Integrand size = 38, antiderivative size = 201 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d^2 x+\frac {3}{2} d (11 d+12 e) x^2+\frac {1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac {1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac {1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5-\frac {1}{6} \left (37 d^2-296 d e-65 e^2\right ) x^6+\frac {37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7-\frac {1}{8} \left (45 d^2-222 d e+37 e^2\right ) x^8+\frac {1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac {1}{2} (40 d-9 e) e x^{10}+\frac {100 e^2 x^{11}}{11} \]
18*d^2*x+3/2*d*(11*d+12*e)*x^2+1/3*(107*d^2+66*d*e+18*e^2)*x^3+1/4*(65*d^2 +214*d*e+33*e^2)*x^4+1/5*(148*d^2+130*d*e+107*e^2)*x^5-1/6*(37*d^2-296*d*e -65*e^2)*x^6+37/7*(3*d^2-2*d*e+4*e^2)*x^7-1/8*(45*d^2-222*d*e+37*e^2)*x^8+ 1/9*(100*d^2-90*d*e+111*e^2)*x^9+1/2*(40*d-9*e)*e*x^10+100/11*e^2*x^11
Time = 0.02 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d^2 x+\frac {3}{2} d (11 d+12 e) x^2+\frac {1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac {1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac {1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5+\frac {1}{6} \left (-37 d^2+296 d e+65 e^2\right ) x^6+\frac {37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7+\frac {1}{8} \left (-45 d^2+222 d e-37 e^2\right ) x^8+\frac {1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac {1}{2} (40 d-9 e) e x^{10}+\frac {100 e^2 x^{11}}{11} \]
18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^ 5)/5 + ((-37*d^2 + 296*d*e + 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2)* x^7)/7 + ((-45*d^2 + 222*d*e - 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e ^2)*x^9)/9 + ((40*d - 9*e)*e*x^10)/2 + (100*e^2*x^11)/11
Time = 0.49 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2159, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (5 x^2+2 x+3\right )^2 \left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (x^8 \left (100 d^2-90 d e+111 e^2\right )-x^7 \left (45 d^2-222 d e+37 e^2\right )+37 x^6 \left (3 d^2-2 d e+4 e^2\right )-x^5 \left (37 d^2-296 d e-65 e^2\right )+x^4 \left (148 d^2+130 d e+107 e^2\right )+x^3 \left (65 d^2+214 d e+33 e^2\right )+x^2 \left (107 d^2+66 d e+18 e^2\right )+18 d^2+5 e x^9 (40 d-9 e)+3 d x (11 d+12 e)+100 e^2 x^{10}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} x^9 \left (100 d^2-90 d e+111 e^2\right )-\frac {1}{8} x^8 \left (45 d^2-222 d e+37 e^2\right )+\frac {37}{7} x^7 \left (3 d^2-2 d e+4 e^2\right )-\frac {1}{6} x^6 \left (37 d^2-296 d e-65 e^2\right )+\frac {1}{5} x^5 \left (148 d^2+130 d e+107 e^2\right )+\frac {1}{4} x^4 \left (65 d^2+214 d e+33 e^2\right )+\frac {1}{3} x^3 \left (107 d^2+66 d e+18 e^2\right )+18 d^2 x+\frac {1}{2} e x^{10} (40 d-9 e)+\frac {3}{2} d x^2 (11 d+12 e)+\frac {100 e^2 x^{11}}{11}\) |
