Integrand size = 20, antiderivative size = 156 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \] Output:
1/5*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^5/e^5-1/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)*(e*x+d)^6/e^5+1/7*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^7/e^5- 1/4*c*(-b*e+2*c*d)*(e*x+d)^8/e^5+1/9*c^2*(e*x+d)^9/e^5
Time = 0.06 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^2 d^4 x+a d^3 (b d+2 a e) x^2+\frac {1}{3} d^2 \left (b^2 d^2+8 a b d e+2 a \left (c d^2+3 a e^2\right )\right ) x^3+\frac {1}{2} d \left (b c d^3+2 b^2 d^2 e+4 a c d^2 e+6 a b d e^2+2 a^2 e^3\right ) x^4+\frac {1}{5} \left (c^2 d^4+4 c d^2 e (2 b d+3 a e)+e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^5+\frac {1}{3} e \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right ) x^6+\frac {1}{7} e^2 \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right ) x^7+\frac {1}{4} c e^3 (2 c d+b e) x^8+\frac {1}{9} c^2 e^4 x^9 \] Input:
Integrate[(d + e*x)^4*(a + b*x + c*x^2)^2,x]
Output:
a^2*d^4*x + a*d^3*(b*d + 2*a*e)*x^2 + (d^2*(b^2*d^2 + 8*a*b*d*e + 2*a*(c*d ^2 + 3*a*e^2))*x^3)/3 + (d*(b*c*d^3 + 2*b^2*d^2*e + 4*a*c*d^2*e + 6*a*b*d* e^2 + 2*a^2*e^3)*x^4)/2 + ((c^2*d^4 + 4*c*d^2*e*(2*b*d + 3*a*e) + e^2*(6*b ^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^5)/5 + (e*(2*c^2*d^3 + b*e^2*(2*b*d + a*e ) + 2*c*d*e*(3*b*d + 2*a*e))*x^6)/3 + (e^2*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(4 *b*d + a*e))*x^7)/7 + (c*e^3*(2*c*d + b*e)*x^8)/4 + (c^2*e^4*x^9)/9
Time = 0.45 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (d+e x)^5 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{e^4}-\frac {2 c (d+e x)^7 (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}\) |
Input:
Int[(d + e*x)^4*(a + b*x + c*x^2)^2,x]
Output:
((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^5)/(5*e^5) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6)/(3*e^5) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^7)/(7*e^5) - (c*(2*c*d - b*e)*(d + e*x)^8)/(4*e^5) + (c ^2*(d + e*x)^9)/(9*e^5)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.84 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {\left (2 e^{4} b c +4 c^{2} d \,e^{3}\right ) x^{8}}{8}+\frac {\left (6 d^{2} e^{2} c^{2}+8 d \,e^{3} b c +e^{4} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (4 d^{3} e \,c^{2}+12 d^{2} e^{2} b c +4 d \,e^{3} \left (2 a c +b^{2}\right )+2 e^{4} a b \right ) x^{6}}{6}+\frac {\left (c^{2} d^{4}+8 b c \,d^{3} e +6 d^{2} e^{2} \left (2 a c +b^{2}\right )+8 d \,e^{3} a b +a^{2} e^{4}\right ) x^{5}}{5}+\frac {\left (2 d^{4} b c +4 d^{3} e \left (2 a c +b^{2}\right )+12 d^{2} e^{2} a b +4 a^{2} d \,e^{3}\right ) x^{4}}{4}+\frac {\left (d^{4} \left (2 a c +b^{2}\right )+8 d^{3} e a b +6 d^{2} e^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (4 a^{2} d^{3} e +2 a b \,d^{4}\right ) x^{2}}{2}+a^{2} d^{4} x\) | \(283\) |
| norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\left (\frac {1}{4} e^{4} b c +\frac {1}{2} c^{2} d \,e^{3}\right ) x^{8}+\left (\frac {2}{7} a c \,e^{4}+\frac {1}{7} b^{2} e^{4}+\frac {8}{7} d \,e^{3} b c +\frac {6}{7} d^{2} e^{2} c^{2}\right ) x^{7}+\left (\frac {1}{3} e^{4} a b +\frac {4}{3} d \,e^{3} a c +\frac {2}{3} b^{2} d \,e^{3}+2 d^{2} e^{2} b c +\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{4}+\frac {8}{5} d \,e^{3} a b +\frac {12}{5} a c \,d^{2} e^{2}+\frac {6}{5} d^{2} e^{2} b^{2}+\frac {8}{5} b c \,d^{3} e +\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (a^{2} d \,e^{3}+3 d^{2} e^{2} a b +2 a c \,d^{3} e +b^{2} d^{3} e +\frac {1}{2} d^{4} b c \right ) x^{4}+\left (2 d^{2} e^{2} a^{2}+\frac {8}{3} d^{3} e a b +\frac {2}{3} a c \,d^{4}+\frac {1}{3} b^{2} d^{4}\right ) x^{3}+\left (2 a^{2} d^{3} e +a b \,d^{4}\right ) x^{2}+a^{2} d^{4} x\) | \(292\) |
| gosper | \(\frac {8}{7} x^{7} d \,e^{3} b c +\frac {4}{3} x^{6} d \,e^{3} a c +a^{2} d^{4} x +\frac {2}{3} a c \,d^{4} x^{3}+\frac {1}{2} b c \,d^{4} x^{4}+2 d^{3} e \,a^{2} x^{2}+\frac {8}{5} x^{5} b c \,d^{3} e +\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {1}{9} e^{4} c^{2} x^{9}+2 x^{4} a c \,d^{3} e +\frac {1}{3} b^{2} d^{4} x^{3}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+a b \,d^{4} x^{2}+\frac {8}{5} x^{5} d \,e^{3} a b +\frac {6}{5} x^{5} d^{2} e^{2} b^{2}+x^{4} a^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+x^{4} b^{2} d^{3} e +2 x^{3} d^{2} e^{2} a^{2}+\frac {8}{3} x^{3} d^{3} e a b +3 x^{4} d^{2} e^{2} a b +2 x^{6} d^{2} e^{2} b c +\frac {1}{4} x^{8} e^{4} b c +\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {1}{3} x^{6} e^{4} a b +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {1}{7} b^{2} e^{4} x^{7}\) | \(341\) |
| risch | \(\frac {8}{7} x^{7} d \,e^{3} b c +\frac {4}{3} x^{6} d \,e^{3} a c +a^{2} d^{4} x +\frac {2}{3} a c \,d^{4} x^{3}+\frac {1}{2} b c \,d^{4} x^{4}+2 d^{3} e \,a^{2} x^{2}+\frac {8}{5} x^{5} b c \,d^{3} e +\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {1}{9} e^{4} c^{2} x^{9}+2 x^{4} a c \,d^{3} e +\frac {1}{3} b^{2} d^{4} x^{3}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+a b \,d^{4} x^{2}+\frac {8}{5} x^{5} d \,e^{3} a b +\frac {6}{5} x^{5} d^{2} e^{2} b^{2}+x^{4} a^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+x^{4} b^{2} d^{3} e +2 x^{3} d^{2} e^{2} a^{2}+\frac {8}{3} x^{3} d^{3} e a b +3 x^{4} d^{2} e^{2} a b +2 x^{6} d^{2} e^{2} b c +\frac {1}{4} x^{8} e^{4} b c +\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {1}{3} x^{6} e^{4} a b +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {1}{7} b^{2} e^{4} x^{7}\) | \(341\) |
