Integrand size = 23, antiderivative size = 151 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{5+r}}{(5+r)^2}-\frac {3 b d e^2 n x^{5+2 r}}{(5+2 r)^2}-\frac {b e^3 n x^{5+3 r}}{(5+3 r)^2}+\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \]
-1/25*b*d^3*n*x^5-3*b*d^2*e*n*x^(5+r)/(5+r)^2-3*b*d*e^2*n*x^(5+2*r)/(5+2*r )^2-b*e^3*n*x^(5+3*r)/(5+3*r)^2+1/5*(d^3*x^5+15*d^2*e*x^(5+r)/(5+r)+15*d*e ^2*x^(5+2*r)/(5+2*r)+5*e^3*x^(5+3*r)/(5+3*r))*(a+b*ln(c*x^n))
Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{25} x^5 \left (b n \left (-d^3-\frac {75 d^2 e x^r}{(5+r)^2}-\frac {75 d e^2 x^{2 r}}{(5+2 r)^2}-\frac {25 e^3 x^{3 r}}{(5+3 r)^2}\right )+5 a \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right )+5 b \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right ) \log \left (c x^n\right )\right ) \]
(x^5*(b*n*(-d^3 - (75*d^2*e*x^r)/(5 + r)^2 - (75*d*e^2*x^(2*r))/(5 + 2*r)^ 2 - (25*e^3*x^(3*r))/(5 + 3*r)^2) + 5*a*(d^3 + (15*d^2*e*x^r)/(5 + r) + (1 5*d*e^2*x^(2*r))/(5 + 2*r) + (5*e^3*x^(3*r))/(5 + 3*r)) + 5*b*(d^3 + (15*d ^2*e*x^r)/(5 + r) + (15*d*e^2*x^(2*r))/(5 + 2*r) + (5*e^3*x^(3*r))/(5 + 3* r))*Log[c*x^n]))/25
Time = 0.58 (sec) , antiderivative size = 150, 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 = {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.
\(\displaystyle \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2771 |
\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{5} x^4 \left (\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}+d^3\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int x^4 \left (\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}+d^3\right )dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int \left (\frac {15 d e^2 x^{2 (r+2)}}{2 r+5}+\frac {15 d^2 e x^{r+4}}{r+5}+\frac {5 e^3 x^{3 r+4}}{3 r+5}+d^3 x^4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \left (\frac {d^3 x^5}{5}+\frac {15 d^2 e x^{r+5}}{(r+5)^2}+\frac {15 d e^2 x^{2 r+5}}{(2 r+5)^2}+\frac {5 e^3 x^{3 r+5}}{(3 r+5)^2}\right )\) |
-1/5*(b*n*((d^3*x^5)/5 + (15*d^2*e*x^(5 + r))/(5 + r)^2 + (15*d*e^2*x^(5 + 2*r))/(5 + 2*r)^2 + (5*e^3*x^(5 + 3*r))/(5 + 3*r)^2)) + ((d^3*x^5 + (15*d ^2*e*x^(5 + r))/(5 + r) + (15*d*e^2*x^(5 + 2*r))/(5 + 2*r) + (5*e^3*x^(5 + 3*r))/(5 + 3*r))*(a + b*Log[c*x^n]))/5
3.4.98.3.1 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. \(1268\) vs. \(2(147)=294\).
Time = 26.56 (sec) , antiderivative size = 1269, normalized size of antiderivative = 8.40
method | result | size |
parallelrisch | \(\text {Expression too large to display}\) | \(1269\) |
risch | \(\text {Expression too large to display}\) | \(4031\) |
