Integrand size = 368, antiderivative size = 272 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{4 e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^7}{7 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{8 e^7}-\frac {c^2 (2 c d-b e) (d+e x)^9}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \] Output:
1/4*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^4/e^7-3/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)^2*(e*x+d)^5/e^7+1/2*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5* b*d))*(e*x+d)^6/e^7-1/7*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b *d))*(e*x+d)^7/e^7+3/8*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^8/e^ 7-1/3*c^2*(-b*e+2*c*d)*(e*x+d)^9/e^7+1/10*c^3*(e*x+d)^10/e^7
Leaf count is larger than twice the leaf count of optimal. \(547\) vs. \(2(272)=544\).
Time = 0.00 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.01 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=a^3 d^3 x+\frac {3}{2} a^2 b d^3 x^2+\frac {3}{2} a^3 d^2 e x^2+a b^2 d^3 x^3+a^2 c d^3 x^3+3 a^2 b d^2 e x^3+a^3 d e^2 x^3+\frac {1}{4} b^3 d^3 x^4+\frac {3}{2} a b c d^3 x^4+\frac {9}{4} a b^2 d^2 e x^4+\frac {9}{4} a^2 c d^2 e x^4+\frac {9}{4} a^2 b d e^2 x^4+\frac {1}{4} a^3 e^3 x^4+\frac {3}{5} b^2 c d^3 x^5+\frac {3}{5} a c^2 d^3 x^5+\frac {3}{5} b^3 d^2 e x^5+\frac {18}{5} a b c d^2 e x^5+\frac {9}{5} a b^2 d e^2 x^5+\frac {9}{5} a^2 c d e^2 x^5+\frac {3}{5} a^2 b e^3 x^5+\frac {1}{2} b c^2 d^3 x^6+\frac {3}{2} b^2 c d^2 e x^6+\frac {3}{2} a c^2 d^2 e x^6+\frac {1}{2} b^3 d e^2 x^6+3 a b c d e^2 x^6+\frac {1}{2} a b^2 e^3 x^6+\frac {1}{2} a^2 c e^3 x^6+\frac {1}{7} c^3 d^3 x^7+\frac {9}{7} b c^2 d^2 e x^7+\frac {9}{7} b^2 c d e^2 x^7+\frac {9}{7} a c^2 d e^2 x^7+\frac {1}{7} b^3 e^3 x^7+\frac {6}{7} a b c e^3 x^7+\frac {3}{8} c^3 d^2 e x^8+\frac {9}{8} b c^2 d e^2 x^8+\frac {3}{8} b^2 c e^3 x^8+\frac {3}{8} a c^2 e^3 x^8+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{3} b c^2 e^3 x^9+\frac {1}{10} c^3 e^3 x^{10} \] Input:
Integrate[a^3*d^3 + 3*a^2*d^2*(b*d + a*e)*x + 3*a*d*(b^2*d^2 + a*c*d^2 + 3 *a*b*d*e + a^2*e^2)*x^2 + (b^3*d^3 + 6*a*b*c*d^3 + 9*a*b^2*d^2*e + 9*a^2*c *d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^3 + 3*(b^2*c*d^3 + a*c^2*d^3 + b^3*d^2 *e + 6*a*b*c*d^2*e + 3*a*b^2*d*e^2 + 3*a^2*c*d*e^2 + a^2*b*e^3)*x^4 + 3*(b *c^2*d^3 + 3*b^2*c*d^2*e + 3*a*c^2*d^2*e + b^3*d*e^2 + 6*a*b*c*d*e^2 + a*b ^2*e^3 + a^2*c*e^3)*x^5 + (c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + 9*a*c ^2*d*e^2 + b^3*e^3 + 6*a*b*c*e^3)*x^6 + 3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e ^2 + a*c*e^2)*x^7 + 3*c^2*e^2*(c*d + b*e)*x^8 + c^3*e^3*x^9,x]
Output:
a^3*d^3*x + (3*a^2*b*d^3*x^2)/2 + (3*a^3*d^2*e*x^2)/2 + a*b^2*d^3*x^3 + a^ 2*c*d^3*x^3 + 3*a^2*b*d^2*e*x^3 + a^3*d*e^2*x^3 + (b^3*d^3*x^4)/4 + (3*a*b *c*d^3*x^4)/2 + (9*a*b^2*d^2*e*x^4)/4 + (9*a^2*c*d^2*e*x^4)/4 + (9*a^2*b*d *e^2*x^4)/4 + (a^3*e^3*x^4)/4 + (3*b^2*c*d^3*x^5)/5 + (3*a*c^2*d^3*x^5)/5 + (3*b^3*d^2*e*x^5)/5 + (18*a*b*c*d^2*e*x^5)/5 + (9*a*b^2*d*e^2*x^5)/5 + ( 9*a^2*c*d*e^2*x^5)/5 + (3*a^2*b*e^3*x^5)/5 + (b*c^2*d^3*x^6)/2 + (3*b^2*c* d^2*e*x^6)/2 + (3*a*c^2*d^2*e*x^6)/2 + (b^3*d*e^2*x^6)/2 + 3*a*b*c*d*e^2*x ^6 + (a*b^2*e^3*x^6)/2 + (a^2*c*e^3*x^6)/2 + (c^3*d^3*x^7)/7 + (9*b*c^2*d^ 2*e*x^7)/7 + (9*b^2*c*d*e^2*x^7)/7 + (9*a*c^2*d*e^2*x^7)/7 + (b^3*e^3*x^7) /7 + (6*a*b*c*e^3*x^7)/7 + (3*c^3*d^2*e*x^8)/8 + (9*b*c^2*d*e^2*x^8)/8 + ( 3*b^2*c*e^3*x^8)/8 + (3*a*c^2*e^3*x^8)/8 + (c^3*d*e^2*x^9)/3 + (b*c^2*e^3* x^9)/3 + (c^3*e^3*x^10)/10
Time = 0.54 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.37, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.003, Rules used = {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.
