Integrand size = 97, antiderivative size = 20 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=(d+e x)^5 \left (a+b x+c x^2\right )^6 \] Output:
(e*x+d)^5*(c*x^2+b*x+a)^6
Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(20)=40\).
Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 8.35 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=x \left (6 a^5 (b+c x) (d+e x)^5+15 a^4 x (b+c x)^2 (d+e x)^5+20 a^3 x^2 (b+c x)^3 (d+e x)^5+15 a^2 x^3 (b+c x)^4 (d+e x)^5+6 a x^4 (b+c x)^5 (d+e x)^5+x^5 (b+c x)^6 (d+e x)^5+a^6 e \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )\right ) \] Input:
Integrate[(d + e*x)^4*(a + b*x + c*x^2)^4*(a*(6*b*d + 5*a*e) + 2*(3*b^2*d + 6*a*c*d + 8*a*b*e)*x + (18*b*c*d + 11*b^2*e + 22*a*c*e)*x^2 + 4*c*(3*c*d + 7*b*e)*x^3 + 17*c^2*e*x^4),x]
Output:
x*(6*a^5*(b + c*x)*(d + e*x)^5 + 15*a^4*x*(b + c*x)^2*(d + e*x)^5 + 20*a^3 *x^2*(b + c*x)^3*(d + e*x)^5 + 15*a^2*x^3*(b + c*x)^4*(d + e*x)^5 + 6*a*x^ 4*(b + c*x)^5*(d + e*x)^5 + x^5*(b + c*x)^6*(d + e*x)^5 + a^6*e*(5*d^4 + 1 0*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4))
Leaf count is larger than twice the leaf count of optimal. \(943\) vs. \(2(20)=40\).
Time = 5.88 (sec) , antiderivative size = 943, normalized size of antiderivative = 47.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2159, 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.
| \(\displaystyle \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (x^2 \left (22 a c e+11 b^2 e+18 b c d\right )+2 x \left (8 a b e+6 a c d+3 b^2 d\right )+a (5 a e+6 b d)+4 c x^3 (7 b e+3 c d)+17 c^2 e x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {17 c^6 (d+e x)^{16}}{e^{11}}-\frac {96 c^5 (2 c d-b e) (d+e x)^{15}}{e^{11}}+\frac {45 c^4 \left (22 c^2 d^2+5 b^2 e^2-2 c e (11 b d-a e)\right ) (d+e x)^{14}}{e^{11}}+\frac {140 c^3 (2 c d-b e) \left (-11 c^2 d^2-2 b^2 e^2+c e (11 b d-3 a e)\right ) (d+e x)^{13}}{e^{11}}+\frac {195 c^2 \left (33 c^4 d^4-6 c^3 e (11 b d-3 a e) d^2+b^4 e^4-4 b^2 c e^3 (3 b d-a e)+c^2 e^2 \left (45 b^2 d^2-18 a b e d+a^2 e^2\right )\right ) (d+e x)^{12}}{e^{11}}+\frac {72 c (2 c d-b e) \left (-66 c^4 d^4+12 c^3 e (11 b d-5 a e) d^2-b^4 e^4+2 b^2 c e^3 (9 b d-5 a e)-2 c^2 e^2 \left (42 b^2 d^2-30 a b e d+5 a^2 e^2\right )\right ) (d+e x)^{11}}{e^{11}}+\frac {11 \left (924 c^6 d^6-252 c^5 e (11 b d-5 a e) d^4+210 c^4 e^2 \left (15 b^2 d^2-12 a b e d+2 a^2 e^2\right ) d^2+b^6 e^6-6 b^4 c e^5 (7 b d-5 a e)+30 b^2 c^2 e^4 \left (14 b^2 d^2-14 a b e d+3 a^2 e^2\right )-20 c^3 e^3 \left (84 b^3 d^3-84 a b^2 e d^2+21 a^2 b e^2 d-a^3 e^3\right )\right ) (d+e x)^{10}}{e^{11}}+\frac {60 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (-66 c^4 d^4+132 b c^3 e d^3-60 a c^3 e^2 d^2-84 b^2 c^2 e^2 d^2+60 a b c^2 e^3 d+18 b^3 c e^3 d-b^4 e^4-10 a^2 c^2 e^4-10 a b^2 c e^4\right ) (d+e