Integrand size = 23, antiderivative size = 191 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {b d^3 n}{4 x^2}-\frac {3 b d e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {3 b d^2 e n x^{-2+r}}{(2-r)^2}-\frac {b e^3 n x^{-2+3 r}}{(2-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {3 d^2 e x^{-2+r} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^3 x^{-2+3 r} \left (a+b \log \left (c x^n\right )\right )}{2-3 r} \]
-1/4*b*d^3*n/x^2-3/4*b*d*e^2*n/(1-r)^2/(x^(2-2*r))-3*b*d^2*e*n*x^(-2+r)/(2 -r)^2-b*e^3*n*x^(-2+3*r)/(2-3*r)^2-1/2*d^3*(a+b*ln(c*x^n))/x^2-3/2*d*e^2*( a+b*ln(c*x^n))/(1-r)/(x^(2-2*r))-3*d^2*e*x^(-2+r)*(a+b*ln(c*x^n))/(2-r)-e^ 3*x^(-2+3*r)*(a+b*ln(c*x^n))/(2-3*r)
Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.95 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {b n \left (-d^3-\frac {12 d^2 e x^r}{(-2+r)^2}-\frac {3 d e^2 x^{2 r}}{(-1+r)^2}-\frac {4 e^3 x^{3 r}}{(2-3 r)^2}\right )+a \left (-2 d^3+\frac {12 d^2 e x^r}{-2+r}+\frac {6 d e^2 x^{2 r}}{-1+r}+\frac {4 e^3 x^{3 r}}{-2+3 r}\right )+2 b \left (-d^3+\frac {6 d^2 e x^r}{-2+r}+\frac {3 d e^2 x^{2 r}}{-1+r}+\frac {2 e^3 x^{3 r}}{-2+3 r}\right ) \log \left (c x^n\right )}{4 x^2} \]
(b*n*(-d^3 - (12*d^2*e*x^r)/(-2 + r)^2 - (3*d*e^2*x^(2*r))/(-1 + r)^2 - (4 *e^3*x^(3*r))/(2 - 3*r)^2) + a*(-2*d^3 + (12*d^2*e*x^r)/(-2 + r) + (6*d*e^ 2*x^(2*r))/(-1 + r) + (4*e^3*x^(3*r))/(-2 + 3*r)) + 2*b*(-d^3 + (6*d^2*e*x ^r)/(-2 + r) + (3*d*e^2*x^(2*r))/(-1 + r) + (2*e^3*x^(3*r))/(-2 + 3*r))*Lo g[c*x^n])/(4*x^2)
Time = 0.64 (sec) , antiderivative size = 190, 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 = {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.
| \(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int -\frac {\frac {6 d^2 e x^r}{2-r}+\frac {3 d e^2 x^{2 r}}{1-r}+\frac {2 e^3 x^{3 r}}{2-3 r}+d^3}{2 x^3}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} b n \int \frac {\frac {6 d^2 e x^r}{2-r}+\frac {3 d e^2 x^{2 r}}{1-r}+\frac {2 e^3 x^{3 r}}{2-3 r}+d^3}{x^3}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{2} b n \int \left (-\frac {6 d^2 e x^{r-3}}{r-2}+\frac {2 e^3 x^{3 (r-1)}}{2-3 r}-\frac {3 d e^2 x^{2 r-3}}{r-1}+\frac {d^3}{x^3}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3 d^2 e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {3 d e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {e^3 x^{3 r-2} \left (a+b \log \left (c x^n\right )\right )}{2-3 r}+\frac {1}{2} b n \left (-\frac {d^3}{2 x^2}-\frac {6 d^2 e x^{r-2}}{(2-r)^2}-\frac {3 d e^2 x^{-2 (1-r)}}{2 (1-r)^2}-\frac {2 e^3 x^{3 r-2}}{(2-3 r)^2}\right )\) |
