Integrand size = 26, antiderivative size = 449 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{2+m}}{e^8 (2+m)}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}-\frac {5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5+m}}{e^8 (5+m)}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac {7 c^3 (2 c d-b e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac {2 c^4 (d+e x)^{8+m}}{e^8 (8+m)} \]
-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(1+m)/e^8/(1+m)+(a*e^2-b*d*e+c *d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(2+m)/e^8/(2+m)- 3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))* (e*x+d)^(3+m)/e^8/(3+m)+(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20* c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d )^(4+m)/e^8/(4+m)-5*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))* (e*x+d)^(5+m)/e^8/(5+m)+3*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e *x+d)^(6+m)/e^8/(6+m)-7*c^3*(-b*e+2*c*d)*(e*x+d)^(7+m)/e^8/(7+m)+2*c^4*(e* x+d)^(8+m)/e^8/(8+m)
Leaf count is larger than twice the leaf count of optimal. \(1112\) vs. \(2(449)=898\).
Time = 2.13 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.48 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (-c^2 (14 c d-b e (14+m)-2 c e (7+m) x) (a+x (b+c x))^3+\frac {3 \left (-2 \left ((2 c d-b e) \left (c d^2+e (-b d+a e)\right ) (2+m) \left (840 c^4 d^4-b^4 e^4 m \left (3+4 m+m^2\right )+4 b^2 c e^3 m (1+m) (-5 b d+a e (11+2 m))-80 c^3 d^2 e \left (21 b d+a e \left (-21+m+m^2\right )\right )+4 c^2 e^2 \left (20 a b d e \left (-21+m+m^2\right )+5 b^2 d^2 \left (42+m+m^2\right )-2 a^2 e^2 \left (-105+16 m+18 m^2+2 m^3\right )\right )\right )+(1+m) \left (-1680 c^6 d^6+b^6 e^6 m \left (6+5 m+m^2\right )-b^4 c e^5 m (2+m) (b d (-11+3 m)+a e (47+9 m))+80 c^5 d^4 e \left (63 b d+a e \left (-63-5 m+2 m^2\right )\right )+b^2 c^2 e^4 m \left (3 b^2 d^2 \left (26-15 m+m^2\right )+12 a^2 e^2 \left (47+20 m+2 m^2\right )+8 a b d e \left (-47+5 m+3 m^2\right )\right )-4 c^4 d^2 e^2 \left (40 a b d e \left (-63-5 m+2 m^2\right )+5 b^2 d^2 \left (252-5 m+2 m^2\right )-12 a^2 e^2 \left (-105-24 m+5 m^2+m^3\right )\right )-8 c^3 e^3 \left (-5 b^3 d^3 \left (42-5 m+2 m^2\right )+3 a b^2 d^2 e \left (210+m-5 m^2+m^3\right )+6 a^2 b d e^2 \left (-105-24 m+5 m^2+m^3\right )+2 a^3 e^3 \left (105+71 m+15 m^2+m^3\right )\right )\right ) (d+e x)+e^2 (1+m) (2+m) \left (c e (4+m) \left (c e (b d-2 a e) (6+m) \left (14 b \left (c d^2+a e^2\right )+4 a c d e m-b^2 d e (14+m)\right )-(b d (5 c d-2 b e)+a e (2 c d m-b e (1+m))) \left (28 c^2 d^2-b^2 e^2 m+4 c e (-7 b d+a e (7+m))\right )\right )-(3 c d-b e) \left (c e (2 c d-b e) (6+m) \left (14 b \left (c d^2+a e^2\right )+4 a c d e m-b^2 d e (14+m)\right )-\left (10 c^2 d^2-b^2 e^2 (3+m)+c e (b d (-4+m)+2 a e (5+m))\right ) \left (28 c^2 d^2-b^2 e^2 m+4 c e (-7 b d+a e (7+m))\right )\right )+c e (3+m) \left (c e (2 c d-b e) (6+m) \left (14 b \left (c d^2+a e^2\right )+4 a c d e m-b^2 d e (14+m)\right )-\left (10 c^2 d^2-b^2 e^2 (3+m)+c e (b d (-4+m)+2 a e (5+m))\right ) \left (28 c^2 d^2-b^2 e^2 m+4 c e (-7 b d+a e (7+m))\right )\right ) x\right ) (a+x (b+c x))\right )+c e^4 (1+m) (2+m) (3+m) (4+m) (a+x (b+c x))^2 \left (-2 b^3 e^3 m-28 c^3 d^2 (5 d-e (5+m) x)-b c e^2 \left (-2 a e (70+11 m)+b d \left (140+15 m+m^2\right )+b e m (5+m) x\right )+2 c^2 e \left (7 b d (d (20+m)-2 e (5+m) x)+2 a e \left (d \left (-35+m+m^2\right )+e \left (35+12 m+m^2\right ) x\right )\right )\right )\right )}{e^6 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m)}\right )}{c^2 e^2 (7+m) (8+m)} \]
((d + e*x)^(1 + m)*(-(c^2*(14*c*d - b*e*(14 + m) - 2*c*e*(7 + m)*x)*(a + x *(b + c*x))^3) + (3*(-2*((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(2 + m)* (840*c^4*d^4 - b^4*e^4*m*(3 + 4*m + m^2) + 4*b^2*c*e^3*m*(1 + m)*(-5*b*d + a*e*(11 + 2*m)) - 80*c^3*d^2*e*(21*b*d + a*e*(-21 + m + m^2)) + 4*c^2*e^2 *(20*a*b*d*e*(-21 + m + m^2) + 5*b^2*d^2*(42 + m + m^2) - 2*a^2*e^2*(-105 + 16*m + 18*m^2 + 2*m^3))) + (1 + m)*(-1680*c^6*d^6 + b^6*e^6*m*(6 + 5*m + m^2) - b^4*c*e^5*m*(2 + m)*(b*d*(-11 + 3*m) + a*e*(47 + 9*m)) + 80*c^5*d^ 4*e*(63*b*d + a*e*(-63 - 5*m + 2*m^2)) + b^2*c^2*e^4*m*(3*b^2*d^2*(26 - 15 *m + m^2) + 12*a^2*e^2*(47 + 20*m + 2*m^2) + 8*a*b*d*e*(-47 + 5*m + 3*m^2) ) - 4*c^4*d^2*e^2*(40*a*b*d*e*(-63 - 5*m + 2*m^2) + 5*b^2*d^2*(252 - 5*m + 2*m^2) - 12*a^2*e^2*(-105 - 24*m + 5*m^2 + m^3)) - 8*c^3*e^3*(-5*b^3*d^3* (42 - 5*m + 2*m^2) + 3*a*b^2*d^2*e*(210 + m - 5*m^2 + m^3) + 6*a^2*b*d*e^2 *(-105 - 24*m + 5*m^2 + m^3) + 2*a^3*e^3*(105 + 71*m + 15*m^2 + m^3)))*(d + e*x) + e^2*(1 + m)*(2 + m)*(c*e*(4 + m)*(c*e*(b*d - 2*a*e)*(6 + m)*(14*b *(c*d^2 + a*e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (b*d*(5*c*d - 2*b*e) + a*e*(2*c*d*m - b*e*(1 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a *e*(7 + m)))) - (3*c*d - b*e)*(c*e*(2*c*d - b*e)*(6 + m)*(14*b*(c*d^2 + a* e^2) + 4*a*c*d*e*m - b^2*d*e*(14 + m)) - (10*c^2*d^2 - b^2*e^2*(3 + m) + c *e*(b*d*(-4 + m) + 2*a*e*(5 + m)))*(28*c^2*d^2 - b^2*e^2*m + 4*c*e*(-7*b*d + a*e*(7 + m)))) + c*e*(3 + m)*(c*e*(2*c*d - b*e)*(6 + m)*(14*b*(c*d^2...
