Integrand size = 20, antiderivative size = 272 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{3 e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^7}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^5}{5 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^6}{6 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^8}{8 e^7}+\frac {c^3 (d+e x)^9}{9 e^7} \] Output:
1/3*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^3/e^7-3/4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)^2*(e*x+d)^4/e^7+3/5*(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)^5/e^7-1/6*(-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)^6/e^7+3/7*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^7/e^ 7-3/8*c^2*(-b*e+2*c*d)*(e*x+d)^8/e^7+1/9*c^3*(e*x+d)^9/e^7
Time = 0.07 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.04 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=a^3 d^2 x+\frac {1}{2} a^2 d (3 b d+2 a e) x^2+\frac {1}{3} a \left (3 b^2 d^2+6 a b d e+a \left (3 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} \left (b^3 d^2+6 a b^2 d e+6 a^2 c d e+3 a b \left (2 c d^2+a e^2\right )\right ) x^4+\frac {1}{5} \left (2 b^3 d e+12 a b c d e+3 b^2 \left (c d^2+a e^2\right )+3 a c \left (c d^2+a e^2\right )\right ) x^5+\frac {1}{6} \left (6 b^2 c d e+6 a c^2 d e+b^3 e^2+3 b c \left (c d^2+2 a e^2\right )\right ) x^6+\frac {1}{7} c \left (c^2 d^2+3 b^2 e^2+3 c e (2 b d+a e)\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \] Input:
Integrate[(d + e*x)^2*(a + b*x + c*x^2)^3,x]
Output:
a^3*d^2*x + (a^2*d*(3*b*d + 2*a*e)*x^2)/2 + (a*(3*b^2*d^2 + 6*a*b*d*e + a* (3*c*d^2 + a*e^2))*x^3)/3 + ((b^3*d^2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b* (2*c*d^2 + a*e^2))*x^4)/4 + ((2*b^3*d*e + 12*a*b*c*d*e + 3*b^2*(c*d^2 + a* e^2) + 3*a*c*(c*d^2 + a*e^2))*x^5)/5 + ((6*b^2*c*d*e + 6*a*c^2*d*e + b^3*e ^2 + 3*b*c*(c*d^2 + 2*a*e^2))*x^6)/6 + (c*(c^2*d^2 + 3*b^2*e^2 + 3*c*e*(2* b*d + a*e))*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9
Time = 0.56 (sec) , antiderivative size = 272, 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.
| \(\displaystyle \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^5 (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {(d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}{e^6}-\frac {3 c^2 (d+e x)^7 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^8}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7}+\frac {3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac {3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac {c^3 (d+e x)^9}{9 e^7}\) |
Input:
Int[(d + e*x)^2*(a + b*x + c*x^2)^3,x]
Output:
((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3)/(3*e^7) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4)/(4*e^7) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^ 2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^5)/(5*e^7) - ((2*c*d - b*e) *(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^6)/(6*e^7) + (3* c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^7)/(7*e^7) - (3*c^2* (2*c*d - b*e)*(d + e*x)^8)/(8*e^7) + (c^3*(d + e*x)^9)/(9*e^7)
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 = 282, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(\frac {c^{3} e^{2} x^{9}}{9}+\left (\frac {3}{8} b \,c^{2} e^{2}+\frac {1}{4} d e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} c^{2} a \,e^{2}+\frac {3}{7} e^{2} b^{2} c +\frac {6}{7} c^{2} d e b +\frac {1}{7} c^{3} d^{2}\right ) x^{7}+\left (a b c \,e^{2}+a \,c^{2} d e +\frac {1}{6} e^{2} b^{3}+b^{2} c d e +\frac {1}{2} b \,c^{2} d^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} c \,e^{2}+\frac {3}{5} a \,b^{2} e^{2}+\frac {12}{5} a b c d e +\frac {3}{5} a \,c^{2} d^{2}+\frac {2}{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {3}{4} a^{2} b \,e^{2}+\frac {3}{2} a^{2} c d e +\frac {3}{2} a \,b^{2} d e +\frac {3}{2} a b c \,d^{2}+\frac {1}{4} b^{3} d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{3} e^{2}+2 a^{2} b d e +a^{2} c \,d^{2}+a \,b^{2} d^{2}\right ) x^{3}+\left (a^{3} d e +\frac {3}{2} a^{2} b \,d^{2}\right ) x^{2}+a^{3} d^{2} x\) | \(282\) |
| gosper | \(\frac {3}{2} a b c \,d^{2} x^{4}+\frac {3}{4} x^{4} a^{2} b \,e^{2}+\frac {6}{7} x^{7} c^{2} d e b +x^{6} a b c \,e^{2}+\frac {3}{7} x^{7} c^{2} a \,e^{2}+\frac {3}{7} x^{7} e^{2} b^{2} c +\frac {1}{2} x^{6} b \,c^{2} d^{2}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} x^{5} a \,b^{2} e^{2}+\frac {3}{5} x^{5} a \,c^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d e +x^{6} b^{2} c d e +a \,b^{2} d^{2} x^{3}+2 x^{3} a^{2} b d e +\frac {1}{4} d e \,c^{3} x^{8}+d e \,a^{3} x^{2}+d e a \,c^{2} x^{6}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {3}{2} x^{2} a^{2} b \,d^{2}+\frac {3}{2} x^{4} a \,b^{2} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{9} c^{3} e^{2} x^{9}+\frac {12}{5} x^{5} a b c d e +\frac {3}{8} x^{8} b \,c^{2} e^{2}+x^{3} a^{2} c \,d^{2}+a^{3} d^{2} x +\frac {1}{4} b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} e^{2} b^{3}\) | \(331\) |
| risch | \(\frac {3}{2} a b c \,d^{2} x^{4}+\frac {3}{4} x^{4} a^{2} b \,e^{2}+\frac {6}{7} x^{7} c^{2} d e b +x^{6} a b c \,e^{2}+\frac {3}{7} x^{7} c^{2} a \,e^{2}+\frac {3}{7} x^{7} e^{2} b^{2} c +\frac {1}{2} x^{6} b \,c^{2} d^{2}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} x^{5} a \,b^{2} e^{2}+\frac {3}{5} x^{5} a \,c^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d e +x^{6} b^{2} c d e +a \,b^{2} d^{2} x^{3}+2 x^{3} a^{2} b d e +\frac {1}{4} d e \,c^{3} x^{8}+d e \,a^{3} x^{2}+d e a \,c^{2} x^{6}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {3}{2} x^{2} a^{2} b \,d^{2}+\frac {3}{2} x^{4} a \,b^{2} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{9} c^{3} e^{2} x^{9}+\frac {12}{5} x^{5} a b c d e +\frac {3}{8} x^{8} b \,c^{2} e^{2}+x^{3} a^{2} c \,d^{2}+a^{3} d^{2} x +\frac {1}{4} b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} e^{2} b^{3}\) | \(331\) |
| parallelrisch | \(\frac {3}{2} a b c \,d^{2} x^{4}+\frac {3}{4} x^{4} a^{2} b \,e^{2}+\frac {6}{7} x^{7} c^{2} d e b +x^{6} a b c \,e^{2}+\frac {3}{7} x^{7} c^{2} a \,e^{2}+\frac {3}{7} x^{7} e^{2} b^{2} c +\frac {1}{2} x^{6} b \,c^{2} d^{2}+\frac {3}{5} x^{5} a^{2} c \,e^{2}+\frac {3}{5} x^{5} a \,b^{2} e^{2}+\frac {3}{5} x^{5} a \,c^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d e +x^{6} b^{2} c d e +a \,b^{2} d^{2} x^{3}+2 x^{3} a^{2} b d e +\frac {1}{4} d e \,c^{3} x^{8}+d e \,a^{3} x^{2}+d e a \,c^{2} x^{6}+\frac {3}{2} d e \,a^{2} c \,x^{4}+\frac {3}{2} x^{2} a^{2} b \,d^{2}+\frac {3}{2} x^{4} a \,b^{2} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{9} c^{3} e^{2} x^{9}+\frac {12}{5} x^{5} a b c d e +\frac {3}{8} x^{8} b \,c^{2} e^{2}+x^{3} a^{2} c \,d^{2}+a^{3} d^{2} x +\frac {1}{4} b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{3} e^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} e^{2} b^{3}\) | \(331\) |
| orering | \(\frac {x \left (280 e^{2} c^{3} x^{8}+945 b \,c^{2} e^{2} x^{7}+630 c^{3} d e \,x^{7}+1080 a \,c^{2} e^{2} x^{6}+1080 b^{2} c \,e^{2} x^{6}+2160 b \,c^{2} d e \,x^{6}+360 c^{3} d^{2} x^{6}+2520 a b c \,e^{2} x^{5}+2520 a \,c^{2} d e \,x^{5}+420 e^{2} b^{3} x^{5}+2520 b^{2} c d e \,x^{5}+1260 b \,c^{2} d^{2} x^{5}+1512 a^{2} c \,e^{2} x^{4}+1512 x^{4} a \,b^{2} e^{2}+6048 a b c d e \,x^{4}+1512 a \,c^{2} d^{2} x^{4}+1008 x^{4} b^{3} d e +1512 b^{2} c \,d^{2} x^{4}+1890 x^{3} a^{2} b \,e^{2}+3780 a^{2} c d e \,x^{3}+3780 x^{3} a \,b^{2} d e +3780 a b c \,d^{2} x^{3}+630 x^{3} b^{3} d^{2}+840 x^{2} a^{3} e^{2}+5040 x^{2} a^{2} b d e +2520 a^{2} c \,d^{2} x^{2}+2520 x^{2} a \,b^{2} d^{2}+2520 x \,a^{3} d e +3780 a^{2} b \,d^{2} x +2520 a^{3} d^{2}\right )}{2520}\) | \(336\) |
| default | \(\frac {c^{3} e^{2} x^{9}}{9}+\frac {\left (3 b \,c^{2} e^{2}+2 d e \,c^{3}\right ) x^{8}}{8}+\frac {\left (c^{3} d^{2}+6 c^{2} d e b +e^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 b \,c^{2} d^{2}+2 d e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+2 d e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{5}}{5}+\frac {\left (d^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+2 d e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 a^{2} b \,e^{2}\right ) x^{4}}{4}+\frac {\left (d^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+6 a^{2} b d e +a^{3} e^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} d e +3 a^{2} b \,d^{2}\right ) x^{2}}{2}+a^{3} d^{2} x\) | \(359\) |
