Integrand size = 23, antiderivative size = 191 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d^3 n}{16 x^4}-\frac {3 b d e^2 n x^{-2 (2-r)}}{4 (2-r)^2}-\frac {3 b d^2 e n x^{-4+r}}{(4-r)^2}-\frac {b e^3 n x^{-4+3 r}}{(4-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {3 d^2 e x^{-4+r} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {e^3 x^{-4+3 r} \left (a+b \log \left (c x^n\right )\right )}{4-3 r} \]
-1/16*b*d^3*n/x^4-3/4*b*d*e^2*n/(2-r)^2/(x^(4-2*r))-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*(a+b*ln(c*x^n))/x^4-3/2*d*e^2* (a+b*ln(c*x^n))/(2-r)/(x^(4-2*r))-3*d^2*e*x^(-4+r)*(a+b*ln(c*x^n))/(4-r)-e ^3*x^(-4+3*r)*(a+b*ln(c*x^n))/(4-3*r)
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\frac {b n \left (-d^3-\frac {48 d^2 e x^r}{(-4+r)^2}-\frac {12 d e^2 x^{2 r}}{(-2+r)^2}-\frac {16 e^3 x^{3 r}}{(4-3 r)^2}\right )+a \left (-4 d^3+\frac {48 d^2 e x^r}{-4+r}+\frac {24 d e^2 x^{2 r}}{-2+r}+\frac {16 e^3 x^{3 r}}{-4+3 r}\right )+4 b \left (-d^3+\frac {12 d^2 e x^r}{-4+r}+\frac {6 d e^2 x^{2 r}}{-2+r}+\frac {4 e^3 x^{3 r}}{-4+3 r}\right ) \log \left (c x^n\right )}{16 x^4} \]
(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) + a*(-4*d^3 + (48*d^2*e*x^r)/(-4 + r) + (24*d *e^2*x^(2*r))/(-2 + r) + (16*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*x^4)
Time = 0.63 (sec) , antiderivative size = 188, 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 = {2772, 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.
| \(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int -\frac {\frac {12 d^2 e x^r}{4-r}+\frac {6 d e^2 x^{2 r}}{2-r}+\frac {4 e^3 x^{3 r}}{4-3 r}+d^3}{4 x^5}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} b n \int \frac {\frac {12 d^2 e x^r}{4-r}+\frac {6 d e^2 x^{2 r}}{2-r}+\frac {4 e^3 x^{3 r}}{4-3 r}+d^3}{x^5}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{4} b n \int \left (-\frac {12 d^2 e x^{r-5}}{r-4}-\frac {6 d e^2 x^{2 r-5}}{r-2}-\frac {4 e^3 x^{3 r-5}}{3 r-4}+\frac {d^3}{x^5}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {3 d^2 e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac {3 d e^2 x^{-2 (2-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (2-r)}-\frac {e^3 x^{3 r-4} \left (a+b \log \left (c x^n\right )\right )}{4-3 r}+\frac {1}{4} b n \left (-\frac {d^3}{4 x^4}-\frac {12 d^2 e x^{r-4}}{(4-r)^2}-\frac {3 d e^2 x^{-2 (2-r)}}{(2-r)^2}-\frac {4 e^3 x^{3 r-4}}{(4-3 r)^2}\right )\) |
(b*n*(-1/4*d^3/x^4 - (3*d*e^2)/((2 - r)^2*x^(2*(2 - r))) - (12*d^2*e*x^(-4 + r))/(4 - r)^2 - (4*e^3*x^(-4 + 3*r))/(4 - 3*r)^2))/4 - (d^3*(a + b*Log[ c*x^n]))/(4*x^4) - (3*d*e^2*(a + b*Log[c*x^n]))/(2*(2 - r)*x^(2*(2 - r))) - (3*d^2*e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r) - (e^3*x^(-4 + 3*r)*(a + b*Log[c*x^n]))/(4 - 3*r)
3.4.97.3.1 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[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(1038\) vs. \(2(183)=366\).