18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^ 5)/5 - ((37*d^2 - 296*d*e - 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2)*x ^7)/7 - ((45*d^2 - 222*d*e + 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e^2 )*x^9)/9 + ((40*d - 9*e)*e*x^10)/2 + (100*e^2*x^11)/11
3.3.97.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.56 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.88
| method | result | size |
| norman | \(\frac {100 e^{2} x^{11}}{11}+\left (20 d e -\frac {9}{2} e^{2}\right ) x^{10}+\left (\frac {100}{9} d^{2}-10 d e +\frac {37}{3} e^{2}\right ) x^{9}+\left (-\frac {45}{8} d^{2}+\frac {111}{4} d e -\frac {37}{8} e^{2}\right ) x^{8}+\left (\frac {111}{7} d^{2}-\frac {74}{7} d e +\frac {148}{7} e^{2}\right ) x^{7}+\left (-\frac {37}{6} d^{2}+\frac {148}{3} d e +\frac {65}{6} e^{2}\right ) x^{6}+\left (\frac {148}{5} d^{2}+26 d e +\frac {107}{5} e^{2}\right ) x^{5}+\left (\frac {65}{4} d^{2}+\frac {107}{2} d e +\frac {33}{4} e^{2}\right ) x^{4}+\left (\frac {107}{3} d^{2}+22 d e +6 e^{2}\right ) x^{3}+\left (\frac {33}{2} d^{2}+18 d e \right ) x^{2}+18 x \,d^{2}\) | \(177\) |
| default | \(\frac {100 e^{2} x^{11}}{11}+\frac {\left (200 d e -45 e^{2}\right ) x^{10}}{10}+\frac {\left (100 d^{2}-90 d e +111 e^{2}\right ) x^{9}}{9}+\frac {\left (-45 d^{2}+222 d e -37 e^{2}\right ) x^{8}}{8}+\frac {\left (111 d^{2}-74 d e +148 e^{2}\right ) x^{7}}{7}+\frac {\left (-37 d^{2}+296 d e +65 e^{2}\right ) x^{6}}{6}+\frac {\left (148 d^{2}+130 d e +107 e^{2}\right ) x^{5}}{5}+\frac {\left (65 d^{2}+214 d e +33 e^{2}\right ) x^{4}}{4}+\frac {\left (107 d^{2}+66 d e +18 e^{2}\right ) x^{3}}{3}+\frac {\left (33 d^{2}+36 d e \right ) x^{2}}{2}+18 x \,d^{2}\) | \(186\) |
| gosper | \(\frac {111}{4} x^{8} d e -\frac {74}{7} x^{7} d e +\frac {148}{3} x^{6} d e +\frac {107}{2} x^{4} d e +22 x^{3} d e +26 x^{5} d e -10 x^{9} d e +20 x^{10} d e -\frac {45}{8} x^{8} d^{2}+\frac {100}{9} x^{9} d^{2}-\frac {9}{2} x^{10} e^{2}+\frac {33}{2} d^{2} x^{2}+\frac {148}{5} x^{5} d^{2}+\frac {65}{4} x^{4} d^{2}+\frac {33}{4} x^{4} e^{2}+\frac {107}{3} x^{3} d^{2}-\frac {37}{6} x^{6} d^{2}+\frac {65}{6} x^{6} e^{2}-\frac {37}{8} x^{8} e^{2}+\frac {111}{7} x^{7} d^{2}+\frac {148}{7} x^{7} e^{2}+18 x \,d^{2}+6 x^{3} e^{2}+18 d e \,x^{2}+\frac {37}{3} e^{2} x^{9}+\frac {100}{11} e^{2} x^{11}+\frac {107}{5} e^{2} x^{5}\) | \(207\) |