(b*n*(-1/2*d^3/x^2 - (3*d*e^2)/(2*(1 - r)^2*x^(2*(1 - r))) - (6*d^2*e*x^(- 2 + r))/(2 - r)^2 - (2*e^3*x^(-2 + 3*r))/(2 - 3*r)^2))/2 - (d^3*(a + b*Log [c*x^n]))/(2*x^2) - (3*d*e^2*(a + b*Log[c*x^n]))/(2*(1 - r)*x^(2*(1 - r))) - (3*d^2*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r) - (e^3*x^(-2 + 3*r)*(a + b*Log[c*x^n]))/(2 - 3*r)
3.4.96.3.1 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 = 3.59 (sec) , antiderivative size = 1039, normalized size of antiderivative = 5.44
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1039\) |
| risch | \(\text {Expression too large to display}\) | \(4027\) |
-1/4*(32*b*ln(c*x^n)*d^3+96*b*d*e^2*ln(c*x^n)*(x^r)^2+32*e^3*(x^r)^3*a-288 *b*d^3*n*r^3+232*b*d^3*n*r^2-96*b*d^3*n*r+912*a*d*e^2*r^2*(x^r)^2-816*a*d* e^2*r^3*(x^r)^2+96*d*e^2*(x^r)^2*a+96*d^2*e*x^r*a+32*a*d^3-204*a*e^3*r^3*( x^r)^3+248*a*e^3*r^2*(x^r)^3-144*a*e^3*r*(x^r)^3-12*a*e^3*r^5*(x^r)^3+80*a *e^3*r^4*(x^r)^3-12*(x^r)^3*ln(c*x^n)*b*e^3*r^5+80*(x^r)^3*ln(c*x^n)*b*e^3 *r^4-204*(x^r)^3*ln(c*x^n)*b*e^3*r^3+248*(x^r)^3*ln(c*x^n)*b*e^3*r^2-144*( x^r)^3*ln(c*x^n)*b*e^3*r+96*b*d^2*e*ln(c*x^n)*x^r+9*b*d^3*n*r^6-66*b*d^3*n *r^5+193*b*d^3*n*r^4-108*x^r*ln(c*x^n)*b*d^2*e*r^5+576*x^r*ln(c*x^n)*b*d^2 *e*r^4-1164*x^r*ln(c*x^n)*b*d^2*e*r^3+1128*x^r*ln(c*x^n)*b*d^2*e*r^2-528*x ^r*ln(c*x^n)*b*d^2*e*r-54*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5+342*(x^r)^2*ln(c*x ^n)*b*d*e^2*r^4-816*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3+912*(x^r)^2*ln(c*x^n)*b* d*e^2*r^2-480*(x^r)^2*ln(c*x^n)*b*d*e^2*r+16*b*d^3*n+18*ln(c*x^n)*b*d^3*r^ 6-132*ln(c*x^n)*b*d^3*r^5+386*ln(c*x^n)*b*d^3*r^4-576*ln(c*x^n)*b*d^3*r^3+ 464*ln(c*x^n)*b*d^3*r^2-192*ln(c*x^n)*b*d^3*r+32*e^3*b*ln(c*x^n)*(x^r)^3-5 76*a*d^3*r^3+464*a*d^3*r^2-192*a*d^3*r+18*a*d^3*r^6-132*a*d^3*r^5+386*a*d^ 3*r^4-1164*a*d^2*e*r^3*x^r+48*b*d*e^2*n*(x^r)^2+48*b*d^2*e*n*x^r+16*b*e^3* n*(x^r)^3-24*b*e^3*n*r^3*(x^r)^3+52*b*e^3*n*r^2*(x^r)^3-48*b*e^3*n*r*(x^r) ^3+1128*a*d^2*e*r^2*x^r+576*a*d^2*e*r^4*x^r-528*a*d^2*e*r*x^r+4*b*e^3*n*r^ 4*(x^r)^3-480*a*d*e^2*r*(x^r)^2-54*a*d*e^2*r^5*(x^r)^2+342*a*d*e^2*r^4*(x^ r)^2-108*a*d^2*e*r^5*x^r+264*b*d*e^2*n*r^2*(x^r)^2+444*b*d^2*e*n*r^2*x^...