| \(\displaystyle \int \left (a^3 d^3+3 a d x^2 \left (a^2 e^2+3 a b d e+a c d^2+b^2 d^2\right )+3 x^5 \left (a^2 c e^3+a b^2 e^3+6 a b c d e^2+3 a c^2 d^2 e+b^3 d e^2+3 b^2 c d^2 e+b c^2 d^3\right )+3 x^4 \left (a^2 b e^3+3 a^2 c d e^2+3 a b^2 d e^2+6 a b c d^2 e+a c^2 d^3+b^3 d^2 e+b^2 c d^3\right )+3 a^2 d^2 x (a e+b d)+x^3 \left (a^3 e^3+9 a^2 b d e^2+9 a^2 c d^2 e+9 a b^2 d^2 e+6 a b c d^3+b^3 d^3\right )+3 c e x^7 \left (a c e^2+b^2 e^2+3 b c d e+c^2 d^2\right )+x^6 \left (6 a b c e^3+9 a c^2 d e^2+b^3 e^3+9 b^2 c d e^2+9 b c^2 d^2 e+c^3 d^3\right )+3 c^2 e^2 x^8 (b e+c d)+c^3 e^3 x^9\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 d^3 x+\frac {1}{4} x^4 \left (a^2 e \left (a e^2+9 c d^2\right )+9 a b^2 d^2 e+3 a b d \left (3 a e^2+2 c d^2\right )+b^3 d^3\right )+\frac {3}{2} a^2 d^2 x^2 (a e+b d)+\frac {1}{7} x^7 \left (9 c^2 d e (a e+b d)+3 b c e^2 (2 a e+3 b d)+b^3 e^3+c^3 d^3\right )+\frac {3}{8} c e x^8 \left (c e (a e+3 b d)+b^2 e^2+c^2 d^2\right )+a d x^3 \left (3 a b d e+a \left (a e^2+c d^2\right )+b^2 d^2\right )+\frac {1}{2} x^6 \left (b^2 \left (a e^3+3 c d^2 e\right )+b c d \left (6 a e^2+c d^2\right )+a c e \left (a e^2+3 c d^2\right )+b^3 d e^2\right )+\frac {3}{5} x^5 \left (b^2 \left (3 a d e^2+c d^3\right )+a b e \left (a e^2+6 c d^2\right )+a c d \left (3 a e^2+c d^2\right )+b^3 d^2 e\right )+\frac {1}{3} c^2 e^2 x^9 (b e+c d)+\frac {1}{10} c^3 e^3 x^{10}\) |
Input:
Int[a^3*d^3 + 3*a^2*d^2*(b*d + a*e)*x + 3*a*d*(b^2*d^2 + a*c*d^2 + 3*a*b*d *e + a^2*e^2)*x^2 + (b^3*d^3 + 6*a*b*c*d^3 + 9*a*b^2*d^2*e + 9*a^2*c*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^3 + 3*(b^2*c*d^3 + a*c^2*d^3 + b^3*d^2*e + 6 *a*b*c*d^2*e + 3*a*b^2*d*e^2 + 3*a^2*c*d*e^2 + a^2*b*e^3)*x^4 + 3*(b*c^2*d ^3 + 3*b^2*c*d^2*e + 3*a*c^2*d^2*e + b^3*d*e^2 + 6*a*b*c*d*e^2 + a*b^2*e^3 + a^2*c*e^3)*x^5 + (c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e^2 + 9*a*c^2*d*e ^2 + b^3*e^3 + 6*a*b*c*e^3)*x^6 + 3*c*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2 + a *c*e^2)*x^7 + 3*c^2*e^2*(c*d + b*e)*x^8 + c^3*e^3*x^9,x]
Output:
a^3*d^3*x + (3*a^2*d^2*(b*d + a*e)*x^2)/2 + a*d*(b^2*d^2 + 3*a*b*d*e + a*( c*d^2 + a*e^2))*x^3 + ((b^3*d^3 + 9*a*b^2*d^2*e + a^2*e*(9*c*d^2 + a*e^2) + 3*a*b*d*(2*c*d^2 + 3*a*e^2))*x^4)/4 + (3*(b^3*d^2*e + a*b*e*(6*c*d^2 + a *e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*x^5)/5 + ((b^3* d*e^2 + a*c*e*(3*c*d^2 + a*e^2) + b*c*d*(c*d^2 + 6*a*e^2) + b^2*(3*c*d^2*e + a*e^3))*x^6)/2 + ((c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(b*d + a*e) + 3*b*c*e^ 2*(3*b*d + 2*a*e))*x^7)/7 + (3*c*e*(c^2*d^2 + b^2*e^2 + c*e*(3*b*d + a*e)) *x^8)/8 + (c^2*e^2*(c*d + b*e)*x^9)/3 + (c^3*e^3*x^10)/10
Time = 0.15 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.49
| method | result | size |