| risch | \(\frac {111}{4} x^{8} d e -\frac {74}{7} x^{7} d e +\frac {148}{3} x^{6} d e +\frac {107}{2} x^{4} d e +22 x^{3} d e +26 x^{5} d e -10 x^{9} d e +20 x^{10} d e -\frac {45}{8} x^{8} d^{2}+\frac {100}{9} x^{9} d^{2}-\frac {9}{2} x^{10} e^{2}+\frac {33}{2} d^{2} x^{2}+\frac {148}{5} x^{5} d^{2}+\frac {65}{4} x^{4} d^{2}+\frac {33}{4} x^{4} e^{2}+\frac {107}{3} x^{3} d^{2}-\frac {37}{6} x^{6} d^{2}+\frac {65}{6} x^{6} e^{2}-\frac {37}{8} x^{8} e^{2}+\frac {111}{7} x^{7} d^{2}+\frac {148}{7} x^{7} e^{2}+18 x \,d^{2}+6 x^{3} e^{2}+18 d e \,x^{2}+\frac {37}{3} e^{2} x^{9}+\frac {100}{11} e^{2} x^{11}+\frac {107}{5} e^{2} x^{5}\) | \(207\) |
| parallelrisch | \(\frac {111}{4} x^{8} d e -\frac {74}{7} x^{7} d e +\frac {148}{3} x^{6} d e +\frac {107}{2} x^{4} d e +22 x^{3} d e +26 x^{5} d e -10 x^{9} d e +20 x^{10} d e -\frac {45}{8} x^{8} d^{2}+\frac {100}{9} x^{9} d^{2}-\frac {9}{2} x^{10} e^{2}+\frac {33}{2} d^{2} x^{2}+\frac {148}{5} x^{5} d^{2}+\frac {65}{4} x^{4} d^{2}+\frac {33}{4} x^{4} e^{2}+\frac {107}{3} x^{3} d^{2}-\frac {37}{6} x^{6} d^{2}+\frac {65}{6} x^{6} e^{2}-\frac {37}{8} x^{8} e^{2}+\frac {111}{7} x^{7} d^{2}+\frac {148}{7} x^{7} e^{2}+18 x \,d^{2}+6 x^{3} e^{2}+18 d e \,x^{2}+\frac {37}{3} e^{2} x^{9}+\frac {100}{11} e^{2} x^{11}+\frac {107}{5} e^{2} x^{5}\) | \(207\) |
100/11*e^2*x^11+(20*d*e-9/2*e^2)*x^10+(100/9*d^2-10*d*e+37/3*e^2)*x^9+(-45 /8*d^2+111/4*d*e-37/8*e^2)*x^8+(111/7*d^2-74/7*d*e+148/7*e^2)*x^7+(-37/6*d ^2+148/3*d*e+65/6*e^2)*x^6+(148/5*d^2+26*d*e+107/5*e^2)*x^5+(65/4*d^2+107/ 2*d*e+33/4*e^2)*x^4+(107/3*d^2+22*d*e+6*e^2)*x^3+(33/2*d^2+18*d*e)*x^2+18* x*d^2
Time = 0.25 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.92 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {100}{11} \, e^{2} x^{11} + \frac {1}{2} \, {\left (40 \, d e - 9 \, e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (100 \, d^{2} - 90 \, d e + 111 \, e^{2}\right )} x^{9} - \frac {1}{8} \, {\left (45 \, d^{2} - 222 \, d e + 37 \, e^{2}\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d^{2} - 2 \, d e + 4 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (37 \, d^{2} - 296 \, d e - 65 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d^{2} + 130 \, d e + 107 \, e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d^{2} + 214 \, d e + 33 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d^{2} + 66 \, d e + 18 \, e^{2}\right )} x^{3} + 18 \, d^{2} x + \frac {3}{2} \, {\left (11 \, d^{2} + 12 \, d e\right )} x^{2} \]