Time = 3.44 (sec) , antiderivative size = 1039, normalized size of antiderivative = 5.44
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1039\) |
| risch | \(\text {Expression too large to display}\) | \(4027\) |
-1/16*(4096*b*ln(c*x^n)*d^3+12288*b*d*e^2*ln(c*x^n)*(x^r)^2+4096*e^3*(x^r) ^3*a-2304*b*d^3*n*r^3+3712*b*d^3*n*r^2-3072*b*d^3*n*r+29184*a*d*e^2*r^2*(x ^r)^2-13056*a*d*e^2*r^3*(x^r)^2+12288*d*e^2*(x^r)^2*a+12288*d^2*e*x^r*a+40 96*a*d^3-3264*a*e^3*r^3*(x^r)^3+7936*a*e^3*r^2*(x^r)^3-9216*a*e^3*r*(x^r)^ 3-48*a*e^3*r^5*(x^r)^3+640*a*e^3*r^4*(x^r)^3-48*(x^r)^3*ln(c*x^n)*b*e^3*r^ 5+640*(x^r)^3*ln(c*x^n)*b*e^3*r^4-3264*(x^r)^3*ln(c*x^n)*b*e^3*r^3+7936*(x ^r)^3*ln(c*x^n)*b*e^3*r^2-9216*(x^r)^3*ln(c*x^n)*b*e^3*r+12288*b*d^2*e*ln( c*x^n)*x^r+9*b*d^3*n*r^6-132*b*d^3*n*r^5+772*b*d^3*n*r^4-432*x^r*ln(c*x^n) *b*d^2*e*r^5+4608*x^r*ln(c*x^n)*b*d^2*e*r^4-18624*x^r*ln(c*x^n)*b*d^2*e*r^ 3+36096*x^r*ln(c*x^n)*b*d^2*e*r^2-33792*x^r*ln(c*x^n)*b*d^2*e*r-216*(x^r)^ 2*ln(c*x^n)*b*d*e^2*r^5+2736*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-13056*(x^r)^2*l n(c*x^n)*b*d*e^2*r^3+29184*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-30720*(x^r)^2*ln( c*x^n)*b*d*e^2*r+1024*b*d^3*n+36*ln(c*x^n)*b*d^3*r^6-528*ln(c*x^n)*b*d^3*r ^5+3088*ln(c*x^n)*b*d^3*r^4-9216*ln(c*x^n)*b*d^3*r^3+14848*ln(c*x^n)*b*d^3 *r^2-12288*ln(c*x^n)*b*d^3*r+4096*e^3*b*ln(c*x^n)*(x^r)^3-9216*a*d^3*r^3+1 4848*a*d^3*r^2-12288*a*d^3*r+36*a*d^3*r^6-528*a*d^3*r^5+3088*a*d^3*r^4-186 24*a*d^2*e*r^3*x^r+3072*b*d*e^2*n*(x^r)^2+3072*b*d^2*e*n*x^r+1024*b*e^3*n* (x^r)^3-192*b*e^3*n*r^3*(x^r)^3+832*b*e^3*n*r^2*(x^r)^3-1536*b*e^3*n*r*(x^ r)^3+36096*a*d^2*e*r^2*x^r+4608*a*d^2*e*r^4*x^r-33792*a*d^2*e*r*x^r+16*b*e ^3*n*r^4*(x^r)^3-30720*a*d*e^2*r*(x^r)^2-216*a*d*e^2*r^5*(x^r)^2+2736*a...
Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (174) = 348\).