Input:
int((e*x+d)^2*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/9*c^3*e^2*x^9+(3/8*b*c^2*e^2+1/4*d*e*c^3)*x^8+(3/7*c^2*a*e^2+3/7*e^2*b^2 *c+6/7*c^2*d*e*b+1/7*c^3*d^2)*x^7+(a*b*c*e^2+a*c^2*d*e+1/6*e^2*b^3+b^2*c*d *e+1/2*b*c^2*d^2)*x^6+(3/5*a^2*c*e^2+3/5*a*b^2*e^2+12/5*a*b*c*d*e+3/5*a*c^ 2*d^2+2/5*b^3*d*e+3/5*b^2*c*d^2)*x^5+(3/4*a^2*b*e^2+3/2*a^2*c*d*e+3/2*a*b^ 2*d*e+3/2*a*b*c*d^2+1/4*b^3*d^2)*x^4+(1/3*a^3*e^2+2*a^2*b*d*e+a^2*c*d^2+a* b^2*d^2)*x^3+(a^3*d*e+3/2*a^2*b*d^2)*x^2+a^3*d^2*x
Time = 0.07 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{6} + a^{3} d^{2} x + \frac {1}{5} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \] Input:
integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/9*c^3*e^2*x^9 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/7*(c^3*d^2 + 6*b*c ^2*d*e + 3*(b^2*c + a*c^2)*e^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*(b^2*c + a*c^2) *d*e + (b^3 + 6*a*b*c)*e^2)*x^6 + a^3*d^2*x + 1/5*(3*(b^2*c + a*c^2)*d^2 + 2*(b^3 + 6*a*b*c)*d*e + 3*(a*b^2 + a^2*c)*e^2)*x^5 + 1/4*(3*a^2*b*e^2 + ( b^3 + 6*a*b*c)*d^2 + 6*(a*b^2 + a^2*c)*d*e)*x^4 + 1/3*(6*a^2*b*d*e + a^3*e ^2 + 3*(a*b^2 + a^2*c)*d^2)*x^3 + 1/2*(3*a^2*b*d^2 + 2*a^3*d*e)*x^2
Time = 0.05 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=a^{3} d^{2} x + \frac {c^{3} e^{2} x^{9}}{9} + x^{8} \cdot \left (\frac {3 b c^{2} e^{2}}{8} + \frac {c^{3} d e}{4}\right ) + x^{7} \cdot \left (\frac {3 a c^{2} e^{2}}{7} + \frac {3 b^{2} c e^{2}}{7} + \frac {6 b c^{2} d e}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{6} \left (a b c e^{2} + a c^{2} d e + \frac {b^{3} e^{2}}{6} + b^{2} c d e + \frac {b c^{2} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c e^{2}}{5} + \frac {3 a b^{2} e^{2}}{5} + \frac {12 a b c d e}{5} + \frac {3 a c^{2} d^{2}}{5} + \frac {2 b^{3} d e}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b e^{2}}{4} + \frac {3 a^{2} c d e}{2} + \frac {3 a b^{2} d e}{2} + \frac {3 a b c d^{2}}{2} + \frac {b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} e^{2}}{3} + 2 a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac {3 a^{2} b d^{2}}{2}\right ) \] Input:
integrate((e*x+d)**2*(c*x**2+b*x+a)**3,x)
Output:
a**3*d**2*x + c**3*e**2*x**9/9 + x**8*(3*b*c**2*e**2/8 + c**3*d*e/4) + x** 7*(3*a*c**2*e**2/7 + 3*b**2*c*e**2/7 + 6*b*c**2*d*e/7 + c**3*d**2/7) + x** 6*(a*b*c*e**2 + a*c**2*d*e + b**3*e**2/6 + b**2*c*d*e + b*c**2*d**2/2) + x **5*(3*a**2*c*e**2/5 + 3*a*b**2*e**2/5 + 12*a*b*c*d*e/5 + 3*a*c**2*d**2/5 + 2*b**3*d*e/5 + 3*b**2*c*d**2/5) + x**4*(3*a**2*b*e**2/4 + 3*a**2*c*d*e/2 + 3*a*b**2*d*e/2 + 3*a*b*c*d**2/2 + b**3*d**2/4) + x**3*(a**3*e**2/3 + 2* a**2*b*d*e + a**2*c*d**2 + a*b**2*d**2) + x**2*(a**3*d*e + 3*a**2*b*d**2/2 )