Time = 0.72 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1195, 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.
| \(\displaystyle \int (b+2 c x) \left (a+b x+c x^2\right )^3 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{m+3} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {5 c (2 c d-b e) (d+e x)^{m+4} \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 c^2 (d+e x)^{m+5} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {(b e-2 c d) (d+e x)^m \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{m+6}}{e^7}+\frac {2 c^4 (d+e x)^{m+7}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^{m+4} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8 (m+4)}+\frac {(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+2)}-\frac {3 (2 c d-b e) (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+3)}-\frac {5 c (2 c d-b e) (d+e x)^{m+5} \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8 (m+5)}+\frac {3 c^2 (d+e x)^{m+6} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 (m+6)}-\frac {(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^8 (m+1)}-\frac {7 c^3 (2 c d-b e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac {2 c^4 (d+e x)^{m+8}}{e^8 (m+8)}\) |
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m) )) + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a *e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^8 *(3 + m)) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3* d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (5*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (7* c^3*(2*c*d - b*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (2*c^4*(d + e*x)^(8 + m))/(e^8*(8 + m))
3.17.54.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(4283\) vs. \(2(449)=898\).
Time = 0.41 (sec) , antiderivative size = 4284, normalized size of antiderivative = 9.54
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(4284\) |
| gosper | \(\text {Expression too large to display}\) | \(5439\) |
| risch | \(\text {Expression too large to display}\) | \(6726\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(9666\) |
d*(a^3*b*e^7*m^7+35*a^3*b*e^7*m^6-2*a^3*c*d*e^6*m^6-3*a^2*b^2*d*e^6*m^6+51 1*a^3*b*e^7*m^5-66*a^3*c*d*e^6*m^5-99*a^2*b^2*d*e^6*m^5+18*a^2*b*c*d^2*e^5 *m^5+6*a*b^3*d^2*e^5*m^5+4025*a^3*b*e^7*m^4-890*a^3*c*d*e^6*m^4-1335*a^2*b ^2*d*e^6*m^4+540*a^2*b*c*d^2*e^5*m^4-36*a^2*c^2*d^3*e^4*m^4+180*a*b^3*d^2* e^5*m^4-72*a*b^2*c*d^3*e^4*m^4-6*b^4*d^3*e^4*m^4+18424*a^3*b*e^7*m^3-6270* a^3*c*d*e^6*m^3-9405*a^2*b^2*d*e^6*m^3+6390*a^2*b*c*d^2*e^5*m^3-936*a^2*c^ 2*d^3*e^4*m^3+2130*a*b^3*d^2*e^5*m^3-1872*a*b^2*c*d^3*e^4*m^3+360*a*b*c^2* d^4*e^3*m^3-156*b^4*d^3*e^4*m^3+120*b^3*c*d^4*e^3*m^3+48860*a^3*b*e^7*m^2- 24308*a^3*c*d*e^6*m^2-36462*a^2*b^2*d*e^6*m^2+37260*a^2*b*c*d^2*e^5*m^2-90 36*a^2*c^2*d^3*e^4*m^2+12420*a*b^3*d^2*e^5*m^2-18072*a*b^2*c*d^3*e^4*m^2+7 560*a*b*c^2*d^4*e^3*m^2-720*a*c^3*d^5*e^2*m^2-1506*b^4*d^3*e^4*m^2+2520*b^ 3*c*d^4*e^3*m^2-1080*b^2*c^2*d^5*e^2*m^2+69264*a^3*b*e^7*m-49104*a^3*c*d*e ^6*m-73656*a^2*b^2*d*e^6*m+106992*a^2*b*c*d^2*e^5*m-38376*a^2*c^2*d^3*e^4* m+35664*a*b^3*d^2*e^5*m-76752*a*b^2*c*d^3*e^4*m+52560*a*b*c^2*d^4*e^3*m-10 800*a*c^3*d^5*e^2*m-6396*b^4*d^3*e^4*m+17520*b^3*c*d^4*e^3*m-16200*b^2*c^2 *d^5*e^2*m+5040*b*c^3*d^6*e*m+40320*a^3*b*e^7-40320*a^3*c*d*e^6-60480*a^2* b^2*d*e^6+120960*a^2*b*c*d^2*e^5-60480*a^2*c^2*d^3*e^4+40320*a*b^3*d^2*e^5 -120960*a*b^2*c*d^3*e^4+120960*a*b*c^2*d^4*e^3-40320*a*c^3*d^5*e^2-10080*b ^4*d^3*e^4+40320*b^3*c*d^4*e^3-60480*b^2*c^2*d^5*e^2+40320*b*c^3*d^6*e-100 80*c^4*d^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+22449*m^4+67284*m^3+118124...