Time = 0.32 (sec) , antiderivative size = 980, normalized size of antiderivative = 5.13 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {9 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r^{6} - 132 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r^{5} + 1024 \, b d^{3} n + 772 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r^{4} + 4096 \, a d^{3} - 2304 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r^{3} + 3712 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r^{2} - 3072 \, {\left (b d^{3} n + 4 \, a d^{3}\right )} r - 16 \, {\left (3 \, a e^{3} r^{5} - 64 \, b e^{3} n - {\left (b e^{3} n + 40 \, a e^{3}\right )} r^{4} - 256 \, a e^{3} + 12 \, {\left (b e^{3} n + 17 \, a e^{3}\right )} r^{3} - 4 \, {\left (13 \, b e^{3} n + 124 \, a e^{3}\right )} r^{2} + 96 \, {\left (b e^{3} n + 6 \, a e^{3}\right )} r + {\left (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}\right )} \log \left (c\right ) + {\left (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\right )} \log \left (x\right )\right )} x^{3 \, r} - 12 \, {\left (18 \, a d e^{2} r^{5} - 256 \, b d e^{2} n - 3 \, {\left (3 \, b d e^{2} n + 76 \, a d e^{2}\right )} r^{4} - 1024 \, a d e^{2} + 32 \, {\left (3 \, b d e^{2} n + 34 \, a d e^{2}\right )} r^{3} - 32 \, {\left (11 \, b d e^{2} n + 76 \, a d e^{2}\right )} r^{2} + 512 \, {\left (b d e^{2} n + 5 \, a d e^{2}\right )} r + 2 \, {\left (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 - 512 \, b d e^{2}\right )} \log \left (c\right ) + 2 \, {\left (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\right )} \log \left (x\right )\right )} x^{2 \, r} - 48 \, {\left (9 \, a d^{2} e r^{5} - 64 \, b d^{2} e n - 3 \, {\left (3 \, b d^{2} e n + 32 \, a d^{2} e\right )} r^{4} - 256 \, a d^{2} e + 4 \, {\left (15 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - 4 \, {\left (37 \, b d^{2} e n + 188 \, a d^{2} e\right )} r^{2} + 32 \, {\left (5 \, b d^{2} e n + 22 \, a d^{2} e\right )} r + {\left (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\right )} \log \left (c\right ) + {\left (9 \, b d^{2} e n r^{5} - 96 \, b d^{2} e n r^{4} + 388 \, b d^{2} e n r^{3} - 752 \, b d^{2} e n r^{2} + 704 \, b d^{2} e n r - 256 \, b d^{2} e n\right )} \log \left (x\right )\right )} x^{r} + 4 \, {\left (9 \, b d^{3} r^{6} - 132 \, b d^{3} r^{5} + 772 \, b d^{3} r^{4} - 2304 \, b d^{3} r^{3} + 3712 \, b d^{3} r^{2} - 3072 \, b d^{3} r + 1024 \, b d^{3}\right )} \log \left (c\right ) + 4 \, {\left (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\right )} \log \left (x\right )}{16 \, {\left (9 \, r^{6} - 132 \, r^{5} + 772 \, r^{4} - 2304 \, r^{3} + 3712 \, r^{2} - 3072 \, r + 1024\right )} x^{4}} \]
-1/16*(9*(b*d^3*n + 4*a*d^3)*r^6 - 132*(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 - 16*(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 + (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)*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)*log(x))*x^(3*r) - 12*(1 8*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 + 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 - 512*b*d*e^2)*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)*log(x))*x^(2*r) - 48*(9*a*d^ 2*e*r^5 - 64*b*d^2*e*n - 3*(3*b*d^2*e*n + 32*a*d^2*e)*r^4 - 256*a*d^2*e + 4*(15*b*d^2*e*n + 97*a*d^2*e)*r^3 - 4*(37*b*d^2*e*n + 188*a*d^2*e)*r^2 + 3 2*(5*b*d^2*e*n + 22*a*d^2*e)*r + (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)*log(c) + (9*b*d^ 2*e*n*r^5 - 96*b*d^2*e*n*r^4 + 388*b*d^2*e*n*r^3 - 752*b*d^2*e*n*r^2 + 704 *b*d^2*e*n*r - 256*b*d^2*e*n)*log(x))*x^r + 4*(9*b*d^3*r^6 - 132*b*d^3*...
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(r-5>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^5} \,d x \]