x)^9}{e^{11}}+\frac {135 \left (c d^2-b e d+a e^2\right )^2 \left (33 c^4 d^4-6 c^3 e (11 b d-3 a e) d^2+b^4 e^4-4 b^2 c e^3 (3 b d-a e)+c^2 e^2 \left (45 b^2 d^2-18 a b e d+a^2 e^2\right )\right ) (d+e x)^8}{e^{11}}+\frac {80 (2 c d-b e) \left (c d^2-b e d+a e^2\right )^3 \left (-11 c^2 d^2-2 b^2 e^2+c e (11 b d-3 a e)\right ) (d+e x)^7}{e^{11}}+\frac {21 \left (c d^2-b e d+a e^2\right )^4 \left (22 c^2 d^2+5 b^2 e^2-2 c e (11 b d-a e)\right ) (d+e x)^6}{e^{11}}+\frac {36 (b e-2 c d) \left (c d^2-b e d+a e^2\right )^5 (d+e x)^5}{e^{11}}+\frac {5 \left (c d^2-b e d+a e^2\right )^6 (d+e x)^4}{e^{11}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^6 (d+e x)^{17}}{e^{12}}-\frac {6 c^5 (2 c d-b e) (d+e x)^{16}}{e^{12}}+\frac {3 c^4 \left (22 c^2 d^2+5 b^2 e^2-2 c e (11 b d-a e)\right ) (d+e x)^{15}}{e^{12}}-\frac {10 c^3 (2 c d-b e) \left (11 c^2 d^2+2 b^2 e^2-c e (11 b d-3 a e)\right ) (d+e x)^{14}}{e^{12}}+\frac {15 c^2 \left (33 c^4 d^4-6 c^3 e (11 b d-3 a e) d^2+b^4 e^4-4 b^2 c e^3 (3 b d-a e)+c^2 e^2 \left (45 b^2 d^2-18 a b e d+a^2 e^2\right )\right ) (d+e x)^{13}}{e^{12}}-\frac {6 c (2 c d-b e) \left (66 c^4 d^4-12 c^3 e (11 b d-5 a e) d^2+b^4 e^4-2 b^2 c e^3 (9 b d-5 a e)+2 c^2 e^2 \left (42 b^2 d^2-30 a b e d+5 a^2 e^2\right )\right ) (d+e x)^{12}}{e^{12}}+\frac {\left (924 c^6 d^6-252 c^5 e (11 b d-5 a e) d^4+210 c^4 e^2 \left (15 b^2 d^2-12 a b e d+2 a^2 e^2\right ) d^2+b^6 e^6-6 b^4 c e^5 (7 b d-5 a e)+30 b^2 c^2 e^4 \left (14 b^2 d^2-14 a b e d+3 a^2 e^2\right )-20 c^3 e^3 \left (84 b^3 d^3-84 a b^2 e d^2+21 a^2 b e^2 d-a^3 e^3\right )\right ) (d+e x)^{11}}{e^{12}}-\frac {6 (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (66 c^4 d^4-12 c^3 e (11 b d-5 a e) d^2+b^4 e^4-2 b^2 c e^3 (9 b d-5 a e)+2 c^2 e^2 \left (42 b^2 d^2-30 a b e d+5 a^2 e^2\right )\right ) (d+e x)^{10}}{e^{12}}+\frac {15 \left (c d^2-b e d+a e^2\right )^2 \left (33 c^4 d^4-6 c^3 e (11 b d-3 a e) d^2+b^4 e^4-4 b^2 c e^3 (3 b d-a e)+c^2 e^2 \left (45 b^2 d^2-18 a b e d+a^2 e^2\right )\right ) (d+e x)^9}{e^{12}}-\frac {10 (2 c d-b e) \left (c d^2-b e d+a e^2\right )^3 \left (11 c^2 d^2+2 b^2 e^2-c e (11 b d-3 a e)\right ) (d+e x)^8}{e^{12}}+\frac {3 \left (c d^2-b e d+a e^2\right )^4 \left (22 c^2 d^2+5 b^2 e^2-2 c e (11 b d-a e)\right ) (d+e x)^7}{e^{12}}-\frac {6 (2 c d-b e) \left (c d^2-b e d+a e^2\right )^5 (d+e x)^6}{e^{12}}+\frac {\left (c d^2-b e d+a e^2\right )^6 (d+e x)^5}{e^{12}}\) |
Input:
Int[(d + e*x)^4*(a + b*x + c*x^2)^4*(a*(6*b*d + 5*a*e) + 2*(3*b^2*d + 6*a* c*d + 8*a*b*e)*x + (18*b*c*d + 11*b^2*e + 22*a*c*e)*x^2 + 4*c*(3*c*d + 7*b *e)*x^3 + 17*c^2*e*x^4),x]
Output:
((c*d^2 - b*d*e + a*e^2)^6*(d + e*x)^5)/e^12 - (6*(2*c*d - b*e)*(c*d^2 - b *d*e + a*e^2)^5*(d + e*x)^6)/e^12 + (3*(c*d^2 - b*d*e + a*e^2)^4*(22*c^2*d ^2 + 5*b^2*e^2 - 2*c*e*(11*b*d - a*e))*(d + e*x)^7)/e^12 - (10*(2*c*d - b* e)*(c*d^2 - b*d*e + a*e^2)^3*(11*c^2*d^2 + 2*b^2*e^2 - c*e*(11*b*d - 3*a*e ))*(d + e*x)^8)/e^12 + (15*(c*d^2 - b*d*e + a*e^2)^2*(33*c^4*d^4 + b^4*e^4 - 6*c^3*d^2*e*(11*b*d - 3*a*e) - 4*b^2*c*e^3*(3*b*d - a*e) + c^2*e^2*(45* b^2*d^2 - 18*a*b*d*e + a^2*e^2))*(d + e*x)^9)/e^12 - (6*(2*c*d - b*e)*(c*d ^2 - b*d*e + a*e^2)*(66*c^4*d^4 + b^4*e^4 - 2*b^2*c*e^3*(9*b*d - 5*a*e) - 12*c^3*d^2*e*(11*b*d - 5*a*e) + 2*c^2*e^2*(42*b^2*d^2 - 30*a*b*d*e + 5*a^2 *e^2))*(d + e*x)^10)/e^12 + ((924*c^6*d^6 + b^6*e^6 - 6*b^4*c*e^5*(7*b*d - 5*a*e) - 252*c^5*d^4*e*(11*b*d - 5*a*e) + 210*c^4*d^2*e^2*(15*b^2*d^2 - 1 2*a*b*d*e + 2*a^2*e^2) + 30*b^2*c^2*e^4*(14*b^2*d^2 - 14*a*b*d*e + 3*a^2*e ^2) - 20*c^3*e^3*(84*b^3*d^3 - 84*a*b^2*d^2*e + 21*a^2*b*d*e^2 - a^3*e^3)) *(d + e*x)^11)/e^12 - (6*c*(2*c*d - b*e)*(66*c^4*d^4 + b^4*e^4 - 2*b^2*c*e ^3*(9*b*d - 5*a*e) - 12*c^3*d^2*e*(11*b*d - 5*a*e) + 2*c^2*e^2*(42*b^2*d^2 - 30*a*b*d*e + 5*a^2*e^2))*(d + e*x)^12)/e^12 + (15*c^2*(33*c^4*d^4 + b^4 *e^4 - 6*c^3*d^2*e*(11*b*d - 3*a*e) - 4*b^2*c*e^3*(3*b*d - a*e) + c^2*e^2* (45*b^2*d^2 - 18*a*b*d*e + a^2*e^2))*(d + e*x)^13)/e^12 - (10*c^3*(2*c*d - b*e)*(11*c^2*d^2 + 2*b^2*e^2 - c*e*(11*b*d - 3*a*e))*(d + e*x)^14)/e^12 + (3*c^4*(22*c^2*d^2 + 5*b^2*e^2 - 2*c*e*(11*b*d - a*e))*(d + e*x)^15)/e...
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(2051\) vs. \(2(20)=40\).
Time = 0.31 (sec) , antiderivative size = 2052, normalized size of antiderivative = 102.60
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(2052\) |
| gosper | \(\text {Expression too large to display}\) | \(2460\) |
| risch | \(\text {Expression too large to display}\) | \(2468\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2468\) |
| orering | \(\text {Expression too large to display}\) | \(2577\) |
| default | \(\text {Expression too large to display}\) | \(5359\) |
Input:
int((e*x+d)^4*(c*x^2+b*x+a)^4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d+3*b^2*d) *x+(22*a*c*e+11*b^2*e+18*b*c*d)*x^2+4*c*(7*b*e+3*c*d)*x^3+17*c^2*e*x^4),x, method=_RETURNVERBOSE)
Output:
(5*a^6*d^4*e+6*a^5*b*d^5)*x+(10*a^6*d^3*e^2+30*a^5*b*d^4*e+6*a^5*c*d^5+15* a^4*b^2*d^5)*x^2+(10*a^6*d^2*e^3+60*a^5*b*d^3*e^2+30*a^5*c*d^4*e+75*a^4*b^ 2*d^4*e+30*a^4*b*c*d^5+20*a^3*b^3*d^5)*x^3+(5*a^6*d*e^4+60*a^5*b*d^2*e^3+6 0*a^5*c*d^3*e^2+150*a^4*b^2*d^3*e^2+150*a^4*b*c*d^4*e+15*a^4*c^2*d^5+100*a ^3*b^3*d^4*e+60*a^3*b^2*c*d^5+15*a^2*b^4*d^5)*x^4+(a^6*e^5+30*a^5*b*d*e^4+ 60*a^5*c*d^2*e^3+150*a^4*b^2*d^2*e^3+300*a^4*b*c*d^3*e^2+75*a^4*c^2*d^4*e+ 200*a^3*b^3*d^3*e^2+300*a^3*b^2*c*d^4*e+60*a^3*b*c^2*d^5+75*a^2*b^4*d^4*e+ 60*a^2*b^3*c*d^5+6*a*b^5*d^5)*x^5+(6*a^5*b*e^5+30*a^5*c*d*e^4+75*a^4*b^2*d *e^4+300*a^4*b*c*d^2*e^3+150*a^4*c^2*d^3*e^2+200*a^3*b^3*d^2*e^3+600*a^3*b ^2*c*d^3*e^2+300*a^3*b*c^2*d^4*e+20*a^3*c^3*d^5+150*a^2*b^4*d^3*e^2+300*a^ 