100/11*e^2*x^11 + 1/2*(40*d*e - 9*e^2)*x^10 + 1/9*(100*d^2 - 90*d*e + 111* e^2)*x^9 - 1/8*(45*d^2 - 222*d*e + 37*e^2)*x^8 + 37/7*(3*d^2 - 2*d*e + 4*e ^2)*x^7 - 1/6*(37*d^2 - 296*d*e - 65*e^2)*x^6 + 1/5*(148*d^2 + 130*d*e + 1 07*e^2)*x^5 + 1/4*(65*d^2 + 214*d*e + 33*e^2)*x^4 + 1/3*(107*d^2 + 66*d*e + 18*e^2)*x^3 + 18*d^2*x + 3/2*(11*d^2 + 12*d*e)*x^2
Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d^{2} x + \frac {100 e^{2} x^{11}}{11} + x^{10} \cdot \left (20 d e - \frac {9 e^{2}}{2}\right ) + x^{9} \cdot \left (\frac {100 d^{2}}{9} - 10 d e + \frac {37 e^{2}}{3}\right ) + x^{8} \left (- \frac {45 d^{2}}{8} + \frac {111 d e}{4} - \frac {37 e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {111 d^{2}}{7} - \frac {74 d e}{7} + \frac {148 e^{2}}{7}\right ) + x^{6} \left (- \frac {37 d^{2}}{6} + \frac {148 d e}{3} + \frac {65 e^{2}}{6}\right ) + x^{5} \cdot \left (\frac {148 d^{2}}{5} + 26 d e + \frac {107 e^{2}}{5}\right ) + x^{4} \cdot \left (\frac {65 d^{2}}{4} + \frac {107 d e}{2} + \frac {33 e^{2}}{4}\right ) + x^{3} \cdot \left (\frac {107 d^{2}}{3} + 22 d e + 6 e^{2}\right ) + x^{2} \cdot \left (\frac {33 d^{2}}{2} + 18 d e\right ) \]
18*d**2*x + 100*e**2*x**11/11 + x**10*(20*d*e - 9*e**2/2) + x**9*(100*d**2 /9 - 10*d*e + 37*e**2/3) + x**8*(-45*d**2/8 + 111*d*e/4 - 37*e**2/8) + x** 7*(111*d**2/7 - 74*d*e/7 + 148*e**2/7) + x**6*(-37*d**2/6 + 148*d*e/3 + 65 *e**2/6) + x**5*(148*d**2/5 + 26*d*e + 107*e**2/5) + x**4*(65*d**2/4 + 107 *d*e/2 + 33*e**2/4) + x**3*(107*d**2/3 + 22*d*e + 6*e**2) + x**2*(33*d**2/ 2 + 18*d*e)
Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.92 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {100}{11} \, e^{2} x^{11} + \frac {1}{2} \, {\left (40 \, d e - 9 \, e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (100 \, d^{2} - 90 \, d e + 111 \, e^{2}\right )} x^{9} - \frac {1}{8} \, {\left (45 \, d^{2} - 222 \, d e + 37 \, e^{2}\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d^{2} - 2 \, d e + 4 \, e^{2}\right )} x^{7} - \frac {1}{6} \, {\left (37 \, d^{2} - 296 \, d e - 65 \, e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d^{2} + 130 \, d e + 107 \, e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d^{2} + 214 \, d e + 33 \, e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d^{2} + 66 \, d e + 18 \, e^{2}\right )} x^{3} + 18 \, d^{2} x + \frac {3}{2} \, {\left (11 \, d^{2} + 12 \, d e\right )} x^{2} \]