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (174) = 348\).
Time = 0.36 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.14 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {9 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{6} - 66 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{5} + 16 \, b d^{3} n + 193 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{4} + 32 \, a d^{3} - 288 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{3} + 232 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r^{2} - 96 \, {\left (b d^{3} n + 2 \, a d^{3}\right )} r - 4 \, {\left (3 \, a e^{3} r^{5} - 4 \, b e^{3} n - {\left (b e^{3} n + 20 \, a e^{3}\right )} r^{4} - 8 \, a e^{3} + 3 \, {\left (2 \, b e^{3} n + 17 \, a e^{3}\right )} r^{3} - {\left (13 \, b e^{3} n + 62 \, a e^{3}\right )} r^{2} + 12 \, {\left (b e^{3} n + 3 \, a e^{3}\right )} r + {\left (3 \, b e^{3} r^{5} - 20 \, b e^{3} r^{4} + 51 \, b e^{3} r^{3} - 62 \, b e^{3} r^{2} + 36 \, b e^{3} r - 8 \, b e^{3}\right )} \log \left (c\right ) + {\left (3 \, b e^{3} n r^{5} - 20 \, b e^{3} n r^{4} + 51 \, b e^{3} n r^{3} - 62 \, b e^{3} n r^{2} + 36 \, b e^{3} n r - 8 \, b e^{3} n\right )} \log \left (x\right )\right )} x^{3 \, r} - 3 \, {\left (18 \, a d e^{2} r^{5} - 16 \, b d e^{2} n - 3 \, {\left (3 \, b d e^{2} n + 38 \, a d e^{2}\right )} r^{4} - 32 \, a d e^{2} + 16 \, {\left (3 \, b d e^{2} n + 17 \, a d e^{2}\right )} r^{3} - 8 \, {\left (11 \, b d e^{2} n + 38 \, a d e^{2}\right )} r^{2} + 32 \, {\left (2 \, b d e^{2} n + 5 \, a d e^{2}\right )} r + 2 \, {\left (9 \, b d e^{2} r^{5} - 57 \, b d e^{2} r^{4} + 136 \, b d e^{2} r^{3} - 152 \, b d e^{2} r^{2} + 80 \, b d e^{2} r - 16 \, b d e^{2}\right )} \log \left (c\right ) + 2 \, {\left (9 \, b d e^{2} n r^{5} - 57 \, b d e^{2} n r^{4} + 136 \, b d e^{2} n r^{3} - 152 \, b d e^{2} n r^{2} + 80 \, b d e^{2} n r - 16 \, b d e^{2} n\right )} \log \left (x\right )\right )} x^{2 \, r} - 12 \, {\left (9 \, a d^{2} e r^{5} - 4 \, b d^{2} e n - 3 \, {\left (3 \, b d^{2} e n + 16 \, a d^{2} e\right )} r^{4} - 8 \, a d^{2} e + {\left (30 \, b d^{2} e n + 97 \, a d^{2} e\right )} r^{3} - {\left (37 \, b d^{2} e n + 94 \, a d^{2} e\right )} r^{2} + 4 \, {\left (5 \, b d^{2} e n + 11 \, a d^{2} e\right )} r + {\left (9 \, b d^{2} e r^{5} - 48 \, b d^{2} e r^{4} + 97 \, b