2*b^3*c*d^4*e+90*a^2*b^2*c^2*d^5+30*a*b^5*d^4*e+30*a*b^4*c*d^5+b^6*d^5)*x^ 6+(6*a^5*c*e^5+15*a^4*b^2*e^5+150*a^4*b*c*d*e^4+150*a^4*c^2*d^2*e^3+100*a^ 3*b^3*d*e^4+600*a^3*b^2*c*d^2*e^3+600*a^3*b*c^2*d^3*e^2+100*a^3*c^3*d^4*e+ 150*a^2*b^4*d^2*e^3+600*a^2*b^3*c*d^3*e^2+450*a^2*b^2*c^2*d^4*e+60*a^2*b*c ^3*d^5+60*a*b^5*d^3*e^2+150*a*b^4*c*d^4*e+60*a*b^3*c^2*d^5+5*b^6*d^4*e+6*b ^5*c*d^5)*x^7+(30*a^4*b*c*e^5+75*a^4*c^2*d*e^4+20*a^3*b^3*e^5+300*a^3*b^2* c*d*e^4+600*a^3*b*c^2*d^2*e^3+200*a^3*c^3*d^3*e^2+75*a^2*b^4*d*e^4+600*a^2 *b^3*c*d^2*e^3+900*a^2*b^2*c^2*d^3*e^2+300*a^2*b*c^3*d^4*e+15*a^2*c^4*d^5+ 60*a*b^5*d^2*e^3+300*a*b^4*c*d^3*e^2+300*a*b^3*c^2*d^4*e+60*a*b^2*c^3*d^5+ 10*b^6*d^3*e^2+30*b^5*c*d^4*e+15*b^4*c^2*d^5)*x^8+(15*a^4*c^2*e^5+60*a^...
Leaf count of result is larger than twice the leaf count of optimal. 1779 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 1779, normalized size of antiderivative = 88.95 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d+3* b^2*d)*x+(22*a*c*e+11*b^2*e+18*b*c*d)*x^2+4*c*(7*b*e+3*c*d)*x^3+17*c^2*e*x ^4),x, algorithm="fricas")
Output:
c^6*e^5*x^17 + (5*c^6*d*e^4 + 6*b*c^5*e^5)*x^16 + (10*c^6*d^2*e^3 + 30*b*c ^5*d*e^4 + 3*(5*b^2*c^4 + 2*a*c^5)*e^5)*x^15 + 5*(2*c^6*d^3*e^2 + 12*b*c^5 *d^2*e^3 + 3*(5*b^2*c^4 + 2*a*c^5)*d*e^4 + 2*(2*b^3*c^3 + 3*a*b*c^4)*e^5)* x^14 + 5*(c^6*d^4*e + 12*b*c^5*d^3*e^2 + 6*(5*b^2*c^4 + 2*a*c^5)*d^2*e^3 + 10*(2*b^3*c^3 + 3*a*b*c^4)*d*e^4 + 3*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*e^ 5)*x^13 + (c^6*d^5 + 30*b*c^5*d^4*e + 30*(5*b^2*c^4 + 2*a*c^5)*d^3*e^2 + 1 00*(2*b^3*c^3 + 3*a*b*c^4)*d^2*e^3 + 75*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)* d*e^4 + 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*e^5)*x^12 + (6*b*c^5*d^5 + 15*(5*b^2*c^4 + 2*a*c^5)*d^4*e + 100*(2*b^3*c^3 + 3*a*b*c^4)*d^3*e^2 + 15 0*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^2*e^3 + 30*(b^5*c + 10*a*b^3*c^2 + 1 0*a^2*b*c^3)*d*e^4 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*e^5) *x^11 + (3*(5*b^2*c^4 + 2*a*c^5)*d^5 + 50*(2*b^3*c^3 + 3*a*b*c^4)*d^4*e + 150*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^3*e^2 + 60*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^2*e^3 + 5*(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3 )*d*e^4 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*e^5)*x^10 + 5*(2*(2*b^3* c^3 + 3*a*b*c^4)*d^5 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^4*e + 12*(b^ 5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^3*e^2 + 2*(b^6 + 30*a*b^4*c + 90*a^2* b^2*c^2 + 20*a^3*c^3)*d^2*e^3 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d* e^4 + 3*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*e^5)*x^9 + 5*(3*(b^4*c^2 + 4*a*b ^2*c^3 + a^2*c^4)*d^5 + 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^4*e +...
