Integrand size = 35, antiderivative size = 111 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \] Output:
-1/6*(-a*e^2+c*d^2)^3*(e*x+d)^6/e^4+3/7*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^7/e^4 -3/8*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^8/e^4+1/9*c^3*d^3*(e*x+d)^9/e^4
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(111)=222\).
Time = 0.05 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.30 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{504} x \left (84 a^3 e^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 c d e^2 x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right ) \] Input:
Integrate[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(x*(84*a^3*e^3*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d *e^4*x^4 + e^5*x^5) + 36*a^2*c*d*e^2*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2* x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 9*a*c^2*d^2*e*x^2*(56*d ^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21* e^5*x^5) + c^3*d^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2* e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)))/504
Time = 0.64 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1121, 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.
| \(\displaystyle \int (d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (-\frac {3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{e^3}+\frac {3 c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{e^3}+\frac {(d+e x)^5 \left (a e^2-c d^2\right )^3}{e^3}+\frac {c^3 d^3 (d+e x)^8}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac {3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4}\) |
Input:
Int[(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
-1/6*((c*d^2 - a*e^2)^3*(d + e*x)^6)/e^4 + (3*c*d*(c*d^2 - a*e^2)^2*(d + e *x)^7)/(7*e^4) - (3*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^8)/(8*e^4) + (c^3*d^ 3*(d + e*x)^9)/(9*e^4)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(103)=206\).
Time = 1.32 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.69
| method | result | size |
| norman | \(\frac {d^{3} c^{3} e^{5} x^{9}}{9}+\left (\frac {3}{8} a \,e^{6} c^{2} d^{2}+\frac {5}{8} d^{4} c^{3} e^{4}\right ) x^{8}+\left (\frac {3}{7} a^{2} e^{7} c d +\frac {15}{7} a \,e^{5} c^{2} d^{3}+\frac {10}{7} d^{5} c^{3} e^{3}\right ) x^{7}+\left (\frac {1}{6} a^{3} e^{8}+\frac {5}{2} a^{2} c \,d^{2} e^{6}+5 a \,c^{2} d^{4} e^{4}+\frac {5}{3} c^{3} d^{6} e^{2}\right ) x^{6}+\left (a^{3} d \,e^{7}+6 a^{2} c \,d^{3} e^{5}+6 a \,c^{2} d^{5} e^{3}+c^{3} d^{7} e \right ) x^{5}+\left (\frac {5}{2} e^{6} a^{3} d^{2}+\frac {15}{2} a^{2} e^{4} c \,d^{4}+\frac {15}{4} a \,e^{2} c^{2} d^{6}+\frac {1}{4} d^{8} c^{3}\right ) x^{4}+\left (\frac {10}{3} e^{5} a^{3} d^{3}+5 a^{2} e^{3} c \,d^{5}+a e \,c^{2} d^{7}\right ) x^{3}+\left (\frac {5}{2} e^{4} a^{3} d^{4}+\frac {3}{2} a^{2} e^{2} c \,d^{6}\right ) x^{2}+e^{3} a^{3} d^{5} x\) | \(299\) |