d^{2} e r^{3} - 94 \, b d^{2} e r^{2} + 44 \, b d^{2} e r - 8 \, b d^{2} e\right )} \log \left (c\right ) + {\left (9 \, b d^{2} e n r^{5} - 48 \, b d^{2} e n r^{4} + 97 \, b d^{2} e n r^{3} - 94 \, b d^{2} e n r^{2} + 44 \, b d^{2} e n r - 8 \, b d^{2} e n\right )} \log \left (x\right )\right )} x^{r} + 2 \, {\left (9 \, b d^{3} r^{6} - 66 \, b d^{3} r^{5} + 193 \, b d^{3} r^{4} - 288 \, b d^{3} r^{3} + 232 \, b d^{3} r^{2} - 96 \, b d^{3} r + 16 \, b d^{3}\right )} \log \left (c\right ) + 2 \, {\left (9 \, b d^{3} n r^{6} - 66 \, b d^{3} n r^{5} + 193 \, b d^{3} n r^{4} - 288 \, b d^{3} n r^{3} + 232 \, b d^{3} n r^{2} - 96 \, b d^{3} n r + 16 \, b d^{3} n\right )} \log \left (x\right )}{4 \, {\left (9 \, r^{6} - 66 \, r^{5} + 193 \, r^{4} - 288 \, r^{3} + 232 \, r^{2} - 96 \, r + 16\right )} x^{2}} \]
-1/4*(9*(b*d^3*n + 2*a*d^3)*r^6 - 66*(b*d^3*n + 2*a*d^3)*r^5 + 16*b*d^3*n + 193*(b*d^3*n + 2*a*d^3)*r^4 + 32*a*d^3 - 288*(b*d^3*n + 2*a*d^3)*r^3 + 2 32*(b*d^3*n + 2*a*d^3)*r^2 - 96*(b*d^3*n + 2*a*d^3)*r - 4*(3*a*e^3*r^5 - 4 *b*e^3*n - (b*e^3*n + 20*a*e^3)*r^4 - 8*a*e^3 + 3*(2*b*e^3*n + 17*a*e^3)*r ^3 - (13*b*e^3*n + 62*a*e^3)*r^2 + 12*(b*e^3*n + 3*a*e^3)*r + (3*b*e^3*r^5 - 20*b*e^3*r^4 + 51*b*e^3*r^3 - 62*b*e^3*r^2 + 36*b*e^3*r - 8*b*e^3)*log( c) + (3*b*e^3*n*r^5 - 20*b*e^3*n*r^4 + 51*b*e^3*n*r^3 - 62*b*e^3*n*r^2 + 3 6*b*e^3*n*r - 8*b*e^3*n)*log(x))*x^(3*r) - 3*(18*a*d*e^2*r^5 - 16*b*d*e^2* n - 3*(3*b*d*e^2*n + 38*a*d*e^2)*r^4 - 32*a*d*e^2 + 16*(3*b*d*e^2*n + 17*a *d*e^2)*r^3 - 8*(11*b*d*e^2*n + 38*a*d*e^2)*r^2 + 32*(2*b*d*e^2*n + 5*a*d* e^2)*r + 2*(9*b*d*e^2*r^5 - 57*b*d*e^2*r^4 + 136*b*d*e^2*r^3 - 152*b*d*e^2 *r^2 + 80*b*d*e^2*r - 16*b*d*e^2)*log(c) + 2*(9*b*d*e^2*n*r^5 - 57*b*d*e^2 *n*r^4 + 136*b*d*e^2*n*r^3 - 152*b*d*e^2*n*r^2 + 80*b*d*e^2*n*r - 16*b*d*e ^2*n)*log(x))*x^(2*r) - 12*(9*a*d^2*e*r^5 - 4*b*d^2*e*n - 3*(3*b*d^2*e*n + 16*a*d^2*e)*r^4 - 8*a*d^2*e + (30*b*d^2*e*n + 97*a*d^2*e)*r^3 - (37*b*d^2 *e*n + 94*a*d^2*e)*r^2 + 4*(5*b*d^2*e*n + 11*a*d^2*e)*r + (9*b*d^2*e*r^5 - 48*b*d^2*e*r^4 + 97*b*d^2*e*r^3 - 94*b*d^2*e*r^2 + 44*b*d^2*e*r - 8*b*d^2 *e)*log(c) + (9*b*d^2*e*n*r^5 - 48*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 - 94*b *d^2*e*n*r^2 + 44*b*d^2*e*n*r - 8*b*d^2*e*n)*log(x))*x^r + 2*(9*b*d^3*r^6 - 66*b*d^3*r^5 + 193*b*d^3*r^4 - 288*b*d^3*r^3 + 232*b*d^3*r^2 - 96*b*d...