Leaf count of result is larger than twice the leaf count of optimal. 2281 vs. \(2 (17) = 34\).
Time = 0.15 (sec) , antiderivative size = 2281, normalized size of antiderivative = 114.05 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**4*(c*x**2+b*x+a)**4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d +3*b**2*d)*x+(22*a*c*e+11*b**2*e+18*b*c*d)*x**2+4*c*(7*b*e+3*c*d)*x**3+17* c**2*e*x**4),x)
Output:
c**6*e**5*x**17 + x**16*(6*b*c**5*e**5 + 5*c**6*d*e**4) + x**15*(6*a*c**5* e**5 + 15*b**2*c**4*e**5 + 30*b*c**5*d*e**4 + 10*c**6*d**2*e**3) + x**14*( 30*a*b*c**4*e**5 + 30*a*c**5*d*e**4 + 20*b**3*c**3*e**5 + 75*b**2*c**4*d*e **4 + 60*b*c**5*d**2*e**3 + 10*c**6*d**3*e**2) + x**13*(15*a**2*c**4*e**5 + 60*a*b**2*c**3*e**5 + 150*a*b*c**4*d*e**4 + 60*a*c**5*d**2*e**3 + 15*b** 4*c**2*e**5 + 100*b**3*c**3*d*e**4 + 150*b**2*c**4*d**2*e**3 + 60*b*c**5*d **3*e**2 + 5*c**6*d**4*e) + x**12*(60*a**2*b*c**3*e**5 + 75*a**2*c**4*d*e* *4 + 60*a*b**3*c**2*e**5 + 300*a*b**2*c**3*d*e**4 + 300*a*b*c**4*d**2*e**3 + 60*a*c**5*d**3*e**2 + 6*b**5*c*e**5 + 75*b**4*c**2*d*e**4 + 200*b**3*c* *3*d**2*e**3 + 150*b**2*c**4*d**3*e**2 + 30*b*c**5*d**4*e + c**6*d**5) + x **11*(20*a**3*c**3*e**5 + 90*a**2*b**2*c**2*e**5 + 300*a**2*b*c**3*d*e**4 + 150*a**2*c**4*d**2*e**3 + 30*a*b**4*c*e**5 + 300*a*b**3*c**2*d*e**4 + 60 0*a*b**2*c**3*d**2*e**3 + 300*a*b*c**4*d**3*e**2 + 30*a*c**5*d**4*e + b**6 *e**5 + 30*b**5*c*d*e**4 + 150*b**4*c**2*d**2*e**3 + 200*b**3*c**3*d**3*e* *2 + 75*b**2*c**4*d**4*e + 6*b*c**5*d**5) + x**10*(60*a**3*b*c**2*e**5 + 1 00*a**3*c**3*d*e**4 + 60*a**2*b**3*c*e**5 + 450*a**2*b**2*c**2*d*e**4 + 60 0*a**2*b*c**3*d**2*e**3 + 150*a**2*c**4*d**3*e**2 + 6*a*b**5*e**5 + 150*a* b**4*c*d*e**4 + 600*a*b**3*c**2*d**2*e**3 + 600*a*b**2*c**3*d**3*e**2 + 15 0*a*b*c**4*d**4*e + 6*a*c**5*d**5 + 5*b**6*d*e**4 + 60*b**5*c*d**2*e**3 + 150*b**4*c**2*d**3*e**2 + 100*b**3*c**3*d**4*e + 15*b**2*c**4*d**5) + x...
Leaf count of result is larger than twice the leaf count of optimal. 1779 vs. \(2 (20) = 40\).