Leaf count of result is larger than twice the leaf count of optimal. 4607 vs. \(2 (449) = 898\).
Time = 0.37 (sec) , antiderivative size = 4607, normalized size of antiderivative = 10.26 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
(a^3*b*d*e^7*m^7 - 10080*c^4*d^8 + 40320*b*c^3*d^7*e + 40320*a^3*b*d*e^7 - 20160*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 40320*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 10080*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 40320*(a*b^3 + 3*a^2*b*c)* d^3*e^5 - 20160*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 2*(c^4*e^8*m^7 + 28*c^4*e^ 8*m^6 + 322*c^4*e^8*m^5 + 1960*c^4*e^8*m^4 + 6769*c^4*e^8*m^3 + 13132*c^4* e^8*m^2 + 13068*c^4*e^8*m + 5040*c^4*e^8)*x^8 + (40320*b*c^3*e^8 + (2*c^4* d*e^7 + 7*b*c^3*e^8)*m^7 + 7*(6*c^4*d*e^7 + 29*b*c^3*e^8)*m^6 + 7*(50*c^4* d*e^7 + 343*b*c^3*e^8)*m^5 + 245*(6*c^4*d*e^7 + 61*b*c^3*e^8)*m^4 + 112*(2 9*c^4*d*e^7 + 469*b*c^3*e^8)*m^3 + 196*(18*c^4*d*e^7 + 527*b*c^3*e^8)*m^2 + 144*(10*c^4*d*e^7 + 721*b*c^3*e^8)*m)*x^7 + (35*a^3*b*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^6 + (20160*(3*b^2*c^2 + 2*a*c^3)*e^8 + (7*b*c^3*d*e ^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^7 - (14*c^4*d^2*e^6 - 161*b*c^3*d*e^7 - 90*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^6 - (210*c^4*d^2*e^6 - 1435*b*c^3*d*e^7 - 1098*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^5 - 5*(238*c^4*d^2*e^6 - 1267*b*c^3*d* e^7 - 1404*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^4 - (3150*c^4*d^2*e^6 - 14518*b*c^ 3*d*e^7 - 25227*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^3 - 2*(1918*c^4*d^2*e^6 - 809 2*b*c^3*d*e^7 - 25245*(3*b^2*c^2 + 2*a*c^3)*e^8)*m^2 - 24*(70*c^4*d^2*e^6 - 280*b*c^3*d*e^7 - 2143*(3*b^2*c^2 + 2*a*c^3)*e^8)*m)*x^6 + (511*a^3*b*d* e^7 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 33*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*m^ 5 + (40320*(b^3*c + 3*a*b*c^2)*e^8 + (3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 5...
Leaf count of result is larger than twice the leaf count of optimal. 76621 vs. \(2 (432) = 864\).
Time = 15.30 (sec) , antiderivative size = 76621, normalized size of antiderivative = 170.65 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**3*b*x + a**3*c*x**2 + 3*a**2*b**2*x**2/2 + 3*a**2*b*c* x**3 + 3*a**2*c**2*x**4/2 + a*b**3*x**3 + 3*a*b**2*c*x**4 + 3*a*b*c**2*x** 5 + a*c**3*x**6 + b**4*x**4/4 + b**3*c*x**5 + 3*b**2*c**2*x**6/2 + b*c**3* x**7 + c**4*x**8/4), Eq(e, 0)), (-60*a**3*b*e**7/(420*d**7*e**8 + 2940*d** 6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12 *x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 20*a* *3*c*d*e**6/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 147 00*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d *e**14*x**6 + 420*e**15*x**7) - 140*a**3*c*e**7*x/(420*d**7*e**8 + 2940*d* *6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**1 2*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 30*a **2*b**2*d*e**6/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13*x**5 + 29 40*d*e**14*x**6 + 420*e**15*x**7) - 210*a**2*b**2*e**7*x/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x**3 + 14700*d* *3*e**12*x**4 + 8820*d**2*e**13*x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 36*a**2*b*c*d**2*e**5/(420*d**7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e** 10*x**2 + 14700*d**4*e**11*x**3 + 14700*d**3*e**12*x**4 + 8820*d**2*e**13* x**5 + 2940*d*e**14*x**6 + 420*e**15*x**7) - 252*a**2*b*c*d*e**6*x/(420*d* *7*e**8 + 2940*d**6*e**9*x + 8820*d**5*e**10*x**2 + 14700*d**4*e**11*x*...