-1/25*(-78125*x^5*a*d^3-31875*x^5*(x^r)^3*ln(c*x^n)*b*e^3*r^3-187500*x^5*l n(c*x^n)*b*d^3*r-300*x^5*(x^r)^3*a*e^3*r^5-96875*x^5*(x^r)^3*a*e^3*r^2-140 625*x^5*(x^r)^3*a*e^3*r-181875*x^5*x^r*r^3*a*d^2*e-515625*e*d^2*b*ln(c*x^n )*x^r*r*x^5-234375*x^5*d^2*e*x^r*b*ln(c*x^n)-234375*x^5*d*e^2*(x^r)^2*b*ln (c*x^n)-78125*x^5*e^3*(x^r)^3*a-78125*x^5*b*ln(c*x^n)*d^3-78125*x^5*e^3*(x ^r)^3*b*ln(c*x^n)-234375*x^5*d^2*e*x^r*a-234375*x^5*d*e^2*(x^r)^2*a-181250 *x^5*ln(c*x^n)*b*d^3*r^2-24125*x^5*ln(c*x^n)*b*d^3*r^4-90000*x^5*ln(c*x^n) *b*d^3*r^3-5000*x^5*(x^r)^3*a*e^3*r^4-468750*x^5*(x^r)^2*a*d*e^2*r-2700*x^ 5*x^r*ln(c*x^n)*b*d^2*e*r^5-36000*x^5*x^r*ln(c*x^n)*b*d^2*e*r^4-181875*x^5 *x^r*ln(c*x^n)*b*d^2*e*r^3-440625*x^5*x^r*ln(c*x^n)*b*d^2*e*r^2+36*x^5*b*d ^3*n*r^6+660*x^5*b*d^3*n*r^5+4825*x^5*b*d^3*n*r^4+18000*x^5*b*d^3*n*r^3+36 250*x^5*b*d^3*n*r^2+37500*x^5*b*d^3*n*r+15625*x^5*(x^r)^3*b*e^3*n-31875*x^ 5*(x^r)^3*a*e^3*r^3-180*x^5*a*d^3*r^6-3300*x^5*a*d^3*r^5-24125*x^5*a*d^3*r ^4-90000*x^5*a*d^3*r^3-181250*x^5*a*d^3*r^2-187500*x^5*a*d^3*r-180*x^5*ln( c*x^n)*b*d^3*r^6-3300*x^5*ln(c*x^n)*b*d^3*r^5+15625*b*d^3*n*x^5-1350*x^5*( x^r)^2*ln(c*x^n)*b*d*e^2*r^5-21375*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-12750 0*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3-356250*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r ^2-1350*x^5*(x^r)^2*a*d*e^2*r^5-96875*x^5*(x^r)^3*ln(c*x^n)*b*e^3*r^2-4687 50*e^2*d*b*ln(c*x^n)*(x^r)^2*x^5*r-356250*x^5*(x^r)^2*a*d*e^2*r^2-440625*x ^5*x^r*a*d^2*e*r^2-515625*x^5*x^r*a*d^2*e*r-300*x^5*(x^r)^3*ln(c*x^n)*b...
Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (147) = 294\).
Time = 0.36 (sec) , antiderivative size = 1023, normalized size of antiderivative = 6.77 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/25*(5*(36*b*d^3*r^6 + 660*b*d^3*r^5 + 4825*b*d^3*r^4 + 18000*b*d^3*r^3 + 36250*b*d^3*r^2 + 37500*b*d^3*r + 15625*b*d^3)*x^5*log(c) + 5*(36*b*d^3*n *r^6 + 660*b*d^3*n*r^5 + 4825*b*d^3*n*r^4 + 18000*b*d^3*n*r^3 + 36250*b*d^ 3*n*r^2 + 37500*b*d^3*n*r + 15625*b*d^3*n)*x^5*log(x) - (36*(b*d^3*n - 5*a *d^3)*r^6 + 660*(b*d^3*n - 5*a*d^3)*r^5 + 15625*b*d^3*n + 4825*(b*d^3*n - 5*a*d^3)*r^4 - 78125*a*d^3 + 18000*(b*d^3*n - 5*a*d^3)*r^3 + 36250*(b*d^3* n - 5*a*d^3)*r^2 + 37500*(b*d^3*n - 5*a*d^3)*r)*x^5 + 25*((12*b*e^3*r^5 + 200*b*e^3*r^4 + 1275*b*e^3*r^3 + 3875*b*e^3*r^2 + 5625*b*e^3*r + 3125*b*e^ 3)*x^5*log(c) + (12*b*e^3*n*r^5 + 200*b*e^3*n*r^4 + 1275*b*e^3*n*r^3 + 387 5*b*e^3*n*r^2 + 5625*b*e^3*n*r + 3125*b*e^3*n)*x^5*log(x) + (12*a*e^3*r^5 - 625*b*e^3*n - 4*(b*e^3*n - 50*a*e^3)*r^4 + 3125*a*e^3 - 15*(4*b*e^3*n - 85*a*e^3)*r^3 - 25*(13*b*e^3*n - 155*a*e^3)*r^2 - 375*(2*b*e^3*n - 15*a*e^ 3)*r)*x^5)*x^(3*r) + 75*((18*b*d*e^2*r^5 + 285*b*d*e^2*r^4 + 1700*b*d*e^2* r^3 + 4750*b*d*e^2*r^2 + 6250*b*d*e^2*r + 3125*b*d*e^2)*x^5*log(c) + (18*b *d*e^2*n*r^5 + 285*b*d*e^2*n*r^4 + 1700*b*d*e^2*n*r^3 + 4750*b*d*e^2*n*r^2 + 6250*b*d*e^2*n*r + 3125*b*d*e^2*n)*x^5*log(x) + (18*a*d*e^2*r^5 - 625*b *d*e^2*n - 3*(3*b*d*e^2*n - 95*a*d*e^2)*r^4 + 3125*a*d*e^2 - 20*(6*b*d*e^2 *n - 85*a*d*e^2)*r^3 - 50*(11*b*d*e^2*n - 95*a*d*e^2)*r^2 - 250*(4*b*d*e^2 *n - 25*a*d*e^2)*r)*x^5)*x^(2*r) + 75*((36*b*d^2*e*r^5 + 480*b*d^2*e*r^4 + 2425*b*d^2*e*r^3 + 5875*b*d^2*e*r^2 + 6875*b*d^2*e*r + 3125*b*d^2*e)*x...