| norman | \(\frac {c^{3} e^{3} x^{10}}{10}+\left (\frac {1}{3} e^{3} b \,c^{2}+\frac {1}{3} d \,e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} a \,c^{2} e^{3}+\frac {3}{8} b^{2} c \,e^{3}+\frac {9}{8} d \,e^{2} b \,c^{2}+\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {6}{7} a b c \,e^{3}+\frac {9}{7} d \,e^{2} a \,c^{2}+\frac {1}{7} b^{3} e^{3}+\frac {9}{7} d \,e^{2} b^{2} c +\frac {9}{7} d^{2} e b \,c^{2}+\frac {1}{7} d^{3} c^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} c \,e^{3}+\frac {1}{2} b^{2} e^{3} a +3 b d \,e^{2} a c +\frac {3}{2} a \,c^{2} d^{2} e +\frac {1}{2} b^{3} e^{2} d +\frac {3}{2} b^{2} c \,d^{2} e +\frac {1}{2} d^{3} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} b \,e^{3}+\frac {9}{5} a^{2} c d \,e^{2}+\frac {9}{5} a \,b^{2} d \,e^{2}+\frac {18}{5} a b c \,d^{2} e +\frac {3}{5} d^{3} a \,c^{2}+\frac {3}{5} b^{3} d^{2} e +\frac {3}{5} b^{2} c \,d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{3}+\frac {9}{4} a^{2} b d \,e^{2}+\frac {9}{4} a^{2} c \,d^{2} e +\frac {9}{4} a \,b^{2} d^{2} e +\frac {3}{2} a b c \,d^{3}+\frac {1}{4} b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{3}+3 d^{2} e \,a^{2} b +d^{3} c \,a^{2}+a \,b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{3}+\frac {3}{2} d^{3} a^{2} b \right ) x^{2}+a^{3} d^{3} x\) | \(406\) |
| risch | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| parallelrisch | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| parts | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| gosper | \(\frac {x \left (84 c^{3} e^{3} x^{9}+280 b \,c^{2} e^{3} x^{8}+280 c^{3} d \,e^{2} x^{8}+315 a \,c^{2} e^{3} x^{7}+315 b^{2} c \,e^{3} x^{7}+945 b \,c^{2} d \,e^{2} x^{7}+315 c^{3} d^{2} e \,x^{7}+720 a b c \,e^{3} x^{6}+1080 a \,c^{2} d \,e^{2} x^{6}+120 b^{3} e^{3} x^{6}+1080 b^{2} c d \,e^{2} x^{6}+1080 b \,c^{2} d^{2} e \,x^{6}+120 c^{3} d^{3} x^{6}+420 a^{2} c \,e^{3} x^{5}+420 x^{5} b^{2} e^{3} a +2520 a b c d \,e^{2} x^{5}+1260 a \,c^{2} d^{2} e \,x^{5}+420 x^{5} d \,e^{2} b^{3}+1260 b^{2} c \,d^{2} e \,x^{5}+420 b \,c^{2} d^{3} x^{5}+504 x^{4} a^{2} b \,e^{3}+1512 a^{2} c d \,e^{2} x^{4}+1512 x^{4} a \,b^{2} d \,e^{2}+3024 a b c \,d^{2} e \,x^{4}+504 a \,c^{2} d^{3} x^{4}+504 x^{4} b^{3} d^{2} e +504 b^{2} c \,d^{3} x^{4}+210 x^{3} e^{3} a^{3}+1890 x^{3} a^{2} b d \,e^{2}+1890 a^{2} c \,d^{2} e \,x^{3}+1890 x^{3} a \,b^{2} d^{2} e +1260 a b c \,d^{3} x^{3}+210 x^{3} b^{3} d^{3}+840 a^{3} d \,e^{2} x^{2}+2520 a^{2} b \,d^{2} e \,x^{2}+840 a^{2} c \,d^{3} x^{2}+840 a \,b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{3}+1260 a^{2} b \,d^{3} x +840 a^{3} d^{3}\right )}{840}\) | \(482\) |
| default | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 d^{2} e \,c^{3}+9 d \,e^{2} b \,c^{2}+e^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (d^{3} c^{3}+9 d^{2} e b \,c^{2}+3 d \,e^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{3} b \,c^{2}+3 d^{2} e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{6}}{6}+\frac {\left (d^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 a^{2} b \,e^{3}\right ) x^{5}}{5}+\frac {\left (d^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 a^{2} b d \,e^{2}+e^{3} a^{3}\right ) x^{4}}{4}+\frac {\left (d^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 d^{2} e \,a^{2} b +3 d \,e^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{3}+3 d^{3} a^{2} b \right ) x^{2}}{2}+a^{3} d^{3} x\) | \(495\) |