Leaf count of result is larger than twice the leaf count of optimal. 1772 vs. \(2 (449) = 898\).
Time = 0.27 (sec) , antiderivative size = 1772, normalized size of antiderivative = 3.95 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*b^2/((m^2 + 3*m + 2)*e ^2) + 2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^3*c/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a^3*b/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a*b^3/((m^3 + 6* m^2 + 11*m + 6)*e^3) + 9*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a^2*b*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*b^4/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 12*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3 *m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e* x + d)^m*a*b^2*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 6*((m^3 + 6*m ^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e ^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*a^2*c^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 5*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12 *(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*b^3*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 15*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^ 5)*(e*x + d)^m*a*b*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)...
Leaf count of result is larger than twice the leaf count of optimal. 9691 vs. \(2 (449) = 898\).
Time = 0.36 (sec) , antiderivative size = 9691, normalized size of antiderivative = 21.58 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
(2*(e*x + d)^m*c^4*e^8*m^7*x^8 + 2*(e*x + d)^m*c^4*d*e^7*m^7*x^7 + 7*(e*x + d)^m*b*c^3*e^8*m^7*x^7 + 56*(e*x + d)^m*c^4*e^8*m^6*x^8 + 7*(e*x + d)^m* b*c^3*d*e^7*m^7*x^6 + 9*(e*x + d)^m*b^2*c^2*e^8*m^7*x^6 + 6*(e*x + d)^m*a* c^3*e^8*m^7*x^6 + 42*(e*x + d)^m*c^4*d*e^7*m^6*x^7 + 203*(e*x + d)^m*b*c^3 *e^8*m^6*x^7 + 644*(e*x + d)^m*c^4*e^8*m^5*x^8 + 9*(e*x + d)^m*b^2*c^2*d*e ^7*m^7*x^5 + 6*(e*x + d)^m*a*c^3*d*e^7*m^7*x^5 + 5*(e*x + d)^m*b^3*c*e^8*m ^7*x^5 + 15*(e*x + d)^m*a*b*c^2*e^8*m^7*x^5 - 14*(e*x + d)^m*c^4*d^2*e^6*m ^6*x^6 + 161*(e*x + d)^m*b*c^3*d*e^7*m^6*x^6 + 270*(e*x + d)^m*b^2*c^2*e^8 *m^6*x^6 + 180*(e*x + d)^m*a*c^3*e^8*m^6*x^6 + 350*(e*x + d)^m*c^4*d*e^7*m ^5*x^7 + 2401*(e*x + d)^m*b*c^3*e^8*m^5*x^7 + 3920*(e*x + d)^m*c^4*e^8*m^4 *x^8 + 5*(e*x + d)^m*b^3*c*d*e^7*m^7*x^4 + 15*(e*x + d)^m*a*b*c^2*d*e^7*m^ 7*x^4 + (e*x + d)^m*b^4*e^8*m^7*x^4 + 12*(e*x + d)^m*a*b^2*c*e^8*m^7*x^4 + 6*(e*x + d)^m*a^2*c^2*e^8*m^7*x^4 - 42*(e*x + d)^m*b*c^3*d^2*e^6*m^6*x^5 + 225*(e*x + d)^m*b^2*c^2*d*e^7*m^6*x^5 + 150*(e*x + d)^m*a*c^3*d*e^7*m^6* x^5 + 155*(e*x + d)^m*b^3*c*e^8*m^6*x^5 + 465*(e*x + d)^m*a*b*c^2*e^8*m^6* x^5 - 210*(e*x + d)^m*c^4*d^2*e^6*m^5*x^6 + 1435*(e*x + d)^m*b*c^3*d*e^7*m ^5*x^6 + 3294*(e*x + d)^m*b^2*c^2*e^8*m^5*x^6 + 2196*(e*x + d)^m*a*c^3*e^8 *m^5*x^6 + 1470*(e*x + d)^m*c^4*d*e^7*m^4*x^7 + 14945*(e*x + d)^m*b*c^3*e^ 8*m^4*x^7 + 13538*(e*x + d)^m*c^4*e^8*m^3*x^8 + (e*x + d)^m*b^4*d*e^7*m^7* x^3 + 12*(e*x + d)^m*a*b^2*c*d*e^7*m^7*x^3 + 6*(e*x + d)^m*a^2*c^2*d*e^...