| parallelrisch | \(\frac {8}{7} x^{7} d \,e^{3} b c +\frac {4}{3} x^{6} d \,e^{3} a c +a^{2} d^{4} x +\frac {2}{3} a c \,d^{4} x^{3}+\frac {1}{2} b c \,d^{4} x^{4}+2 d^{3} e \,a^{2} x^{2}+\frac {8}{5} x^{5} b c \,d^{3} e +\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {1}{9} e^{4} c^{2} x^{9}+2 x^{4} a c \,d^{3} e +\frac {1}{3} b^{2} d^{4} x^{3}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+a b \,d^{4} x^{2}+\frac {8}{5} x^{5} d \,e^{3} a b +\frac {6}{5} x^{5} d^{2} e^{2} b^{2}+x^{4} a^{2} d \,e^{3}+\frac {1}{5} c^{2} d^{4} x^{5}+x^{4} b^{2} d^{3} e +2 x^{3} d^{2} e^{2} a^{2}+\frac {8}{3} x^{3} d^{3} e a b +3 x^{4} d^{2} e^{2} a b +2 x^{6} d^{2} e^{2} b c +\frac {1}{4} x^{8} e^{4} b c +\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {1}{3} x^{6} e^{4} a b +\frac {2}{3} x^{6} b^{2} d \,e^{3}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {1}{7} b^{2} e^{4} x^{7}\) | \(341\) |
| orering | \(\frac {x \left (140 e^{4} c^{2} x^{8}+315 b c \,e^{4} x^{7}+630 c^{2} d \,e^{3} x^{7}+360 a c \,e^{4} x^{6}+180 b^{2} e^{4} x^{6}+1440 b c d \,e^{3} x^{6}+1080 c^{2} d^{2} e^{2} x^{6}+420 a b \,e^{4} x^{5}+1680 a c d \,e^{3} x^{5}+840 b^{2} d \,e^{3} x^{5}+2520 b c \,d^{2} e^{2} x^{5}+840 c^{2} d^{3} e \,x^{5}+252 a^{2} e^{4} x^{4}+2016 a b d \,e^{3} x^{4}+3024 a c \,d^{2} e^{2} x^{4}+1512 b^{2} d^{2} e^{2} x^{4}+2016 b c \,d^{3} e \,x^{4}+252 c^{2} d^{4} x^{4}+1260 a^{2} d \,e^{3} x^{3}+3780 a b \,d^{2} e^{2} x^{3}+2520 a c \,d^{3} e \,x^{3}+1260 b^{2} d^{3} e \,x^{3}+630 b c \,d^{4} x^{3}+2520 a^{2} d^{2} e^{2} x^{2}+3360 a b \,d^{3} e \,x^{2}+840 a c \,d^{4} x^{2}+420 b^{2} d^{4} x^{2}+2520 a^{2} d^{3} e x +1260 a b \,d^{4} x +1260 a^{2} d^{4}\right )}{1260}\) | \(343\) |
Input:
int((e*x+d)^4*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/9*e^4*c^2*x^9+1/8*(2*b*c*e^4+4*c^2*d*e^3)*x^8+1/7*(6*d^2*e^2*c^2+8*d*e^3 *b*c+e^4*(2*a*c+b^2))*x^7+1/6*(4*d^3*e*c^2+12*d^2*e^2*b*c+4*d*e^3*(2*a*c+b ^2)+2*e^4*a*b)*x^6+1/5*(c^2*d^4+8*b*c*d^3*e+6*d^2*e^2*(2*a*c+b^2)+8*d*e^3* a*b+a^2*e^4)*x^5+1/4*(2*d^4*b*c+4*d^3*e*(2*a*c+b^2)+12*d^2*e^2*a*b+4*a^2*d *e^3)*x^4+1/3*(d^4*(2*a*c+b^2)+8*d^3*e*a*b+6*d^2*e^2*a^2)*x^3+1/2*(4*a^2*d ^3*e+2*a*b*d^4)*x^2+a^2*d^4*x
Time = 0.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/9*c^2*e^4*x^9 + 1/4*(2*c^2*d*e^3 + b*c*e^4)*x^8 + 1/7*(6*c^2*d^2*e^2 + 8 *b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^7 + a^2*d^4*x + 1/3*(2*c^2*d^3*e + 6*b*c *d^2*e^2 + a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x^6 + 1/5*(c^2*d^4 + 8*b*c*d^3 *e + 8*a*b*d*e^3 + a^2*e^4 + 6*(b^2 + 2*a*c)*d^2*e^2)*x^5 + 1/2*(b*c*d^4 + 6*a*b*d^2*e^2 + 2*a^2*d*e^3 + 2*(b^2 + 2*a*c)*d^3*e)*x^4 + 1/3*(8*a*b*d^3 *e + 6*a^2*d^2*e^2 + (b^2 + 2*a*c)*d^4)*x^3 + (a*b*d^4 + 2*a^2*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (146) = 292\).