| risch | \(\frac {1}{9} d^{3} c^{3} e^{5} x^{9}+\frac {3}{8} x^{8} a \,e^{6} c^{2} d^{2}+\frac {5}{8} x^{8} d^{4} c^{3} e^{4}+\frac {3}{7} x^{7} a^{2} e^{7} c d +\frac {15}{7} x^{7} a \,e^{5} c^{2} d^{3}+\frac {10}{7} x^{7} d^{5} c^{3} e^{3}+\frac {1}{6} x^{6} a^{3} e^{8}+\frac {5}{2} x^{6} a^{2} c \,d^{2} e^{6}+5 x^{6} a \,c^{2} d^{4} e^{4}+\frac {5}{3} x^{6} c^{3} d^{6} e^{2}+a^{3} d \,e^{7} x^{5}+6 a^{2} c \,d^{3} e^{5} x^{5}+6 a \,c^{2} d^{5} e^{3} x^{5}+c^{3} d^{7} e \,x^{5}+\frac {5}{2} x^{4} e^{6} a^{3} d^{2}+\frac {15}{2} x^{4} a^{2} e^{4} c \,d^{4}+\frac {15}{4} x^{4} a \,e^{2} c^{2} d^{6}+\frac {1}{4} x^{4} d^{8} c^{3}+\frac {10}{3} x^{3} e^{5} a^{3} d^{3}+5 x^{3} a^{2} e^{3} c \,d^{5}+x^{3} a e \,c^{2} d^{7}+\frac {5}{2} x^{2} e^{4} a^{3} d^{4}+\frac {3}{2} x^{2} a^{2} e^{2} c \,d^{6}+e^{3} a^{3} d^{5} x\) | \(330\) |
| parallelrisch | \(\frac {1}{9} d^{3} c^{3} e^{5} x^{9}+\frac {3}{8} x^{8} a \,e^{6} c^{2} d^{2}+\frac {5}{8} x^{8} d^{4} c^{3} e^{4}+\frac {3}{7} x^{7} a^{2} e^{7} c d +\frac {15}{7} x^{7} a \,e^{5} c^{2} d^{3}+\frac {10}{7} x^{7} d^{5} c^{3} e^{3}+\frac {1}{6} x^{6} a^{3} e^{8}+\frac {5}{2} x^{6} a^{2} c \,d^{2} e^{6}+5 x^{6} a \,c^{2} d^{4} e^{4}+\frac {5}{3} x^{6} c^{3} d^{6} e^{2}+a^{3} d \,e^{7} x^{5}+6 a^{2} c \,d^{3} e^{5} x^{5}+6 a \,c^{2} d^{5} e^{3} x^{5}+c^{3} d^{7} e \,x^{5}+\frac {5}{2} x^{4} e^{6} a^{3} d^{2}+\frac {15}{2} x^{4} a^{2} e^{4} c \,d^{4}+\frac {15}{4} x^{4} a \,e^{2} c^{2} d^{6}+\frac {1}{4} x^{4} d^{8} c^{3}+\frac {10}{3} x^{3} e^{5} a^{3} d^{3}+5 x^{3} a^{2} e^{3} c \,d^{5}+x^{3} a e \,c^{2} d^{7}+\frac {5}{2} x^{2} e^{4} a^{3} d^{4}+\frac {3}{2} x^{2} a^{2} e^{2} c \,d^{6}+e^{3} a^{3} d^{5} x\) | \(330\) |
| gosper | \(\frac {x \left (56 d^{3} c^{3} e^{5} x^{8}+189 x^{7} a \,e^{6} c^{2} d^{2}+315 x^{7} d^{4} c^{3} e^{4}+216 x^{6} a^{2} e^{7} c d +1080 x^{6} a \,e^{5} c^{2} d^{3}+720 x^{6} d^{5} c^{3} e^{3}+84 x^{5} a^{3} e^{8}+1260 x^{5} a^{2} c \,d^{2} e^{6}+2520 x^{5} a \,c^{2} d^{4} e^{4}+840 x^{5} c^{3} d^{6} e^{2}+504 a^{3} d \,e^{7} x^{4}+3024 a^{2} c \,d^{3} e^{5} x^{4}+3024 a \,c^{2} d^{5} e^{3} x^{4}+504 c^{3} d^{7} e \,x^{4}+1260 x^{3} e^{6} a^{3} d^{2}+3780 x^{3} a^{2} e^{4} c \,d^{4}+1890 x^{3} a \,e^{2} c^{2} d^{6}+126 x^{3} d^{8} c^{3}+1680 x^{2} e^{5} a^{3} d^{3}+2520 x^{2} a^{2} e^{3} c \,d^{5}+504 x^{2} a e \,c^{2} d^{7}+1260 x \,e^{4} a^{3} d^{4}+756 x \,a^{2} e^{2} c \,d^{6}+504 e^{3} a^{3} d^{5}\right )}{504}\) | \(332\) |