| orering | \(\frac {x \left (84 c^{3} e^{3} x^{9}+280 b \,c^{2} e^{3} x^{8}+280 c^{3} d \,e^{2} x^{8}+315 a \,c^{2} e^{3} x^{7}+315 b^{2} c \,e^{3} x^{7}+945 b \,c^{2} d \,e^{2} x^{7}+315 c^{3} d^{2} e \,x^{7}+720 a b c \,e^{3} x^{6}+1080 a \,c^{2} d \,e^{2} x^{6}+120 b^{3} e^{3} x^{6}+1080 b^{2} c d \,e^{2} x^{6}+1080 b \,c^{2} d^{2} e \,x^{6}+120 c^{3} d^{3} x^{6}+420 a^{2} c \,e^{3} x^{5}+420 x^{5} b^{2} e^{3} a +2520 a b c d \,e^{2} x^{5}+1260 a \,c^{2} d^{2} e \,x^{5}+420 x^{5} d \,e^{2} b^{3}+1260 b^{2} c \,d^{2} e \,x^{5}+420 b \,c^{2} d^{3} x^{5}+504 x^{4} a^{2} b \,e^{3}+1512 a^{2} c d \,e^{2} x^{4}+1512 x^{4} a \,b^{2} d \,e^{2}+3024 a b c \,d^{2} e \,x^{4}+504 a \,c^{2} d^{3} x^{4}+504 x^{4} b^{3} d^{2} e +504 b^{2} c \,d^{3} x^{4}+210 x^{3} e^{3} a^{3}+1890 x^{3} a^{2} b d \,e^{2}+1890 a^{2} c \,d^{2} e \,x^{3}+1890 x^{3} a \,b^{2} d^{2} e +1260 a b c \,d^{3} x^{3}+210 x^{3} b^{3} d^{3}+840 a^{3} d \,e^{2} x^{2}+2520 a^{2} b \,d^{2} e \,x^{2}+840 a^{2} c \,d^{3} x^{2}+840 a \,b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{3}+1260 a^{2} b \,d^{3} x +840 a^{3} d^{3}\right ) \left (a^{3} d^{3}+3 a^{2} d^{2} \left (a e +b d \right ) x +3 a d \left (e^{2} a^{2}+3 a b d e +a \,d^{2} c +b^{2} d^{2}\right ) x^{2}+\left (e^{3} a^{3}+9 a^{2} b d \,e^{2}+9 a^{2} c \,d^{2} e +9 a \,b^{2} d^{2} e +6 a b c \,d^{3}+b^{3} d^{3}\right ) x^{3}+3 \left (a^{2} b \,e^{3}+3 a^{2} c d \,e^{2}+3 a \,b^{2} d \,e^{2}+6 a b c \,d^{2} e +d^{3} a \,c^{2}+b^{3} d^{2} e +b^{2} c \,d^{3}\right ) x^{4}+3 \left (a^{2} c \,e^{3}+b^{2} e^{3} a +6 b d \,e^{2} a c +3 a \,c^{2} d^{2} e +b^{3} e^{2} d +3 b^{2} c \,d^{2} e +d^{3} b \,c^{2}\right ) x^{5}+\left (6 a b c \,e^{3}+9 d \,e^{2} a \,c^{2}+b^{3} e^{3}+9 d \,e^{2} b^{2} c +9 d^{2} e b \,c^{2}+d^{3} c^{3}\right ) x^{6}+3 c e \left (a c \,e^{2}+b^{2} e^{2}+3 b c d e +c^{2} d^{2}\right ) x^{7}+3 c^{2} e^{2} \left (b e +c d \right ) x^{8}+c^{3} e^{3} x^{9}\right )}{840 \left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{3}}\) | \(869\) |
Input:
int(a^3*d^3+3*a^2*d^2*(a*e+b*d)*x+3*a*d*(a^2*e^2+3*a*b*d*e+a*c*d^2+b^2*d^2 )*x^2+(a^3*e^3+9*a^2*b*d*e^2+9*a^2*c*d^2*e+9*a*b^2*d^2*e+6*a*b*c*d^3+b^3*d ^3)*x^3+3*(a^2*b*e^3+3*a^2*c*d*e^2+3*a*b^2*d*e^2+6*a*b*c*d^2*e+a*c^2*d^3+b ^3*d^2*e+b^2*c*d^3)*x^4+3*(a^2*c*e^3+a*b^2*e^3+6*a*b*c*d*e^2+3*a*c^2*d^2*e +b^3*d*e^2+3*b^2*c*d^2*e+b*c^2*d^3)*x^5+(6*a*b*c*e^3+9*a*c^2*d*e^2+b^3*e^3 +9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^6+3*c*e*(a*c*e^2+b^2*e^2+3*b*c*d*e +c^2*d^2)*x^7+3*c^2*e^2*(b*e+c*d)*x^8+c^3*e^3*x^9,x,method=_RETURNVERBOSE)
Output:
1/10*c^3*e^3*x^10+(1/3*e^3*b*c^2+1/3*d*e^2*c^3)*x^9+(3/8*a*c^2*e^3+3/8*b^2 *c*e^3+9/8*d*e^2*b*c^2+3/8*d^2*e*c^3)*x^8+(6/7*a*b*c*e^3+9/7*d*e^2*a*c^2+1 /7*b^3*e^3+9/7*d*e^2*b^2*c+9/7*d^2*e*b*c^2+1/7*d^3*c^3)*x^7+(1/2*a^2*c*e^3 +1/2*b^2*e^3*a+3*b*d*e^2*a*c+3/2*a*c^2*d^2*e+1/2*b^3*e^2*d+3/2*b^2*c*d^2*e +1/2*d^3*b*c^2)*x^6+(3/5*a^2*b*e^3+9/5*a^2*c*d*e^2+9/5*a*b^2*d*e^2+18/5*a* b*c*d^2*e+3/5*d^3*a*c^2+3/5*b^3*d^2*e+3/5*b^2*c*d^3)*x^5+(1/4*e^3*a^3+9/4* a^2*b*d*e^2+9/4*a^2*c*d^2*e+9/4*a*b^2*d^2*e+3/2*a*b*c*d^3+1/4*b^3*d^3)*x^4 +(a^3*d*e^2+3*a^2*b*d^2*e+a^2*c*d^3+a*b^2*d^3)*x^3+(3/2*d^2*e*a^3+3/2*d^3* a^2*b)*x^2+a^3*d^3*x
Time = 0.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \] Input:
integrate(a^3*d^3+3*a^2*d^2*(a*e+b*d)*x+3*a*d*(a^2*e^2+3*a*b*d*e+a*c*d^2+b ^2*d^2)*x^2+(a^3*e^3+9*a^2*b*d*e^2+9*a^2*c*d^2*e+9*a*b^2*d^2*e+6*a*b*c*d^3 +b^3*d^3)*x^3+3*(a^2*b*e^3+3*a^2*c*d*e^2+3*a*b^2*d*e^2+6*a*b*c*d^2*e+a*c^2 *d^3+b^3*d^2*e+b^2*c*d^3)*x^4+3*(a^2*c*e^3+a*b^2*e^3+6*a*b*c*d*e^2+3*a*c^2 *d^2*e+b^3*d*e^2+3*b^2*c*d^2*e+b*c^2*d^3)*x^5+(6*a*b*c*e^3+9*a*c^2*d*e^2+b ^3*e^3+9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^6+3*c*e*(a*c*e^2+b^2*e^2+3*b *c*d*e+c^2*d^2)*x^7+3*c^2*e^2*(b*e+c*d)*x^8+c^3*e^3*x^9,x, algorithm="fric as")
Output:
1/10*c^3*e^3*x^10 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b *c^2*d*e^2 + (b^2*c + a*c^2)*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*( b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^7 + a^3*d^3*x + 1/2*(b*c^2*d ^3 + 3*(b^2*c + a*c^2)*d^2*e + (b^3 + 6*a*b*c)*d*e^2 + (a*b^2 + a^2*c)*e^3 )*x^6 + 3/5*(a^2*b*e^3 + (b^2*c + a*c^2)*d^3 + (b^3 + 6*a*b*c)*d^2*e + 3*( a*b^2 + a^2*c)*d*e^2)*x^5 + 1/4*(9*a^2*b*d*e^2 + a^3*e^3 + (b^3 + 6*a*b*c) *d^3 + 9*(a*b^2 + a^2*c)*d^2*e)*x^4 + (3*a^2*b*d^2*e + a^3*d*e^2 + (a*b^2 + a^2*c)*d^3)*x^3 + 3/2*(a^2*b*d^3 + a^3*d^2*e)*x^2
Time = 0.05 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.78 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=a^{3} d^{3} x + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {6 a b c e^{3}}{7} + \frac {9 a c^{2} d e^{2}}{7} + \frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {a b^{2} e^{3}}{2} + 3 a b c d e^{2} + \frac {3 a c^{2} d^{2} e}{2} + \frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a^{2} c d e^{2}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {18 a b c d^{2} e}{5} + \frac {3 a c^{2} d^{3}}{5} + \frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a^{2} c d^{2} e}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {3 a b c d^{3}}{2} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a^{2} c d^{3} + a b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\right ) \] Input:
integrate(a**3*d**3+3*a**2*d**2*(a*e+b*d)*x+3*a*d*(a**2*e**2+3*a*b*d*e+a*c *d**2+b**2*d**2)*x**2+(a**3*e**3+9*a**2*b*d*e**2+9*a**2*c*d**2*e+9*a*b**2* d**2*e+6*a*b*c*d**3+b**3*d**3)*x**3+3*(a**2*b*e**3+3*a**2*c*d*e**2+3*a*b** 2*d*e**2+6*a*b*c*d**2*e+a*c**2*d**3+b**3*d**2*e+b**2*c*d**3)*x**4+3*(a**2* c*e**3+a*b**2*e**3+6*a*b*c*d*e**2+3*a*c**2*d**2*e+b**3*d*e**2+3*b**2*c*d** 2*e+b*c**2*d**3)*x**5+(6*a*b*c*e**3+9*a*c**2*d*e**2+b**3*e**3+9*b**2*c*d*e **2+9*b*c**2*d**2*e+c**3*d**3)*x**6+3*c*e*(a*c*e**2+b**2*e**2+3*b*c*d*e+c* *2*d**2)*x**7+3*c**2*e**2*(b*e+c*d)*x**8+c**3*e**3*x**9,x)
Output:
a**3*d**3*x + c**3*e**3*x**10/10 + x**9*(b*c**2*e**3/3 + c**3*d*e**2/3) + x**8*(3*a*c**2*e**3/8 + 3*b**2*c*e**3/8 + 9*b*c**2*d*e**2/8 + 3*c**3*d**2* e/8) + x**7*(6*a*b*c*e**3/7 + 9*a*c**2*d*e**2/7 + b**3*e**3/7 + 9*b**2*c*d *e**2/7 + 9*b*c**2*d**2*e/7 + c**3*d**3/7) + x**6*(a**2*c*e**3/2 + a*b**2* e**3/2 + 3*a*b*c*d*e**2 + 3*a*c**2*d**2*e/2 + b**3*d*e**2/2 + 3*b**2*c*d** 2*e/2 + b*c**2*d**3/2) + x**5*(3*a**2*b*e**3/5 + 9*a**2*c*d*e**2/5 + 9*a*b **2*d*e**2/5 + 18*a*b*c*d**2*e/5 + 3*a*c**2*d**3/5 + 3*b**3*d**2*e/5 + 3*b **2*c*d**3/5) + x**4*(a**3*e**3/4 + 9*a**2*b*d*e**2/4 + 9*a**2*c*d**2*e/4 + 9*a*b**2*d**2*e/4 + 3*a*b*c*d**3/2 + b**3*d**3/4) + x**3*(a**3*d*e**2 + 3*a**2*b*d**2*e + a**2*c*d**3 + a*b**2*d**3) + x**2*(3*a**3*d**2*e/2 + 3*a **2*b*d**3/2)