Time = 0.06 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.16 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^{2} d^{4} x + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \cdot \left (\frac {2 a c e^{4}}{7} + \frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {a b e^{4}}{3} + \frac {4 a c d e^{3}}{3} + \frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {12 a c d^{2} e^{2}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + 2 a c d^{3} e + b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {2 a c d^{4}}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \cdot \left (2 a^{2} d^{3} e + a b d^{4}\right ) \] Input:
integrate((e*x+d)**4*(c*x**2+b*x+a)**2,x)
Output:
a**2*d**4*x + c**2*e**4*x**9/9 + x**8*(b*c*e**4/4 + c**2*d*e**3/2) + x**7* (2*a*c*e**4/7 + b**2*e**4/7 + 8*b*c*d*e**3/7 + 6*c**2*d**2*e**2/7) + x**6* (a*b*e**4/3 + 4*a*c*d*e**3/3 + 2*b**2*d*e**3/3 + 2*b*c*d**2*e**2 + 2*c**2* d**3*e/3) + x**5*(a**2*e**4/5 + 8*a*b*d*e**3/5 + 12*a*c*d**2*e**2/5 + 6*b* *2*d**2*e**2/5 + 8*b*c*d**3*e/5 + c**2*d**4/5) + x**4*(a**2*d*e**3 + 3*a*b *d**2*e**2 + 2*a*c*d**3*e + b**2*d**3*e + b*c*d**4/2) + x**3*(2*a**2*d**2* e**2 + 8*a*b*d**3*e/3 + 2*a*c*d**4/3 + b**2*d**4/3) + x**2*(2*a**2*d**3*e + a*b*d**4)
Time = 0.04 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.78 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/9*c^2*e^4*x^9 + 1/4*(2*c^2*d*e^3 + b*c*e^4)*x^8 + 1/7*(6*c^2*d^2*e^2 + 8 *b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^7 + a^2*d^4*x + 1/3*(2*c^2*d^3*e + 6*b*c *d^2*e^2 + a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x^6 + 1/5*(c^2*d^4 + 8*b*c*d^3 *e + 8*a*b*d*e^3 + a^2*e^4 + 6*(b^2 + 2*a*c)*d^2*e^2)*x^5 + 1/2*(b*c*d^4 + 6*a*b*d^2*e^2 + 2*a^2*d*e^3 + 2*(b^2 + 2*a*c)*d^3*e)*x^4 + 1/3*(8*a*b*d^3 *e + 6*a^2*d^2*e^2 + (b^2 + 2*a*c)*d^4)*x^3 + (a*b*d^4 + 2*a^2*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (146) = 292\).