Time = 0.03 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{6} + a^{3} d^{2} x + \frac {1}{5} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \] Input:
integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
1/9*c^3*e^2*x^9 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/7*(c^3*d^2 + 6*b*c ^2*d*e + 3*(b^2*c + a*c^2)*e^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*(b^2*c + a*c^2) *d*e + (b^3 + 6*a*b*c)*e^2)*x^6 + a^3*d^2*x + 1/5*(3*(b^2*c + a*c^2)*d^2 + 2*(b^3 + 6*a*b*c)*d*e + 3*(a*b^2 + a^2*c)*e^2)*x^5 + 1/4*(3*a^2*b*e^2 + ( b^3 + 6*a*b*c)*d^2 + 6*(a*b^2 + a^2*c)*d*e)*x^4 + 1/3*(6*a^2*b*d*e + a^3*e ^2 + 3*(a*b^2 + a^2*c)*d^2)*x^3 + 1/2*(3*a^2*b*d^2 + 2*a^3*d*e)*x^2
Time = 0.32 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, c^{3} d e x^{8} + \frac {3}{8} \, b c^{2} e^{2} x^{8} + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {6}{7} \, b c^{2} d e x^{7} + \frac {3}{7} \, b^{2} c e^{2} x^{7} + \frac {3}{7} \, a c^{2} e^{2} x^{7} + \frac {1}{2} \, b c^{2} d^{2} x^{6} + b^{2} c d e x^{6} + a c^{2} d e x^{6} + \frac {1}{6} \, b^{3} e^{2} x^{6} + a b c e^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {3}{5} \, a c^{2} d^{2} x^{5} + \frac {2}{5} \, b^{3} d e x^{5} + \frac {12}{5} \, a b c d e x^{5} + \frac {3}{5} \, a b^{2} e^{2} x^{5} + \frac {3}{5} \, a^{2} c e^{2} x^{5} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {3}{2} \, a b^{2} d e x^{4} + \frac {3}{2} \, a^{2} c d e x^{4} + \frac {3}{4} \, a^{2} b e^{2} x^{4} + a b^{2} d^{2} x^{3} + a^{2} c d^{2} x^{3} + 2 \, a^{2} b d e x^{3} + \frac {1}{3} \, a^{3} e^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{2} x^{2} + a^{3} d e x^{2} + a^{3} d^{2} x \] Input:
integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
1/9*c^3*e^2*x^9 + 1/4*c^3*d*e*x^8 + 3/8*b*c^2*e^2*x^8 + 1/7*c^3*d^2*x^7 + 6/7*b*c^2*d*e*x^7 + 3/7*b^2*c*e^2*x^7 + 3/7*a*c^2*e^2*x^7 + 1/2*b*c^2*d^2* x^6 + b^2*c*d*e*x^6 + a*c^2*d*e*x^6 + 1/6*b^3*e^2*x^6 + a*b*c*e^2*x^6 + 3/ 5*b^2*c*d^2*x^5 + 3/5*a*c^2*d^2*x^5 + 2/5*b^3*d*e*x^5 + 12/5*a*b*c*d*e*x^5 + 3/5*a*b^2*e^2*x^5 + 3/5*a^2*c*e^2*x^5 + 1/4*b^3*d^2*x^4 + 3/2*a*b*c*d^2 *x^4 + 3/2*a*b^2*d*e*x^4 + 3/2*a^2*c*d*e*x^4 + 3/4*a^2*b*e^2*x^4 + a*b^2*d ^2*x^3 + a^2*c*d^2*x^3 + 2*a^2*b*d*e*x^3 + 1/3*a^3*e^2*x^3 + 3/2*a^2*b*d^2 *x^2 + a^3*d*e*x^2 + a^3*d^2*x