Timed out. \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.51 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} + \frac {b e^{3} x^{3 \, r + 5} \log \left (c x^{n}\right )}{3 \, r + 5} + \frac {3 \, b d e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {3 \, b d^{2} e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{3} n x^{3 \, r + 5}}{{\left (3 \, r + 5\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 5}}{3 \, r + 5} - \frac {3 \, b d e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {3 \, b d^{2} e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 5}}{r + 5} \]
-1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5 + b*e^3*x^(3* r + 5)*log(c*x^n)/(3*r + 5) + 3*b*d*e^2*x^(2*r + 5)*log(c*x^n)/(2*r + 5) + 3*b*d^2*e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^3*n*x^(3*r + 5)/(3*r + 5)^2 + a*e^3*x^(3*r + 5)/(3*r + 5) - 3*b*d*e^2*n*x^(2*r + 5)/(2*r + 5)^2 + 3*a* d*e^2*x^(2*r + 5)/(2*r + 5) - 3*b*d^2*e*n*x^(r + 5)/(r + 5)^2 + 3*a*d^2*e* x^(r + 5)/(r + 5)
Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (147) = 294\).
Time = 0.35 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.67 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/25*(300*b*e^3*n*r^5*x^5*x^(3*r)*log(x) + 1350*b*d*e^2*n*r^5*x^5*x^(2*r)* log(x) + 2700*b*d^2*e*n*r^5*x^5*x^r*log(x) + 180*b*d^3*n*r^6*x^5*log(x) - 36*b*d^3*n*r^6*x^5 + 300*b*e^3*r^5*x^5*x^(3*r)*log(c) + 1350*b*d*e^2*r^5*x ^5*x^(2*r)*log(c) + 2700*b*d^2*e*r^5*x^5*x^r*log(c) + 180*b*d^3*r^6*x^5*lo g(c) + 5000*b*e^3*n*r^4*x^5*x^(3*r)*log(x) + 21375*b*d*e^2*n*r^4*x^5*x^(2* r)*log(x) + 36000*b*d^2*e*n*r^4*x^5*x^r*log(x) + 3300*b*d^3*n*r^5*x^5*log( x) - 100*b*e^3*n*r^4*x^5*x^(3*r) + 300*a*e^3*r^5*x^5*x^(3*r) - 675*b*d*e^2 *n*r^4*x^5*x^(2*r) + 1350*a*d*e^2*r^5*x^5*x^(2*r) - 2700*b*d^2*e*n*r^4*x^5 *x^r + 2700*a*d^2*e*r^5*x^5*x^r - 660*b*d^3*n*r^5*x^5 + 180*a*d^3*r^6*x^5 + 5000*b*e^3*r^4*x^5*x^(3*r)*log(c) + 21375*b*d*e^2*r^4*x^5*x^(2*r)*log(c) + 36000*b*d^2*e*r^4*x^5*x^r*log(c) + 3300*b*d^3*r^5*x^5*log(c) + 31875*b* e^3*n*r^3*x^5*x^(3*r)*log(x) + 127500*b*d*e^2*n*r^3*x^5*x^(2*r)*log(x) + 1 81875*b*d^2*e*n*r^3*x^5*x^r*log(x) + 24125*b*d^3*n*r^4*x^5*log(x) - 1500*b *e^3*n*r^3*x^5*x^(3*r) + 5000*a*e^3*r^4*x^5*x^(3*r) - 9000*b*d*e^2*n*r^3*x ^5*x^(2*r) + 21375*a*d*e^2*r^4*x^5*x^(2*r) - 22500*b*d^2*e*n*r^3*x^5*x^r + 36000*a*d^2*e*r^4*x^5*x^r - 4825*b*d^3*n*r^4*x^5 + 3300*a*d^3*r^5*x^5 + 3 1875*b*e^3*r^3*x^5*x^(3*r)*log(c) + 127500*b*d*e^2*r^3*x^5*x^(2*r)*log(c) + 181875*b*d^2*e*r^3*x^5*x^r*log(c) + 24125*b*d^3*r^4*x^5*log(c) + 96875*b *e^3*n*r^2*x^5*x^(3*r)*log(x) + 356250*b*d*e^2*n*r^2*x^5*x^(2*r)*log(x) + 440625*b*d^2*e*n*r^2*x^5*x^r*log(x) + 90000*b*d^3*n*r^3*x^5*log(x) - 81...
Timed out. \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^4\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]