Time = 0.33 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.18 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + \frac {1}{4} \, b c e^{4} x^{8} + \frac {6}{7} \, c^{2} d^{2} e^{2} x^{7} + \frac {8}{7} \, b c d e^{3} x^{7} + \frac {1}{7} \, b^{2} e^{4} x^{7} + \frac {2}{7} \, a c e^{4} x^{7} + \frac {2}{3} \, c^{2} d^{3} e x^{6} + 2 \, b c d^{2} e^{2} x^{6} + \frac {2}{3} \, b^{2} d e^{3} x^{6} + \frac {4}{3} \, a c d e^{3} x^{6} + \frac {1}{3} \, a b e^{4} x^{6} + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {8}{5} \, b c d^{3} e x^{5} + \frac {6}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {12}{5} \, a c d^{2} e^{2} x^{5} + \frac {8}{5} \, a b d e^{3} x^{5} + \frac {1}{5} \, a^{2} e^{4} x^{5} + \frac {1}{2} \, b c d^{4} x^{4} + b^{2} d^{3} e x^{4} + 2 \, a c d^{3} e x^{4} + 3 \, a b d^{2} e^{2} x^{4} + a^{2} d e^{3} x^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {2}{3} \, a c d^{4} x^{3} + \frac {8}{3} \, a b d^{3} e x^{3} + 2 \, a^{2} d^{2} e^{2} x^{3} + a b d^{4} x^{2} + 2 \, a^{2} d^{3} e x^{2} + a^{2} d^{4} x \] Input:
integrate((e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 1/4*b*c*e^4*x^8 + 6/7*c^2*d^2*e^2*x^ 7 + 8/7*b*c*d*e^3*x^7 + 1/7*b^2*e^4*x^7 + 2/7*a*c*e^4*x^7 + 2/3*c^2*d^3*e* x^6 + 2*b*c*d^2*e^2*x^6 + 2/3*b^2*d*e^3*x^6 + 4/3*a*c*d*e^3*x^6 + 1/3*a*b* e^4*x^6 + 1/5*c^2*d^4*x^5 + 8/5*b*c*d^3*e*x^5 + 6/5*b^2*d^2*e^2*x^5 + 12/5 *a*c*d^2*e^2*x^5 + 8/5*a*b*d*e^3*x^5 + 1/5*a^2*e^4*x^5 + 1/2*b*c*d^4*x^4 + b^2*d^3*e*x^4 + 2*a*c*d^3*e*x^4 + 3*a*b*d^2*e^2*x^4 + a^2*d*e^3*x^4 + 1/3 *b^2*d^4*x^3 + 2/3*a*c*d^4*x^3 + 8/3*a*b*d^3*e*x^3 + 2*a^2*d^2*e^2*x^3 + a *b*d^4*x^2 + 2*a^2*d^3*e*x^2 + a^2*d^4*x
Time = 5.84 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=x^4\,\left (a^2\,d\,e^3+3\,a\,b\,d^2\,e^2+2\,c\,a\,d^3\,e+b^2\,d^3\,e+\frac {c\,b\,d^4}{2}\right )+x^6\,\left (\frac {2\,b^2\,d\,e^3}{3}+2\,b\,c\,d^2\,e^2+\frac {a\,b\,e^4}{3}+\frac {2\,c^2\,d^3\,e}{3}+\frac {4\,a\,c\,d\,e^3}{3}\right )+x^5\,\left (\frac {a^2\,e^4}{5}+\frac {8\,a\,b\,d\,e^3}{5}+\frac {12\,a\,c\,d^2\,e^2}{5}+\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^3\,\left (2\,a^2\,d^2\,e^2+\frac {8\,a\,b\,d^3\,e}{3}+\frac {2\,c\,a\,d^4}{3}+\frac {b^2\,d^4}{3}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}+\frac {2\,a\,c\,e^4}{7}\right )+a^2\,d^4\,x+\frac {c^2\,e^4\,x^9}{9}+a\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4} \] Input:
int((d + e*x)^4*(a + b*x + c*x^2)^2,x)
Output:
x^4*(a^2*d*e^3 + b^2*d^3*e + (b*c*d^4)/2 + 2*a*c*d^3*e + 3*a*b*d^2*e^2) + x^6*((2*b^2*d*e^3)/3 + (2*c^2*d^3*e)/3 + (a*b*e^4)/3 + (4*a*c*d*e^3)/3 + 2 *b*c*d^2*e^2) + x^5*((a^2*e^4)/5 + (c^2*d^4)/5 + (6*b^2*d^2*e^2)/5 + (8*a* b*d*e^3)/5 + (8*b*c*d^3*e)/5 + (12*a*c*d^2*e^2)/5) + x^3*((b^2*d^4)/3 + 2* a^2*d^2*e^2 + (2*a*c*d^4)/3 + (8*a*b*d^3*e)/3) + x^7*((b^2*e^4)/7 + (6*c^2 *d^2*e^2)/7 + (2*a*c*e^4)/7 + (8*b*c*d*e^3)/7) + a^2*d^4*x + (c^2*e^4*x^9) /9 + a*d^3*x^2*(2*a*e + b*d) + (c*e^3*x^8*(b*e + 2*c*d))/4
Time = 0.27 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.19 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {x \left (140 c^{2} e^{4} x^{8}+315 b c \,e^{4} x^{7}+630 c^{2} d \,e^{3} x^{7}+360 a c \,e^{4} x^{6}+180 b^{2} e^{4} x^{6}+1440 b c d \,e^{3} x^{6}+1080 c^{2} d^{2} e^{2} x^{6}+420 a b \,e^{4} x^{5}+1680 a c d \,e^{3} x^{5}+840 b^{2} d \,e^{3} x^{5}+2520 b c \,d^{2} e^{2} x^{5}+840 c^{2} d^{3} e \,x^{5}+252 a^{2} e^{4} x^{4}+2016 a b d \,e^{3} x^{4}+3024 a c \,d^{2} e^{2} x^{4}+1512 b^{2} d^{2} e^{2} x^{4}+2016 b c \,d^{3} e \,x^{4}+252 c^{2} d^{4} x^{4}+1260 a^{2} d \,e^{3} x^{3}+3780 a b \,d^{2} e^{2} x^{3}+2520 a c \,d^{3} e \,x^{3}+1260 b^{2} d^{3} e \,x^{3}+630 b c \,d^{4} x^{3}+2520 a^{2} d^{2} e^{2} x^{2}+3360 a b \,d^{3} e \,x^{2}+840 a c \,d^{4} x^{2}+420 b^{2} d^{4} x^{2}+2520 a^{2} d^{3} e x +1260 a b \,d^{4} x +1260 a^{2} d^{4}\right )}{1260} \] Input:
int((e*x+d)^4*(c*x^2+b*x+a)^2,x)
Output:
(x*(1260*a**2*d**4 + 2520*a**2*d**3*e*x + 2520*a**2*d**2*e**2*x**2 + 1260* a**2*d*e**3*x**3 + 252*a**2*e**4*x**4 + 1260*a*b*d**4*x + 3360*a*b*d**3*e* x**2 + 3780*a*b*d**2*e**2*x**3 + 2016*a*b*d*e**3*x**4 + 420*a*b*e**4*x**5 + 840*a*c*d**4*x**2 + 2520*a*c*d**3*e*x**3 + 3024*a*c*d**2*e**2*x**4 + 168 0*a*c*d*e**3*x**5 + 360*a*c*e**4*x**6 + 420*b**2*d**4*x**2 + 1260*b**2*d** 3*e*x**3 + 1512*b**2*d**2*e**2*x**4 + 840*b**2*d*e**3*x**5 + 180*b**2*e**4 *x**6 + 630*b*c*d**4*x**3 + 2016*b*c*d**3*e*x**4 + 2520*b*c*d**2*e**2*x**5 + 1440*b*c*d*e**3*x**6 + 315*b*c*e**4*x**7 + 252*c**2*d**4*x**4 + 840*c** 2*d**3*e*x**5 + 1080*c**2*d**2*e**2*x**6 + 630*c**2*d*e**3*x**7 + 140*c**2 *e**4*x**8))/1260