Time = 5.36 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=x^3\,\left (\frac {a^3\,e^2}{3}+2\,a^2\,b\,d\,e+c\,a^2\,d^2+a\,b^2\,d^2\right )+x^7\,\left (\frac {3\,b^2\,c\,e^2}{7}+\frac {6\,b\,c^2\,d\,e}{7}+\frac {c^3\,d^2}{7}+\frac {3\,a\,c^2\,e^2}{7}\right )+x^4\,\left (\frac {3\,a^2\,b\,e^2}{4}+\frac {3\,c\,a^2\,d\,e}{2}+\frac {3\,a\,b^2\,d\,e}{2}+\frac {3\,c\,a\,b\,d^2}{2}+\frac {b^3\,d^2}{4}\right )+x^6\,\left (\frac {b^3\,e^2}{6}+b^2\,c\,d\,e+\frac {b\,c^2\,d^2}{2}+a\,b\,c\,e^2+a\,c^2\,d\,e\right )+x^5\,\left (\frac {3\,a^2\,c\,e^2}{5}+\frac {3\,a\,b^2\,e^2}{5}+\frac {12\,a\,b\,c\,d\,e}{5}+\frac {3\,a\,c^2\,d^2}{5}+\frac {2\,b^3\,d\,e}{5}+\frac {3\,b^2\,c\,d^2}{5}\right )+a^3\,d^2\,x+\frac {c^3\,e^2\,x^9}{9}+\frac {a^2\,d\,x^2\,\left (2\,a\,e+3\,b\,d\right )}{2}+\frac {c^2\,e\,x^8\,\left (3\,b\,e+2\,c\,d\right )}{8} \] Input:
int((d + e*x)^2*(a + b*x + c*x^2)^3,x)
Output:
x^3*((a^3*e^2)/3 + a*b^2*d^2 + a^2*c*d^2 + 2*a^2*b*d*e) + x^7*((c^3*d^2)/7 + (3*a*c^2*e^2)/7 + (3*b^2*c*e^2)/7 + (6*b*c^2*d*e)/7) + x^4*((b^3*d^2)/4 + (3*a^2*b*e^2)/4 + (3*a*b*c*d^2)/2 + (3*a*b^2*d*e)/2 + (3*a^2*c*d*e)/2) + x^6*((b^3*e^2)/6 + (b*c^2*d^2)/2 + a*b*c*e^2 + a*c^2*d*e + b^2*c*d*e) + x^5*((3*a*b^2*e^2)/5 + (3*a*c^2*d^2)/5 + (3*a^2*c*e^2)/5 + (3*b^2*c*d^2)/5 + (2*b^3*d*e)/5 + (12*a*b*c*d*e)/5) + a^3*d^2*x + (c^3*e^2*x^9)/9 + (a^2* d*x^2*(2*a*e + 3*b*d))/2 + (c^2*e*x^8*(3*b*e + 2*c*d))/8
Time = 0.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.23 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx=\frac {x \left (280 c^{3} e^{2} x^{8}+945 b \,c^{2} e^{2} x^{7}+630 c^{3} d e \,x^{7}+1080 a \,c^{2} e^{2} x^{6}+1080 b^{2} c \,e^{2} x^{6}+2160 b \,c^{2} d e \,x^{6}+360 c^{3} d^{2} x^{6}+2520 a b c \,e^{2} x^{5}+2520 a \,c^{2} d e \,x^{5}+420 b^{3} e^{2} x^{5}+2520 b^{2} c d e \,x^{5}+1260 b \,c^{2} d^{2} x^{5}+1512 a^{2} c \,e^{2} x^{4}+1512 a \,b^{2} e^{2} x^{4}+6048 a b c d e \,x^{4}+1512 a \,c^{2} d^{2} x^{4}+1008 b^{3} d e \,x^{4}+1512 b^{2} c \,d^{2} x^{4}+1890 a^{2} b \,e^{2} x^{3}+3780 a^{2} c d e \,x^{3}+3780 a \,b^{2} d e \,x^{3}+3780 a b c \,d^{2} x^{3}+630 b^{3} d^{2} x^{3}+840 a^{3} e^{2} x^{2}+5040 a^{2} b d e \,x^{2}+2520 a^{2} c \,d^{2} x^{2}+2520 a \,b^{2} d^{2} x^{2}+2520 a^{3} d e x +3780 a^{2} b \,d^{2} x +2520 a^{3} d^{2}\right )}{2520} \] Input:
int((e*x+d)^2*(c*x^2+b*x+a)^3,x)
Output:
(x*(2520*a**3*d**2 + 2520*a**3*d*e*x + 840*a**3*e**2*x**2 + 3780*a**2*b*d* *2*x + 5040*a**2*b*d*e*x**2 + 1890*a**2*b*e**2*x**3 + 2520*a**2*c*d**2*x** 2 + 3780*a**2*c*d*e*x**3 + 1512*a**2*c*e**2*x**4 + 2520*a*b**2*d**2*x**2 + 3780*a*b**2*d*e*x**3 + 1512*a*b**2*e**2*x**4 + 3780*a*b*c*d**2*x**3 + 604 8*a*b*c*d*e*x**4 + 2520*a*b*c*e**2*x**5 + 1512*a*c**2*d**2*x**4 + 2520*a*c **2*d*e*x**5 + 1080*a*c**2*e**2*x**6 + 630*b**3*d**2*x**3 + 1008*b**3*d*e* x**4 + 420*b**3*e**2*x**5 + 1512*b**2*c*d**2*x**4 + 2520*b**2*c*d*e*x**5 + 1080*b**2*c*e**2*x**6 + 1260*b*c**2*d**2*x**5 + 2160*b*c**2*d*e*x**6 + 94 5*b*c**2*e**2*x**7 + 360*c**3*d**2*x**6 + 630*c**3*d*e*x**7 + 280*c**3*e** 2*x**8))/2520