Time = 0.04 (sec) , antiderivative size = 1779, normalized size of antiderivative = 88.95 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d+3* b^2*d)*x+(22*a*c*e+11*b^2*e+18*b*c*d)*x^2+4*c*(7*b*e+3*c*d)*x^3+17*c^2*e*x ^4),x, algorithm="maxima")
Output:
c^6*e^5*x^17 + (5*c^6*d*e^4 + 6*b*c^5*e^5)*x^16 + (10*c^6*d^2*e^3 + 30*b*c ^5*d*e^4 + 3*(5*b^2*c^4 + 2*a*c^5)*e^5)*x^15 + 5*(2*c^6*d^3*e^2 + 12*b*c^5 *d^2*e^3 + 3*(5*b^2*c^4 + 2*a*c^5)*d*e^4 + 2*(2*b^3*c^3 + 3*a*b*c^4)*e^5)* x^14 + 5*(c^6*d^4*e + 12*b*c^5*d^3*e^2 + 6*(5*b^2*c^4 + 2*a*c^5)*d^2*e^3 + 10*(2*b^3*c^3 + 3*a*b*c^4)*d*e^4 + 3*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*e^ 5)*x^13 + (c^6*d^5 + 30*b*c^5*d^4*e + 30*(5*b^2*c^4 + 2*a*c^5)*d^3*e^2 + 1 00*(2*b^3*c^3 + 3*a*b*c^4)*d^2*e^3 + 75*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)* d*e^4 + 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*e^5)*x^12 + (6*b*c^5*d^5 + 15*(5*b^2*c^4 + 2*a*c^5)*d^4*e + 100*(2*b^3*c^3 + 3*a*b*c^4)*d^3*e^2 + 15 0*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^2*e^3 + 30*(b^5*c + 10*a*b^3*c^2 + 1 0*a^2*b*c^3)*d*e^4 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*e^5) *x^11 + (3*(5*b^2*c^4 + 2*a*c^5)*d^5 + 50*(2*b^3*c^3 + 3*a*b*c^4)*d^4*e + 150*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^3*e^2 + 60*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^2*e^3 + 5*(b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3 )*d*e^4 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*e^5)*x^10 + 5*(2*(2*b^3* c^3 + 3*a*b*c^4)*d^5 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^4*e + 12*(b^ 5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^3*e^2 + 2*(b^6 + 30*a*b^4*c + 90*a^2* b^2*c^2 + 20*a^3*c^3)*d^2*e^3 + 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d* e^4 + 3*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*e^5)*x^9 + 5*(3*(b^4*c^2 + 4*a*b ^2*c^3 + a^2*c^4)*d^5 + 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^4*e +...
Leaf count of result is larger than twice the leaf count of optimal. 2467 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 2467, normalized size of antiderivative = 123.35 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d+3* b^2*d)*x+(22*a*c*e+11*b^2*e+18*b*c*d)*x^2+4*c*(7*b*e+3*c*d)*x^3+17*c^2*e*x ^4),x, algorithm="giac")
Output:
c^6*e^5*x^17 + 5*c^6*d*e^4*x^16 + 6*b*c^5*e^5*x^16 + 10*c^6*d^2*e^3*x^15 + 30*b*c^5*d*e^4*x^15 + 15*b^2*c^4*e^5*x^15 + 6*a*c^5*e^5*x^15 + 10*c^6*d^3 *e^2*x^14 + 60*b*c^5*d^2*e^3*x^14 + 75*b^2*c^4*d*e^4*x^14 + 30*a*c^5*d*e^4 *x^14 + 20*b^3*c^3*e^5*x^14 + 30*a*b*c^4*e^5*x^14 + 5*c^6*d^4*e*x^13 + 60* b*c^5*d^3*e^2*x^13 + 150*b^2*c^4*d^2*e^3*x^13 + 60*a*c^5*d^2*e^3*x^13 + 10 0*b^3*c^3*d*e^4*x^13 + 150*a*b*c^4*d*e^4*x^13 + 15*b^4*c^2*e^5*x^13 + 60*a *b^2*c^3*e^5*x^13 + 15*a^2*c^4*e^5*x^13 + c^6*d^5*x^12 + 30*b*c^5*d^4*e*x^ 12 + 