100/11*e^2*x^11 + 1/2*(40*d*e - 9*e^2)*x^10 + 1/9*(100*d^2 - 90*d*e + 111* e^2)*x^9 - 1/8*(45*d^2 - 222*d*e + 37*e^2)*x^8 + 37/7*(3*d^2 - 2*d*e + 4*e ^2)*x^7 - 1/6*(37*d^2 - 296*d*e - 65*e^2)*x^6 + 1/5*(148*d^2 + 130*d*e + 1 07*e^2)*x^5 + 1/4*(65*d^2 + 214*d*e + 33*e^2)*x^4 + 1/3*(107*d^2 + 66*d*e + 18*e^2)*x^3 + 18*d^2*x + 3/2*(11*d^2 + 12*d*e)*x^2
Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {100}{11} \, e^{2} x^{11} + 20 \, d e x^{10} - \frac {9}{2} \, e^{2} x^{10} + \frac {100}{9} \, d^{2} x^{9} - 10 \, d e x^{9} + \frac {37}{3} \, e^{2} x^{9} - \frac {45}{8} \, d^{2} x^{8} + \frac {111}{4} \, d e x^{8} - \frac {37}{8} \, e^{2} x^{8} + \frac {111}{7} \, d^{2} x^{7} - \frac {74}{7} \, d e x^{7} + \frac {148}{7} \, e^{2} x^{7} - \frac {37}{6} \, d^{2} x^{6} + \frac {148}{3} \, d e x^{6} + \frac {65}{6} \, e^{2} x^{6} + \frac {148}{5} \, d^{2} x^{5} + 26 \, d e x^{5} + \frac {107}{5} \, e^{2} x^{5} + \frac {65}{4} \, d^{2} x^{4} + \frac {107}{2} \, d e x^{4} + \frac {33}{4} \, e^{2} x^{4} + \frac {107}{3} \, d^{2} x^{3} + 22 \, d e x^{3} + 6 \, e^{2} x^{3} + \frac {33}{2} \, d^{2} x^{2} + 18 \, d e x^{2} + 18 \, d^{2} x \]
100/11*e^2*x^11 + 20*d*e*x^10 - 9/2*e^2*x^10 + 100/9*d^2*x^9 - 10*d*e*x^9 + 37/3*e^2*x^9 - 45/8*d^2*x^8 + 111/4*d*e*x^8 - 37/8*e^2*x^8 + 111/7*d^2*x ^7 - 74/7*d*e*x^7 + 148/7*e^2*x^7 - 37/6*d^2*x^6 + 148/3*d*e*x^6 + 65/6*e^ 2*x^6 + 148/5*d^2*x^5 + 26*d*e*x^5 + 107/5*e^2*x^5 + 65/4*d^2*x^4 + 107/2* d*e*x^4 + 33/4*e^2*x^4 + 107/3*d^2*x^3 + 22*d*e*x^3 + 6*e^2*x^3 + 33/2*d^2 *x^2 + 18*d*e*x^2 + 18*d^2*x
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.87 \[ \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=x^3\,\left (\frac {107\,d^2}{3}+22\,d\,e+6\,e^2\right )+x^9\,\left (\frac {100\,d^2}{9}-10\,d\,e+\frac {37\,e^2}{3}\right )+x^4\,\left (\frac {65\,d^2}{4}+\frac {107\,d\,e}{2}+\frac {33\,e^2}{4}\right )-x^8\,\left (\frac {45\,d^2}{8}-\frac {111\,d\,e}{4}+\frac {37\,e^2}{8}\right )+x^6\,\left (-\frac {37\,d^2}{6}+\frac {148\,d\,e}{3}+\frac {65\,e^2}{6}\right )+x^5\,\left (\frac {148\,d^2}{5}+26\,d\,e+\frac {107\,e^2}{5}\right )+x^7\,\left (\frac {111\,d^2}{7}-\frac {74\,d\,e}{7}+\frac {148\,e^2}{7}\right )+18\,d^2\,x+\frac {100\,e^2\,x^{11}}{11}+\frac {3\,d\,x^2\,\left (11\,d+12\,e\right )}{2}+\frac {e\,x^{10}\,\left (40\,d-9\,e\right )}{2} \]
x^3*(22*d*e + (107*d^2)/3 + 6*e^2) + x^9*((100*d^2)/9 - 10*d*e + (37*e^2)/ 3) + x^4*((107*d*e)/2 + (65*d^2)/4 + (33*e^2)/4) - x^8*((45*d^2)/8 - (111* d*e)/4 + (37*e^2)/8) + x^6*((148*d*e)/3 - (37*d^2)/6 + (65*e^2)/6) + x^5*( 26*d*e + (148*d^2)/5 + (107*e^2)/5) + x^7*((111*d^2)/7 - (74*d*e)/7 + (148 *e^2)/7) + 18*d^2*x + (100*e^2*x^11)/11 + (3*d*x^2*(11*d + 12*e))/2 + (e*x ^10*(40*d - 9*e))/2