Time = 46.14 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.82 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=- \frac {a d^{3}}{2 x^{2}} + 3 a d^{2} e \left (\begin {cases} \frac {x^{r}}{r x^{2} - 2 x^{2}} & \text {for}\: r \neq 2 \\\frac {x^{r} \log {\left (x \right )}}{x^{2}} & \text {otherwise} \end {cases}\right ) + 3 a d e^{2} \left (\begin {cases} \frac {x^{2 r}}{2 r x^{2} - 2 x^{2}} & \text {for}\: r \neq 1 \\\frac {x^{2 r} \log {\left (x \right )}}{x^{2}} & \text {otherwise} \end {cases}\right ) + a e^{3} \left (\begin {cases} \frac {x^{3 r}}{3 r x^{2} - 2 x^{2}} & \text {for}\: r \neq \frac {2}{3} \\\frac {x^{3 r} \log {\left (x \right )}}{x^{2}} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} n}{4 x^{2}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{2 x^{2}} - 3 b d^{2} e n \left (\begin {cases} \frac {\begin {cases} \frac {x^{r - 2}}{r - 2} & \text {for}\: r \neq 2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 2 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d^{2} e \left (\begin {cases} \frac {x^{r - 2}}{r - 2} & \text {for}\: r \neq 2 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - 3 b d e^{2} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{2 r - 2}}{2 r - 2} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{2 r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq 1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + 3 b d e^{2} \left (\begin {cases} \frac {x^{2 r - 2}}{2 r - 2} & \text {for}\: r \neq 1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e^{3} n \left (\begin {cases} \frac {\begin {cases} \frac {x^{3 r - 2}}{3 r - 2} & \text {for}\: r \neq \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{3 r - 2} & \text {for}\: r > -\infty \wedge r < \infty \wedge r \neq \frac {2}{3} \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases}\right ) + b e^{3} \left (\begin {cases} \frac {x^{3 r - 2}}{3 r - 2} & \text {for}\: r \neq \frac {2}{3} \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
-a*d**3/(2*x**2) + 3*a*d**2*e*Piecewise((x**r/(r*x**2 - 2*x**2), Ne(r, 2)) , (x**r*log(x)/x**2, True)) + 3*a*d*e**2*Piecewise((x**(2*r)/(2*r*x**2 - 2 *x**2), Ne(r, 1)), (x**(2*r)*log(x)/x**2, True)) + a*e**3*Piecewise((x**(3 *r)/(3*r*x**2 - 2*x**2), Ne(r, 2/3)), (x**(3*r)*log(x)/x**2, True)) - b*d* *3*n/(4*x**2) - b*d**3*log(c*x**n)/(2*x**2) - 3*b*d**2*e*n*Piecewise((Piec ewise((x**(r - 2)/(r - 2), Ne(r, 2)), (log(x), True))/(r - 2), (r > -oo) & (r < oo) & Ne(r, 2)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x**(r - 2)/(r - 2), Ne(r, 2)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n*Piecew ise((Piecewise((x**(2*r - 2)/(2*r - 2), Ne(r, 1)), (log(x), True))/(2*r - 2), (r > -oo) & (r < oo) & Ne(r, 1)), (log(x)**2/2, True)) + 3*b*d*e**2*Pi ecewise((x**(2*r - 2)/(2*r - 2), Ne(r, 1)), (log(x), True))*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r - 2)/(3*r - 2), Ne(r, 2/3)), (log(x ), True))/(3*r - 2), (r > -oo) & (r < oo) & Ne(r, 2/3)), (log(x)**2/2, Tru e)) + b*e**3*Piecewise((x**(3*r - 2)/(3*r - 2), Ne(r, 2/3)), (log(x), True ))*log(c*x**n)
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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-3>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]