150*b^2*c^4*d^3*e^2*x^12 + 60*a*c^5*d^3*e^2*x^12 + 200*b^3*c^3*d^2*e^ 3*x^12 + 300*a*b*c^4*d^2*e^3*x^12 + 75*b^4*c^2*d*e^4*x^12 + 300*a*b^2*c^3* d*e^4*x^12 + 75*a^2*c^4*d*e^4*x^12 + 6*b^5*c*e^5*x^12 + 60*a*b^3*c^2*e^5*x ^12 + 60*a^2*b*c^3*e^5*x^12 + 6*b*c^5*d^5*x^11 + 75*b^2*c^4*d^4*e*x^11 + 3 0*a*c^5*d^4*e*x^11 + 200*b^3*c^3*d^3*e^2*x^11 + 300*a*b*c^4*d^3*e^2*x^11 + 150*b^4*c^2*d^2*e^3*x^11 + 600*a*b^2*c^3*d^2*e^3*x^11 + 150*a^2*c^4*d^2*e ^3*x^11 + 30*b^5*c*d*e^4*x^11 + 300*a*b^3*c^2*d*e^4*x^11 + 300*a^2*b*c^3*d *e^4*x^11 + b^6*e^5*x^11 + 30*a*b^4*c*e^5*x^11 + 90*a^2*b^2*c^2*e^5*x^11 + 20*a^3*c^3*e^5*x^11 + 15*b^2*c^4*d^5*x^10 + 6*a*c^5*d^5*x^10 + 100*b^3*c^ 3*d^4*e*x^10 + 150*a*b*c^4*d^4*e*x^10 + 150*b^4*c^2*d^3*e^2*x^10 + 600*a*b ^2*c^3*d^3*e^2*x^10 + 150*a^2*c^4*d^3*e^2*x^10 + 60*b^5*c*d^2*e^3*x^10 + 6 00*a*b^3*c^2*d^2*e^3*x^10 + 600*a^2*b*c^3*d^2*e^3*x^10 + 5*b^6*d*e^4*x^10 + 150*a*b^4*c*d*e^4*x^10 + 450*a^2*b^2*c^2*d*e^4*x^10 + 100*a^3*c^3*d*e...
Time = 0.63 (sec) , antiderivative size = 2026, normalized size of antiderivative = 101.30 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx=\text {Too large to display} \] Input:
int((d + e*x)^4*(a + b*x + c*x^2)^4*(x^2*(11*b^2*e + 22*a*c*e + 18*b*c*d) + a*(5*a*e + 6*b*d) + 2*x*(3*b^2*d + 8*a*b*e + 6*a*c*d) + 4*c*x^3*(7*b*e + 3*c*d) + 17*c^2*e*x^4),x)
Output:
x^6*(b^6*d^5 + 6*a^5*b*e^5 + 20*a^3*c^3*d^5 + 75*a^4*b^2*d*e^4 + 90*a^2*b^ 2*c^2*d^5 + 150*a^2*b^4*d^3*e^2 + 200*a^3*b^3*d^2*e^3 + 150*a^4*c^2*d^3*e^ 2 + 30*a*b^4*c*d^5 + 30*a*b^5*d^4*e + 30*a^5*c*d*e^4 + 300*a^2*b^3*c*d^4*e + 300*a^3*b*c^2*d^4*e + 300*a^4*b*c*d^2*e^3 + 600*a^3*b^2*c*d^3*e^2) + x^ 11*(b^6*e^5 + 6*b*c^5*d^5 + 20*a^3*c^3*e^5 + 75*b^2*c^4*d^4*e + 90*a^2*b^2 *c^2*e^5 + 150*a^2*c^4*d^2*e^3 + 200*b^3*c^3*d^3*e^2 + 150*b^4*c^2*d^2*e^3 + 30*a*b^4*c*e^5 + 30*a*c^5*d^4*e + 30*b^5*c*d*e^4 + 300*a*b*c^4*d^3*e^2 + 300*a*b^3*c^2*d*e^4 + 300*a^2*b*c^3*d*e^4 + 600*a*b^2*c^3*d^2*e^3) + x^5 *(a^6*e^5 + 6*a*b^5*d^5 + 60*a^2*b^3*c*d^5 + 60*a^3*b*c^2*d^5 + 75*a^2*b^4 *d^4*e + 75*a^4*c^2*d^4*e + 60*a^5*c*d^2*e^3 + 200*a^3*b^3*d^3*e^2 + 150*a ^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + 300*a^3*b^2*c*d^4*e + 300*a^4*b*c*d^3*e^ 2) + x^3*(20*a^3*b^3*d^5 + 10*a^6*d^2*e^3 + 75*a^4*b^2*d^4*e + 60*a^5*b*d^ 3*e^2 + 30*a^4*b*c*d^5 + 30*a^5*c*d^4*e) + x^12*(c^6*d^5 + 6*b^5*c*e^5 + 6 0*a*b^3*c^2*e^5 + 60*a^2*b*c^3*e^5 + 60*a*c^5*d^3*e^2 + 75*a^2*c^4*d*e^4 + 75*b^4*c^2*d*e^4 + 150*b^2*c^4*d^3*e^2 + 200*b^3*c^3*d^2*e^3 + 30*b*c^5*d ^4*e + 300*a*b*c^4*d^2*e^3 + 300*a*b^2*c^3*d*e^4) + x^7*(6*a^5*c*e^5 + 6*b ^5*c*d^5 + 5*b^6*d^4*e + 15*a^4*b^2*e^5 + 60*a*b^3*c^2*d^5 + 60*a^2*b*c^3* d^5 + 60*a*b^5*d^3*e^2 + 100*a^3*b^3*d*e^4 + 100*a^3*c^3*d^4*e + 150*a^2*b ^4*d^2*e^3 + 150*a^4*c^2*d^2*e^3 + 150*a*b^4*c*d^4*e + 150*a^4*b*c*d*e^4 + 450*a^2*b^2*c^2*d^4*e + 600*a^2*b^3*c*d^3*e^2 + 600*a^3*b*c^2*d^3*e^2 ...