Time = 13.24 (sec) , antiderivative size = 4573, normalized size of antiderivative = 10.18 \[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
(x*(d + e*x)^m*(40320*a^3*b*e^8 + 48860*a^3*b*e^8*m^2 + 18424*a^3*b*e^8*m^ 3 + 4025*a^3*b*e^8*m^4 + 511*a^3*b*e^8*m^5 + 35*a^3*b*e^8*m^6 + a^3*b*e^8* m^7 + 10080*b^4*d^3*e^5*m + 6396*b^4*d^3*e^5*m^2 + 1506*b^4*d^3*e^5*m^3 + 156*b^4*d^3*e^5*m^4 + 6*b^4*d^3*e^5*m^5 + 69264*a^3*b*e^8*m + 10080*c^4*d^ 7*e*m + 40320*a^3*c*d*e^7*m + 38376*a^2*c^2*d^3*e^5*m^2 + 9036*a^2*c^2*d^3 *e^5*m^3 + 936*a^2*c^2*d^3*e^5*m^4 + 36*a^2*c^2*d^3*e^5*m^5 + 16200*b^2*c^ 2*d^5*e^3*m^2 + 1080*b^2*c^2*d^5*e^3*m^3 - 40320*a*b^3*d^2*e^6*m + 60480*a ^2*b^2*d*e^7*m + 40320*a*c^3*d^5*e^3*m + 49104*a^3*c*d*e^7*m^2 + 24308*a^3 *c*d*e^7*m^3 + 6270*a^3*c*d*e^7*m^4 + 890*a^3*c*d*e^7*m^5 + 66*a^3*c*d*e^7 *m^6 + 2*a^3*c*d*e^7*m^7 - 40320*b*c^3*d^6*e^2*m - 40320*b^3*c*d^4*e^4*m - 35664*a*b^3*d^2*e^6*m^2 + 73656*a^2*b^2*d*e^7*m^2 - 12420*a*b^3*d^2*e^6*m ^3 + 36462*a^2*b^2*d*e^7*m^3 - 2130*a*b^3*d^2*e^6*m^4 + 9405*a^2*b^2*d*e^7 *m^4 - 180*a*b^3*d^2*e^6*m^5 + 1335*a^2*b^2*d*e^7*m^5 - 6*a*b^3*d^2*e^6*m^ 6 + 99*a^2*b^2*d*e^7*m^6 + 3*a^2*b^2*d*e^7*m^7 + 60480*a^2*c^2*d^3*e^5*m + 10800*a*c^3*d^5*e^3*m^2 + 720*a*c^3*d^5*e^3*m^3 + 60480*b^2*c^2*d^5*e^3*m - 5040*b*c^3*d^6*e^2*m^2 - 17520*b^3*c*d^4*e^4*m^2 - 2520*b^3*c*d^4*e^4*m ^3 - 120*b^3*c*d^4*e^4*m^4 - 52560*a*b*c^2*d^4*e^4*m^2 + 76752*a*b^2*c*d^3 *e^5*m^2 - 106992*a^2*b*c*d^2*e^6*m^2 - 7560*a*b*c^2*d^4*e^4*m^3 + 18072*a *b^2*c*d^3*e^5*m^3 - 37260*a^2*b*c*d^2*e^6*m^3 - 360*a*b*c^2*d^4*e^4*m^4 + 1872*a*b^2*c*d^3*e^5*m^4 - 6390*a^2*b*c*d^2*e^6*m^4 + 72*a*b^2*c*d^3*e...