Time = 0.04 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.37 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c d + b e\right )} c^{2} e^{2} x^{9} + \frac {3}{8} \, {\left (c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2} + a c e^{2}\right )} c e x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + 9 \, a c^{2} d e^{2} + b^{3} e^{3} + 6 \, a b c e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {3}{2} \, {\left (b d + a e\right )} a^{2} d^{2} x^{2} + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + 3 \, a c^{2} d^{2} e + b^{3} d e^{2} + 6 \, a b c d e^{2} + a b^{2} e^{3} + a^{2} c e^{3}\right )} x^{6} + {\left (b^{2} d^{2} + a c d^{2} + 3 \, a b d e + a^{2} e^{2}\right )} a d x^{3} + \frac {3}{5} \, {\left (b^{2} c d^{3} + a c^{2} d^{3} + b^{3} d^{2} e + 6 \, a b c d^{2} e + 3 \, a b^{2} d e^{2} + 3 \, a^{2} c d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{3} + 6 \, a b c d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} c d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} \] Input:
integrate(a^3*d^3+3*a^2*d^2*(a*e+b*d)*x+3*a*d*(a^2*e^2+3*a*b*d*e+a*c*d^2+b ^2*d^2)*x^2+(a^3*e^3+9*a^2*b*d*e^2+9*a^2*c*d^2*e+9*a*b^2*d^2*e+6*a*b*c*d^3 +b^3*d^3)*x^3+3*(a^2*b*e^3+3*a^2*c*d*e^2+3*a*b^2*d*e^2+6*a*b*c*d^2*e+a*c^2 *d^3+b^3*d^2*e+b^2*c*d^3)*x^4+3*(a^2*c*e^3+a*b^2*e^3+6*a*b*c*d*e^2+3*a*c^2 *d^2*e+b^3*d*e^2+3*b^2*c*d^2*e+b*c^2*d^3)*x^5+(6*a*b*c*e^3+9*a*c^2*d*e^2+b ^3*e^3+9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^6+3*c*e*(a*c*e^2+b^2*e^2+3*b *c*d*e+c^2*d^2)*x^7+3*c^2*e^2*(b*e+c*d)*x^8+c^3*e^3*x^9,x, algorithm="maxi ma")
Output:
1/10*c^3*e^3*x^10 + 1/3*(c*d + b*e)*c^2*e^2*x^9 + 3/8*(c^2*d^2 + 3*b*c*d*e + b^2*e^2 + a*c*e^2)*c*e*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e ^2 + 9*a*c^2*d*e^2 + b^3*e^3 + 6*a*b*c*e^3)*x^7 + a^3*d^3*x + 3/2*(b*d + a *e)*a^2*d^2*x^2 + 1/2*(b*c^2*d^3 + 3*b^2*c*d^2*e + 3*a*c^2*d^2*e + b^3*d*e ^2 + 6*a*b*c*d*e^2 + a*b^2*e^3 + a^2*c*e^3)*x^6 + (b^2*d^2 + a*c*d^2 + 3*a *b*d*e + a^2*e^2)*a*d*x^3 + 3/5*(b^2*c*d^3 + a*c^2*d^3 + b^3*d^2*e + 6*a*b *c*d^2*e + 3*a*b^2*d*e^2 + 3*a^2*c*d*e^2 + a^2*b*e^3)*x^5 + 1/4*(b^3*d^3 + 6*a*b*c*d^3 + 9*a*b^2*d^2*e + 9*a^2*c*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^ 4
Time = 0.38 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.37 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c d + b e\right )} c^{2} e^{2} x^{9} + \frac {3}{8} \, {\left (c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2} + a c e^{2}\right )} c e x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + 9 \, a c^{2} d e^{2} + b^{3} e^{3} + 6 \, a b c e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {3}{2} \, {\left (b d + a e\right )} a^{2} d^{2} x^{2} + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + 3 \, a c^{2} d^{2} e + b^{3} d e^{2} + 6 \, a b c d e^{2} + a b^{2} e^{3} + a^{2} c e^{3}\right )} x^{6} + {\left (b^{2} d^{2} + a c d^{2} + 3 \, a b d e + a^{2} e^{2}\right )} a d x^{3} + \frac {3}{5} \, {\left (b^{2} c d^{3} + a c^{2} d^{3} + b^{3} d^{2} e + 6 \, a b c d^{2} e + 3 \, a b^{2} d e^{2} + 3 \, a^{2} c d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{3} + 6 \, a b c d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} c d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} \] Input:
integrate(a^3*d^3+3*a^2*d^2*(a*e+b*d)*x+3*a*d*(a^2*e^2+3*a*b*d*e+a*c*d^2+b ^2*d^2)*x^2+(a^3*e^3+9*a^2*b*d*e^2+9*a^2*c*d^2*e+9*a*b^2*d^2*e+6*a*b*c*d^3 +b^3*d^3)*x^3+3*(a^2*b*e^3+3*a^2*c*d*e^2+3*a*b^2*d*e^2+6*a*b*c*d^2*e+a*c^2 *d^3+b^3*d^2*e+b^2*c*d^3)*x^4+3*(a^2*c*e^3+a*b^2*e^3+6*a*b*c*d*e^2+3*a*c^2 *d^2*e+b^3*d*e^2+3*b^2*c*d^2*e+b*c^2*d^3)*x^5+(6*a*b*c*e^3+9*a*c^2*d*e^2+b ^3*e^3+9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^6+3*c*e*(a*c*e^2+b^2*e^2+3*b *c*d*e+c^2*d^2)*x^7+3*c^2*e^2*(b*e+c*d)*x^8+c^3*e^3*x^9,x, algorithm="giac ")
Output:
1/10*c^3*e^3*x^10 + 1/3*(c*d + b*e)*c^2*e^2*x^9 + 3/8*(c^2*d^2 + 3*b*c*d*e + b^2*e^2 + a*c*e^2)*c*e*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*b^2*c*d*e ^2 + 9*a*c^2*d*e^2 + b^3*e^3 + 6*a*b*c*e^3)*x^7 + a^3*d^3*x + 3/2*(b*d + a *e)*a^2*d^2*x^2 + 1/2*(b*c^2*d^3 + 3*b^2*c*d^2*e + 3*a*c^2*d^2*e + b^3*d*e ^2 + 6*a*b*c*d*e^2 + a*b^2*e^3 + a^2*c*e^3)*x^6 + (b^2*d^2 + a*c*d^2 + 3*a *b*d*e + a^2*e^2)*a*d*x^3 + 3/5*(b^2*c*d^3 + a*c^2*d^3 + b^3*d^2*e + 6*a*b *c*d^2*e + 3*a*b^2*d*e^2 + 3*a^2*c*d*e^2 + a^2*b*e^3)*x^5 + 1/4*(b^3*d^3 + 6*a*b*c*d^3 + 9*a*b^2*d^2*e + 9*a^2*c*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^ 4
Time = 5.67 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.40 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {3\,c\,a\,b\,d^3}{2}+\frac {b^3\,d^3}{4}\right )+x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {6\,a\,b\,c\,e^3}{7}+\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^5\,\left (\frac {3\,a^2\,b\,e^3}{5}+\frac {9\,a^2\,c\,d\,e^2}{5}+\frac {9\,a\,b^2\,d\,e^2}{5}+\frac {18\,a\,b\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d^3}{5}+\frac {3\,b^3\,d^2\,e}{5}+\frac {3\,b^2\,c\,d^3}{5}\right )+x^6\,\left (\frac {a^2\,c\,e^3}{2}+\frac {a\,b^2\,e^3}{2}+3\,a\,b\,c\,d\,e^2+\frac {3\,a\,c^2\,d^2\,e}{2}+\frac {b^3\,d\,e^2}{2}+\frac {3\,b^2\,c\,d^2\,e}{2}+\frac {b\,c^2\,d^3}{2}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2+a\,c\,e^2\right )}{8} \] Input:
int(x^3*(a^3*e^3 + b^3*d^3 + 6*a*b*c*d^3 + 9*a*b^2*d^2*e + 9*a^2*b*d*e^2 + 9*a^2*c*d^2*e) + x^6*(b^3*e^3 + c^3*d^3 + 6*a*b*c*e^3 + 9*a*c^2*d*e^2 + 9 *b*c^2*d^2*e + 9*b^2*c*d*e^2) + 3*x^4*(a*c^2*d^3 + a^2*b*e^3 + b^2*c*d^3 + b^3*d^2*e + 3*a*b^2*d*e^2 + 3*a^2*c*d*e^2 + 6*a*b*c*d^2*e) + 3*x^5*(a*b^2 *e^3 + b*c^2*d^3 + a^2*c*e^3 + b^3*d*e^2 + 3*a*c^2*d^2*e + 3*b^2*c*d^2*e + 6*a*b*c*d*e^2) + a^3*d^3 + c^3*e^3*x^9 + 3*a^2*d^2*x*(a*e + b*d) + 3*c^2* e^2*x^8*(b*e + c*d) + 3*a*d*x^2*(a^2*e^2 + b^2*d^2 + a*c*d^2 + 3*a*b*d*e) + 3*c*e*x^7*(b^2*e^2 + c^2*d^2 + a*c*e^2 + 3*b*c*d*e),x)
Output:
x^4*((a^3*e^3)/4 + (b^3*d^3)/4 + (3*a*b*c*d^3)/2 + (9*a*b^2*d^2*e)/4 + (9* a^2*b*d*e^2)/4 + (9*a^2*c*d^2*e)/4) + x^7*((b^3*e^3)/7 + (c^3*d^3)/7 + (6* a*b*c*e^3)/7 + (9*a*c^2*d*e^2)/7 + (9*b*c^2*d^2*e)/7 + (9*b^2*c*d*e^2)/7) + x^5*((3*a*c^2*d^3)/5 + (3*a^2*b*e^3)/5 + (3*b^2*c*d^3)/5 + (3*b^3*d^2*e) /5 + (9*a*b^2*d*e^2)/5 + (9*a^2*c*d*e^2)/5 + (18*a*b*c*d^2*e)/5) + x^6*((a *b^2*e^3)/2 + (b*c^2*d^3)/2 + (a^2*c*e^3)/2 + (b^3*d*e^2)/2 + (3*a*c^2*d^2 *e)/2 + (3*b^2*c*d^2*e)/2 + 3*a*b*c*d*e^2) + a^3*d^3*x + (c^3*e^3*x^10)/10 + (3*a^2*d^2*x^2*(a*e + b*d))/2 + (c^2*e^2*x^9*(b*e + c*d))/3 + a*d*x^3*( a^2*e^2 + b^2*d^2 + a*c*d^2 + 3*a*b*d*e) + (3*c*e*x^8*(b^2*e^2 + c^2*d^2 + a*c*e^2 + 3*b*c*d*e))/8
