Integrand size = 23, antiderivative size = 183 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^3 n}{49 x^7}-\frac {3 b d^2 e n x^{-7+r}}{(7-r)^2}-\frac {3 b d e^2 n x^{-7+2 r}}{(7-2 r)^2}-\frac {b e^3 n x^{-7+3 r}}{(7-3 r)^2}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{-7+r} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{-7+2 r} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{-7+3 r} \left (a+b \log \left (c x^n\right )\right )}{7-3 r} \]
-1/49*b*d^3*n/x^7-3*b*d^2*e*n*x^(-7+r)/(7-r)^2-3*b*d*e^2*n*x^(-7+2*r)/(7-2 *r)^2-b*e^3*n*x^(-7+3*r)/(7-3*r)^2-1/7*d^3*(a+b*ln(c*x^n))/x^7-3*d^2*e*x^( -7+r)*(a+b*ln(c*x^n))/(7-r)-3*d*e^2*x^(-7+2*r)*(a+b*ln(c*x^n))/(7-2*r)-e^3 *x^(-7+3*r)*(a+b*ln(c*x^n))/(7-3*r)
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\frac {b n \left (-d^3-\frac {147 d^2 e x^r}{(-7+r)^2}-\frac {147 d e^2 x^{2 r}}{(7-2 r)^2}-\frac {49 e^3 x^{3 r}}{(7-3 r)^2}\right )+7 a \left (-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}\right )+7 b \left (-d^3+\frac {21 d^2 e x^r}{-7+r}+\frac {21 d e^2 x^{2 r}}{-7+2 r}+\frac {7 e^3 x^{3 r}}{-7+3 r}\right ) \log \left (c x^n\right )}{49 x^7} \]
(b*n*(-d^3 - (147*d^2*e*x^r)/(-7 + r)^2 - (147*d*e^2*x^(2*r))/(7 - 2*r)^2 - (49*e^3*x^(3*r))/(7 - 3*r)^2) + 7*a*(-d^3 + (21*d^2*e*x^r)/(-7 + r) + (2 1*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(-7 + 3*r)) + 7*b*(-d^3 + (2 1*d^2*e*x^r)/(-7 + r) + (21*d*e^2*x^(2*r))/(-7 + 2*r) + (7*e^3*x^(3*r))/(- 7 + 3*r))*Log[c*x^n])/(49*x^7)
Time = 0.62 (sec) , antiderivative size = 182, 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.
| \(\displaystyle \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle -b n \int -\frac {\frac {21 d^2 e x^r}{7-r}+\frac {21 d e^2 x^{2 r}}{7-2 r}+\frac {7 e^3 x^{3 r}}{7-3 r}+d^3}{7 x^8}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} b n \int \frac {\frac {21 d^2 e x^r}{7-r}+\frac {21 d e^2 x^{2 r}}{7-2 r}+\frac {7 e^3 x^{3 r}}{7-3 r}+d^3}{x^8}dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{7} b n \int \left (-\frac {21 d^2 e x^{r-8}}{r-7}+\frac {21 d e^2 x^{2 (r-4)}}{7-2 r}-\frac {7 e^3 x^{3 r-8}}{3 r-7}+\frac {d^3}{x^8}\right )dx-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {3 d^2 e x^{r-7} \left (a+b \log \left (c x^n\right )\right )}{7-r}-\frac {3 d e^2 x^{2 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-2 r}-\frac {e^3 x^{3 r-7} \left (a+b \log \left (c x^n\right )\right )}{7-3 r}+\frac {1}{7} b n \left (-\frac {d^3}{7 x^7}-\frac {21 d^2 e x^{r-7}}{(7-r)^2}-\frac {21 d e^2 x^{2 r-7}}{(7-2 r)^2}-\frac {7 e^3 x^{3 r-7}}{(7-3 r)^2}\right )\) |
(b*n*(-1/7*d^3/x^7 - (21*d^2*e*x^(-7 + r))/(7 - r)^2 - (21*d*e^2*x^(-7 + 2 *r))/(7 - 2*r)^2 - (7*e^3*x^(-7 + 3*r))/(7 - 3*r)^2))/7 - (d^3*(a + b*Log[ c*x^n]))/(7*x^7) - (3*d^2*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r) - (3*d* e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 - 2*r) - (e^3*x^(-7 + 3*r)*(a + b* Log[c*x^n]))/(7 - 3*r)
3.5.4.3.1 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. \(1040\) vs. \(2(179)=358\).