Time = 0.19 (sec) , antiderivative size = 2459, normalized size of antiderivative = 122.95 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \left (a (6 b d+5 a e)+2 \left (3 b^2 d+6 a c d+8 a b e\right ) x+\left (18 b c d+11 b^2 e+22 a c e\right ) x^2+4 c (3 c d+7 b e) x^3+17 c^2 e x^4\right ) \, dx =\text {Too large to display} \] Input:
int((e*x+d)^4*(c*x^2+b*x+a)^4*(a*(5*a*e+6*b*d)+2*(8*a*b*e+6*a*c*d+3*b^2*d) *x+(22*a*c*e+11*b^2*e+18*b*c*d)*x^2+4*c*(7*b*e+3*c*d)*x^3+17*c^2*e*x^4),x)
Output:
x*(5*a**6*d**4*e + 10*a**6*d**3*e**2*x + 10*a**6*d**2*e**3*x**2 + 5*a**6*d *e**4*x**3 + a**6*e**5*x**4 + 6*a**5*b*d**5 + 30*a**5*b*d**4*e*x + 60*a**5 *b*d**3*e**2*x**2 + 60*a**5*b*d**2*e**3*x**3 + 30*a**5*b*d*e**4*x**4 + 6*a **5*b*e**5*x**5 + 6*a**5*c*d**5*x + 30*a**5*c*d**4*e*x**2 + 60*a**5*c*d**3 *e**2*x**3 + 60*a**5*c*d**2*e**3*x**4 + 30*a**5*c*d*e**4*x**5 + 6*a**5*c*e **5*x**6 + 15*a**4*b**2*d**5*x + 75*a**4*b**2*d**4*e*x**2 + 150*a**4*b**2* d**3*e**2*x**3 + 150*a**4*b**2*d**2*e**3*x**4 + 75*a**4*b**2*d*e**4*x**5 + 15*a**4*b**2*e**5*x**6 + 30*a**4*b*c*d**5*x**2 + 150*a**4*b*c*d**4*e*x**3 + 300*a**4*b*c*d**3*e**2*x**4 + 300*a**4*b*c*d**2*e**3*x**5 + 150*a**4*b* c*d*e**4*x**6 + 30*a**4*b*c*e**5*x**7 + 15*a**4*c**2*d**5*x**3 + 75*a**4*c **2*d**4*e*x**4 + 150*a**4*c**2*d**3*e**2*x**5 + 150*a**4*c**2*d**2*e**3*x **6 + 75*a**4*c**2*d*e**4*x**7 + 15*a**4*c**2*e**5*x**8 + 20*a**3*b**3*d** 5*x**2 + 100*a**3*b**3*d**4*e*x**3 + 200*a**3*b**3*d**3*e**2*x**4 + 200*a* *3*b**3*d**2*e**3*x**5 + 100*a**3*b**3*d*e**4*x**6 + 20*a**3*b**3*e**5*x** 7 + 60*a**3*b**2*c*d**5*x**3 + 300*a**3*b**2*c*d**4*e*x**4 + 600*a**3*b**2 *c*d**3*e**2*x**5 + 600*a**3*b**2*c*d**2*e**3*x**6 + 300*a**3*b**2*c*d*e** 4*x**7 + 60*a**3*b**2*c*e**5*x**8 + 60*a**3*b*c**2*d**5*x**4 + 300*a**3*b* c**2*d**4*e*x**5 + 600*a**3*b*c**2*d**3*e**2*x**6 + 600*a**3*b*c**2*d**2*e **3*x**7 + 300*a**3*b*c**2*d*e**4*x**8 + 60*a**3*b*c**2*e**5*x**9 + 20*a** 3*c**3*d**5*x**5 + 100*a**3*c**3*d**4*e*x**6 + 200*a**3*c**3*d**3*e**2*...