| orering | \(\frac {x \left (56 d^{3} c^{3} e^{5} x^{8}+189 x^{7} a \,e^{6} c^{2} d^{2}+315 x^{7} d^{4} c^{3} e^{4}+216 x^{6} a^{2} e^{7} c d +1080 x^{6} a \,e^{5} c^{2} d^{3}+720 x^{6} d^{5} c^{3} e^{3}+84 x^{5} a^{3} e^{8}+1260 x^{5} a^{2} c \,d^{2} e^{6}+2520 x^{5} a \,c^{2} d^{4} e^{4}+840 x^{5} c^{3} d^{6} e^{2}+504 a^{3} d \,e^{7} x^{4}+3024 a^{2} c \,d^{3} e^{5} x^{4}+3024 a \,c^{2} d^{5} e^{3} x^{4}+504 c^{3} d^{7} e \,x^{4}+1260 x^{3} e^{6} a^{3} d^{2}+3780 x^{3} a^{2} e^{4} c \,d^{4}+1890 x^{3} a \,e^{2} c^{2} d^{6}+126 x^{3} d^{8} c^{3}+1680 x^{2} e^{5} a^{3} d^{3}+2520 x^{2} a^{2} e^{3} c \,d^{5}+504 x^{2} a e \,c^{2} d^{7}+1260 x \,e^{4} a^{3} d^{4}+756 x \,a^{2} e^{2} c \,d^{6}+504 e^{3} a^{3} d^{5}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{504 \left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\) | \(376\) |
| default | \(\frac {d^{3} c^{3} e^{5} x^{9}}{9}+\frac {\left (2 d^{4} c^{3} e^{4}+3 e^{4} \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2}\right ) x^{8}}{8}+\frac {\left (d^{5} c^{3} e^{3}+6 d^{3} e^{3} \left (a \,e^{2}+c \,d^{2}\right ) c^{2}+e^{2} \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{4} \left (a \,e^{2}+c \,d^{2}\right ) e^{2} c^{2}+2 d e \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d^{2} \left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+2 d e \left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right )\right ) x^{5}}{5}+\frac {\left (d^{2} \left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right )+2 d e \left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right )+3 e^{4} a^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (d^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right )+6 d^{3} e^{3} a^{2} \left (a \,e^{2}+c \,d^{2}\right )+e^{5} a^{3} d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{4} a^{2} e^{2} \left (a \,e^{2}+c \,d^{2}\right )+2 e^{4} a^{3} d^{4}\right ) x^{2}}{2}+e^{3} a^{3} d^{5} x\) | \(801\) |
Input:
int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
Output:
1/9*d^3*c^3*e^5*x^9+(3/8*a*e^6*c^2*d^2+5/8*d^4*c^3*e^4)*x^8+(3/7*a^2*e^7*c *d+15/7*a*e^5*c^2*d^3+10/7*d^5*c^3*e^3)*x^7+(1/6*a^3*e^8+5/2*a^2*c*d^2*e^6 +5*a*c^2*d^4*e^4+5/3*c^3*d^6*e^2)*x^6+(a^3*d*e^7+6*a^2*c*d^3*e^5+6*a*c^2*d ^5*e^3+c^3*d^7*e)*x^5+(5/2*e^6*a^3*d^2+15/2*a^2*e^4*c*d^4+15/4*a*e^2*c^2*d ^6+1/4*d^8*c^3)*x^4+(10/3*e^5*a^3*d^3+5*a^2*e^3*c*d^5+a*e*c^2*d^7)*x^3+(5/ 2*e^4*a^3*d^4+3/2*a^2*e^2*c*d^6)*x^2+e^3*a^3*d^5*x
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (103) = 206\).
