^r*log(c) + 3088*b*d^3*r^4*x^4*log(c) + 7936*b*e^3*n*r^2*x^4*x^(3*r)*log(x) + 29184*b *d*e^2*n*r^2*x^4*x^(2*r)*log(x) + 36096*b*d^2*e*n*r^2*x^4*x^r*log(x) + 921 6*b*d^3*n*r^3*x^4*log(x) - 832*b*e^3*n*r^2*x^4*x^(3*r) + 3264*a*e^3*r^3...
Timed out. \[ \int x^3 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^3\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]