Time = 3.29 (sec) , antiderivative size = 1041, normalized size of antiderivative = 5.69
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1041\) |
| risch | \(\text {Expression too large to display}\) | \(4031\) |
-1/49*(823543*b*ln(c*x^n)*d^3+2470629*b*d*e^2*ln(c*x^n)*(x^r)^2+823543*e^3 *(x^r)^3*a-49392*b*d^3*n*r^3+139258*b*d^3*n*r^2-201684*b*d^3*n*r+1915998*a *d*e^2*r^2*(x^r)^2-489804*a*d*e^2*r^3*(x^r)^2+2470629*d*e^2*(x^r)^2*a+2470 629*d^2*e*x^r*a+823543*a*d^3-122451*a*e^3*r^3*(x^r)^3+521017*a*e^3*r^2*(x^ r)^3-1058841*a*e^3*r*(x^r)^3-588*a*e^3*r^5*(x^r)^3+13720*a*e^3*r^4*(x^r)^3 -588*(x^r)^3*ln(c*x^n)*b*e^3*r^5+13720*(x^r)^3*ln(c*x^n)*b*e^3*r^4-122451* (x^r)^3*ln(c*x^n)*b*e^3*r^3+521017*(x^r)^3*ln(c*x^n)*b*e^3*r^2-1058841*(x^ r)^3*ln(c*x^n)*b*e^3*r+2470629*b*d^2*e*ln(c*x^n)*x^r+36*b*d^3*n*r^6-924*b* d^3*n*r^5+9457*b*d^3*n*r^4-5292*x^r*ln(c*x^n)*b*d^2*e*r^5+98784*x^r*ln(c*x ^n)*b*d^2*e*r^4-698691*x^r*ln(c*x^n)*b*d^2*e*r^3+2369787*x^r*ln(c*x^n)*b*d ^2*e*r^2-3882417*x^r*ln(c*x^n)*b*d^2*e*r-2646*(x^r)^2*ln(c*x^n)*b*d*e^2*r^ 5+58653*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-489804*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3 +1915998*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-3529470*(x^r)^2*ln(c*x^n)*b*d*e^2*r +117649*b*d^3*n+252*ln(c*x^n)*b*d^3*r^6-6468*ln(c*x^n)*b*d^3*r^5+66199*ln( c*x^n)*b*d^3*r^4-345744*ln(c*x^n)*b*d^3*r^3+974806*ln(c*x^n)*b*d^3*r^2-141 1788*ln(c*x^n)*b*d^3*r+823543*e^3*b*ln(c*x^n)*(x^r)^3-345744*a*d^3*r^3+974 806*a*d^3*r^2-1411788*a*d^3*r+252*a*d^3*r^6-6468*a*d^3*r^5+66199*a*d^3*r^4 -698691*a*d^2*e*r^3*x^r+352947*b*d*e^2*n*(x^r)^2+352947*b*d^2*e*n*x^r+1176 49*b*e^3*n*(x^r)^3-4116*b*e^3*n*r^3*(x^r)^3+31213*b*e^3*n*r^2*(x^r)^3-1008 42*b*e^3*n*r*(x^r)^3+2369787*a*d^2*e*r^2*x^r+98784*a*d^2*e*r^4*x^r-3882...
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (174) = 348\).
Time = 0.30 (sec) , antiderivative size = 981, normalized size of antiderivative = 5.36 \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Too large to display} \]
-1/49*(36*(b*d^3*n + 7*a*d^3)*r^6 - 924*(b*d^3*n + 7*a*d^3)*r^5 + 117649*b *d^3*n + 9457*(b*d^3*n + 7*a*d^3)*r^4 + 823543*a*d^3 - 49392*(b*d^3*n + 7* a*d^3)*r^3 + 139258*(b*d^3*n + 7*a*d^3)*r^2 - 201684*(b*d^3*n + 7*a*d^3)*r - 49*(12*a*e^3*r^5 - 2401*b*e^3*n - 4*(b*e^3*n + 70*a*e^3)*r^4 - 16807*a* e^3 + 21*(4*b*e^3*n + 119*a*e^3)*r^3 - 49*(13*b*e^3*n + 217*a*e^3)*r^2 + 1 029*(2*b*e^3*n + 21*a*e^3)*r + (12*b*e^3*r^5 - 280*b*e^3*r^4 + 2499*b*e^3* r^3 - 10633*b*e^3*r^2 + 21609*b*e^3*r - 16807*b*e^3)*log(c) + (12*b*e^3*n* r^5 - 280*b*e^3*n*r^4 + 2499*b*e^3*n*r^3 - 10633*b*e^3*n*r^2 + 21609*b*e^3 *n*r - 16807*b*e^3*n)*log(x))*x^(3*r) - 147*(18*a*d*e^2*r^5 - 2401*b*d*e^2 *n - 3*(3*b*d*e^2*n + 133*a*d*e^2)*r^4 - 16807*a*d*e^2 + 28*(6*b*d*e^2*n + 119*a*d*e^2)*r^3 - 98*(11*b*d*e^2*n + 133*a*d*e^2)*r^2 + 686*(4*b*d*e^2*n + 35*a*d*e^2)*r + (18*b*d*e^2*r^5 - 399*b*d*e^2*r^4 + 3332*b*d*e^2*r^3 - 13034*b*d*e^2*r^2 + 24010*b*d*e^2*r - 16807*b*d*e^2)*log(c) + (18*b*d*e^2* n*r^5 - 399*b*d*e^2*n*r^4 + 3332*b*d*e^2*n*r^3 - 13034*b*d*e^2*n*r^2 + 240 10*b*d*e^2*n*r - 16807*b*d*e^2*n)*log(x))*x^(2*r) - 147*(36*a*d^2*e*r^5 - 2401*b*d^2*e*n - 12*(3*b*d^2*e*n + 56*a*d^2*e)*r^4 - 16807*a*d^2*e + 7*(60 *b*d^2*e*n + 679*a*d^2*e)*r^3 - 49*(37*b*d^2*e*n + 329*a*d^2*e)*r^2 + 343* (10*b*d^2*e*n + 77*a*d^2*e)*r + (36*b*d^2*e*r^5 - 672*b*d^2*e*r^4 + 4753*b *d^2*e*r^3 - 16121*b*d^2*e*r^2 + 26411*b*d^2*e*r - 16807*b*d^2*e)*log(c) + (36*b*d^2*e*n*r^5 - 672*b*d^2*e*n*r^4 + 4753*b*d^2*e*n*r^3 - 16121*b*d...
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]
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-8>0)', see `assume?` for more details)Is
\[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{3} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=\int \frac {{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^8} \,d x \]