Time = 0.07 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.73 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fric as")
Output:
1/9*c^3*d^3*e^5*x^9 + a^3*d^5*e^3*x + 1/8*(5*c^3*d^4*e^4 + 3*a*c^2*d^2*e^6 )*x^8 + 1/7*(10*c^3*d^5*e^3 + 15*a*c^2*d^3*e^5 + 3*a^2*c*d*e^7)*x^7 + 1/6* (10*c^3*d^6*e^2 + 30*a*c^2*d^4*e^4 + 15*a^2*c*d^2*e^6 + a^3*e^8)*x^6 + (c^ 3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c*d^3*e^5 + a^3*d*e^7)*x^5 + 1/4*(c^3*d^ 8 + 15*a*c^2*d^6*e^2 + 30*a^2*c*d^4*e^4 + 10*a^3*d^2*e^6)*x^4 + 1/3*(3*a*c ^2*d^7*e + 15*a^2*c*d^5*e^3 + 10*a^3*d^3*e^5)*x^3 + 1/2*(3*a^2*c*d^6*e^2 + 5*a^3*d^4*e^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (100) = 200\).
Time = 0.05 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.02 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=a^{3} d^{5} e^{3} x + \frac {c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \cdot \left (\frac {3 a c^{2} d^{2} e^{6}}{8} + \frac {5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c d e^{7}}{7} + \frac {15 a c^{2} d^{3} e^{5}}{7} + \frac {10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{8}}{6} + \frac {5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac {5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \cdot \left (\frac {5 a^{3} d^{2} e^{6}}{2} + \frac {15 a^{2} c d^{4} e^{4}}{2} + \frac {15 a c^{2} d^{6} e^{2}}{4} + \frac {c^{3} d^{8}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \cdot \left (\frac {5 a^{3} d^{4} e^{4}}{2} + \frac {3 a^{2} c d^{6} e^{2}}{2}\right ) \] Input:
integrate((e*x+d)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
a**3*d**5*e**3*x + c**3*d**3*e**5*x**9/9 + x**8*(3*a*c**2*d**2*e**6/8 + 5* c**3*d**4*e**4/8) + x**7*(3*a**2*c*d*e**7/7 + 15*a*c**2*d**3*e**5/7 + 10*c **3*d**5*e**3/7) + x**6*(a**3*e**8/6 + 5*a**2*c*d**2*e**6/2 + 5*a*c**2*d** 4*e**4 + 5*c**3*d**6*e**2/3) + x**5*(a**3*d*e**7 + 6*a**2*c*d**3*e**5 + 6* a*c**2*d**5*e**3 + c**3*d**7*e) + x**4*(5*a**3*d**2*e**6/2 + 15*a**2*c*d** 4*e**4/2 + 15*a*c**2*d**6*e**2/4 + c**3*d**8/4) + x**3*(10*a**3*d**3*e**5/ 3 + 5*a**2*c*d**5*e**3 + a*c**2*d**7*e) + x**2*(5*a**3*d**4*e**4/2 + 3*a** 2*c*d**6*e**2/2)
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (103) = 206\).
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.73 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxi ma")
Output:
1/9*c^3*d^3*e^5*x^9 + a^3*d^5*e^3*x + 1/8*(5*c^3*d^4*e^4 + 3*a*c^2*d^2*e^6 )*x^8 + 1/7*(10*c^3*d^5*e^3 + 15*a*c^2*d^3*e^5 + 3*a^2*c*d*e^7)*x^7 + 1/6* (10*c^3*d^6*e^2 + 30*a*c^2*d^4*e^4 + 15*a^2*c*d^2*e^6 + a^3*e^8)*x^6 + (c^ 3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c*d^3*e^5 + a^3*d*e^7)*x^5 + 1/4*(c^3*d^ 8 + 15*a*c^2*d^6*e^2 + 30*a^2*c*d^4*e^4 + 10*a^3*d^2*e^6)*x^4 + 1/3*(3*a*c ^2*d^7*e + 15*a^2*c*d^5*e^3 + 10*a^3*d^3*e^5)*x^3 + 1/2*(3*a^2*c*d^6*e^2 + 5*a^3*d^4*e^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (103) = 206\).