Time = 0.23 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.77 \[ \int \left (a^3 d^3+3 a^2 d^2 (b d+a e) x+3 a d \left (b^2 d^2+a c d^2+3 a b d e+a^2 e^2\right ) x^2+\left (b^3 d^3+6 a b c d^3+9 a b^2 d^2 e+9 a^2 c d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^3+3 \left (b^2 c d^3+a c^2 d^3+b^3 d^2 e+6 a b c d^2 e+3 a b^2 d e^2+3 a^2 c d e^2+a^2 b e^3\right ) x^4+3 \left (b c^2 d^3+3 b^2 c d^2 e+3 a c^2 d^2 e+b^3 d e^2+6 a b c d e^2+a b^2 e^3+a^2 c e^3\right ) x^5+\left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+9 a c^2 d e^2+b^3 e^3+6 a b c e^3\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2+a c e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx=\frac {x \left (84 c^{3} e^{3} x^{9}+280 b \,c^{2} e^{3} x^{8}+280 c^{3} d \,e^{2} x^{8}+315 a \,c^{2} e^{3} x^{7}+315 b^{2} c \,e^{3} x^{7}+945 b \,c^{2} d \,e^{2} x^{7}+315 c^{3} d^{2} e \,x^{7}+720 a b c \,e^{3} x^{6}+1080 a \,c^{2} d \,e^{2} x^{6}+120 b^{3} e^{3} x^{6}+1080 b^{2} c d \,e^{2} x^{6}+1080 b \,c^{2} d^{2} e \,x^{6}+120 c^{3} d^{3} x^{6}+420 a^{2} c \,e^{3} x^{5}+420 a \,b^{2} e^{3} x^{5}+2520 a b c d \,e^{2} x^{5}+1260 a \,c^{2} d^{2} e \,x^{5}+420 b^{3} d \,e^{2} x^{5}+1260 b^{2} c \,d^{2} e \,x^{5}+420 b \,c^{2} d^{3} x^{5}+504 a^{2} b \,e^{3} x^{4}+1512 a^{2} c d \,e^{2} x^{4}+1512 a \,b^{2} d \,e^{2} x^{4}+3024 a b c \,d^{2} e \,x^{4}+504 a \,c^{2} d^{3} x^{4}+504 b^{3} d^{2} e \,x^{4}+504 b^{2} c \,d^{3} x^{4}+210 a^{3} e^{3} x^{3}+1890 a^{2} b d \,e^{2} x^{3}+1890 a^{2} c \,d^{2} e \,x^{3}+1890 a \,b^{2} d^{2} e \,x^{3}+1260 a b c \,d^{3} x^{3}+210 b^{3} d^{3} x^{3}+840 a^{3} d \,e^{2} x^{2}+2520 a^{2} b \,d^{2} e \,x^{2}+840 a^{2} c \,d^{3} x^{2}+840 a \,b^{2} d^{3} x^{2}+1260 a^{3} d^{2} e x +1260 a^{2} b \,d^{3} x +840 a^{3} d^{3}\right )}{840} \] Input:
int(a^3*d^3+3*a^2*d^2*(a*e+b*d)*x+3*a*d*(a^2*e^2+3*a*b*d*e+a*c*d^2+b^2*d^2 )*x^2+(a^3*e^3+9*a^2*b*d*e^2+9*a^2*c*d^2*e+9*a*b^2*d^2*e+6*a*b*c*d^3+b^3*d ^3)*x^3+3*(a^2*b*e^3+3*a^2*c*d*e^2+3*a*b^2*d*e^2+6*a*b*c*d^2*e+a*c^2*d^3+b ^3*d^2*e+b^2*c*d^3)*x^4+3*(a^2*c*e^3+a*b^2*e^3+6*a*b*c*d*e^2+3*a*c^2*d^2*e +b^3*d*e^2+3*b^2*c*d^2*e+b*c^2*d^3)*x^5+(6*a*b*c*e^3+9*a*c^2*d*e^2+b^3*e^3 +9*b^2*c*d*e^2+9*b*c^2*d^2*e+c^3*d^3)*x^6+3*c*e*(a*c*e^2+b^2*e^2+3*b*c*d*e +c^2*d^2)*x^7+3*c^2*e^2*(b*e+c*d)*x^8+c^3*e^3*x^9,x)
Output:
(x*(840*a**3*d**3 + 1260*a**3*d**2*e*x + 840*a**3*d*e**2*x**2 + 210*a**3*e **3*x**3 + 1260*a**2*b*d**3*x + 2520*a**2*b*d**2*e*x**2 + 1890*a**2*b*d*e* *2*x**3 + 504*a**2*b*e**3*x**4 + 840*a**2*c*d**3*x**2 + 1890*a**2*c*d**2*e *x**3 + 1512*a**2*c*d*e**2*x**4 + 420*a**2*c*e**3*x**5 + 840*a*b**2*d**3*x **2 + 1890*a*b**2*d**2*e*x**3 + 1512*a*b**2*d*e**2*x**4 + 420*a*b**2*e**3* x**5 + 1260*a*b*c*d**3*x**3 + 3024*a*b*c*d**2*e*x**4 + 2520*a*b*c*d*e**2*x **5 + 720*a*b*c*e**3*x**6 + 504*a*c**2*d**3*x**4 + 1260*a*c**2*d**2*e*x**5 + 1080*a*c**2*d*e**2*x**6 + 315*a*c**2*e**3*x**7 + 210*b**3*d**3*x**3 + 5 04*b**3*d**2*e*x**4 + 420*b**3*d*e**2*x**5 + 120*b**3*e**3*x**6 + 504*b**2 *c*d**3*x**4 + 1260*b**2*c*d**2*e*x**5 + 1080*b**2*c*d*e**2*x**6 + 315*b** 2*c*e**3*x**7 + 420*b*c**2*d**3*x**5 + 1080*b*c**2*d**2*e*x**6 + 945*b*c** 2*d*e**2*x**7 + 280*b*c**2*e**3*x**8 + 120*c**3*d**3*x**6 + 315*c**3*d**2* e*x**7 + 280*c**3*d*e**2*x**8 + 84*c**3*e**3*x**9))/840