Time = 0.14 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.96 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + \frac {5}{8} \, c^{3} d^{4} e^{4} x^{8} + \frac {3}{8} \, a c^{2} d^{2} e^{6} x^{8} + \frac {10}{7} \, c^{3} d^{5} e^{3} x^{7} + \frac {15}{7} \, a c^{2} d^{3} e^{5} x^{7} + \frac {3}{7} \, a^{2} c d e^{7} x^{7} + \frac {5}{3} \, c^{3} d^{6} e^{2} x^{6} + 5 \, a c^{2} d^{4} e^{4} x^{6} + \frac {5}{2} \, a^{2} c d^{2} e^{6} x^{6} + \frac {1}{6} \, a^{3} e^{8} x^{6} + c^{3} d^{7} e x^{5} + 6 \, a c^{2} d^{5} e^{3} x^{5} + 6 \, a^{2} c d^{3} e^{5} x^{5} + a^{3} d e^{7} x^{5} + \frac {1}{4} \, c^{3} d^{8} x^{4} + \frac {15}{4} \, a c^{2} d^{6} e^{2} x^{4} + \frac {15}{2} \, a^{2} c d^{4} e^{4} x^{4} + \frac {5}{2} \, a^{3} d^{2} e^{6} x^{4} + a c^{2} d^{7} e x^{3} + 5 \, a^{2} c d^{5} e^{3} x^{3} + \frac {10}{3} \, a^{3} d^{3} e^{5} x^{3} + \frac {3}{2} \, a^{2} c d^{6} e^{2} x^{2} + \frac {5}{2} \, a^{3} d^{4} e^{4} x^{2} + a^{3} d^{5} e^{3} x \] Input:
integrate((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac ")
Output:
1/9*c^3*d^3*e^5*x^9 + 5/8*c^3*d^4*e^4*x^8 + 3/8*a*c^2*d^2*e^6*x^8 + 10/7*c ^3*d^5*e^3*x^7 + 15/7*a*c^2*d^3*e^5*x^7 + 3/7*a^2*c*d*e^7*x^7 + 5/3*c^3*d^ 6*e^2*x^6 + 5*a*c^2*d^4*e^4*x^6 + 5/2*a^2*c*d^2*e^6*x^6 + 1/6*a^3*e^8*x^6 + c^3*d^7*e*x^5 + 6*a*c^2*d^5*e^3*x^5 + 6*a^2*c*d^3*e^5*x^5 + a^3*d*e^7*x^ 5 + 1/4*c^3*d^8*x^4 + 15/4*a*c^2*d^6*e^2*x^4 + 15/2*a^2*c*d^4*e^4*x^4 + 5/ 2*a^3*d^2*e^6*x^4 + a*c^2*d^7*e*x^3 + 5*a^2*c*d^5*e^3*x^3 + 10/3*a^3*d^3*e ^5*x^3 + 3/2*a^2*c*d^6*e^2*x^2 + 5/2*a^3*d^4*e^4*x^2 + a^3*d^5*e^3*x
Time = 5.45 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.66 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=x^4\,\left (\frac {5\,a^3\,d^2\,e^6}{2}+\frac {15\,a^2\,c\,d^4\,e^4}{2}+\frac {15\,a\,c^2\,d^6\,e^2}{4}+\frac {c^3\,d^8}{4}\right )+x^6\,\left (\frac {a^3\,e^8}{6}+\frac {5\,a^2\,c\,d^2\,e^6}{2}+5\,a\,c^2\,d^4\,e^4+\frac {5\,c^3\,d^6\,e^2}{3}\right )+x^5\,\left (a^3\,d\,e^7+6\,a^2\,c\,d^3\,e^5+6\,a\,c^2\,d^5\,e^3+c^3\,d^7\,e\right )+a^3\,d^5\,e^3\,x+\frac {c^3\,d^3\,e^5\,x^9}{9}+\frac {a\,d^3\,e\,x^3\,\left (10\,a^2\,e^4+15\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{3}+\frac {c\,d\,e^3\,x^7\,\left (3\,a^2\,e^4+15\,a\,c\,d^2\,e^2+10\,c^2\,d^4\right )}{7}+\frac {a^2\,d^4\,e^2\,x^2\,\left (3\,c\,d^2+5\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^4\,x^8\,\left (5\,c\,d^2+3\,a\,e^2\right )}{8} \] Input:
int((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
x^4*((c^3*d^8)/4 + (5*a^3*d^2*e^6)/2 + (15*a*c^2*d^6*e^2)/4 + (15*a^2*c*d^ 4*e^4)/2) + x^6*((a^3*e^8)/6 + (5*c^3*d^6*e^2)/3 + 5*a*c^2*d^4*e^4 + (5*a^ 2*c*d^2*e^6)/2) + x^5*(a^3*d*e^7 + c^3*d^7*e + 6*a*c^2*d^5*e^3 + 6*a^2*c*d ^3*e^5) + a^3*d^5*e^3*x + (c^3*d^3*e^5*x^9)/9 + (a*d^3*e*x^3*(10*a^2*e^4 + 3*c^2*d^4 + 15*a*c*d^2*e^2))/3 + (c*d*e^3*x^7*(3*a^2*e^4 + 10*c^2*d^4 + 1 5*a*c*d^2*e^2))/7 + (a^2*d^4*e^2*x^2*(5*a*e^2 + 3*c*d^2))/2 + (c^2*d^2*e^4 *x^8*(3*a*e^2 + 5*c*d^2))/8
Time = 0.24 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.98 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {x \left (56 c^{3} d^{3} e^{5} x^{8}+189 a \,c^{2} d^{2} e^{6} x^{7}+315 c^{3} d^{4} e^{4} x^{7}+216 a^{2} c d \,e^{7} x^{6}+1080 a \,c^{2} d^{3} e^{5} x^{6}+720 c^{3} d^{5} e^{3} x^{6}+84 a^{3} e^{8} x^{5}+1260 a^{2} c \,d^{2} e^{6} x^{5}+2520 a \,c^{2} d^{4} e^{4} x^{5}+840 c^{3} d^{6} e^{2} x^{5}+504 a^{3} d \,e^{7} x^{4}+3024 a^{2} c \,d^{3} e^{5} x^{4}+3024 a \,c^{2} d^{5} e^{3} x^{4}+504 c^{3} d^{7} e \,x^{4}+1260 a^{3} d^{2} e^{6} x^{3}+3780 a^{2} c \,d^{4} e^{4} x^{3}+1890 a \,c^{2} d^{6} e^{2} x^{3}+126 c^{3} d^{8} x^{3}+1680 a^{3} d^{3} e^{5} x^{2}+2520 a^{2} c \,d^{5} e^{3} x^{2}+504 a \,c^{2} d^{7} e \,x^{2}+1260 a^{3} d^{4} e^{4} x +756 a^{2} c \,d^{6} e^{2} x +504 a^{3} d^{5} e^{3}\right )}{504} \] Input:
int((e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(x*(504*a**3*d**5*e**3 + 1260*a**3*d**4*e**4*x + 1680*a**3*d**3*e**5*x**2 + 1260*a**3*d**2*e**6*x**3 + 504*a**3*d*e**7*x**4 + 84*a**3*e**8*x**5 + 75 6*a**2*c*d**6*e**2*x + 2520*a**2*c*d**5*e**3*x**2 + 3780*a**2*c*d**4*e**4* x**3 + 3024*a**2*c*d**3*e**5*x**4 + 1260*a**2*c*d**2*e**6*x**5 + 216*a**2* c*d*e**7*x**6 + 504*a*c**2*d**7*e*x**2 + 1890*a*c**2*d**6*e**2*x**3 + 3024 *a*c**2*d**5*e**3*x**4 + 2520*a*c**2*d**4*e**4*x**5 + 1080*a*c**2*d**3*e** 5*x**6 + 189*a*c**2*d**2*e**6*x**7 + 126*c**3*d**8*x**3 + 504*c**3*d**7*e* x**4 + 840*c**3*d**6*e**2*x**5 + 720*c**3*d**5*e**3*x**6 + 315*c**3*d**4*e **4*x**7 + 56*c**3*d**3*e**5*x**8))/504