Integrand size = 31, antiderivative size = 143 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {(b d-a e)^5 (d+e x)^7}{7 e^6}+\frac {5 b (b d-a e)^4 (d+e x)^8}{8 e^6}-\frac {10 b^2 (b d-a e)^3 (d+e x)^9}{9 e^6}+\frac {b^3 (b d-a e)^2 (d+e x)^{10}}{e^6}-\frac {5 b^4 (b d-a e) (d+e x)^{11}}{11 e^6}+\frac {b^5 (d+e x)^{12}}{12 e^6} \] Output:
-1/7*(-a*e+b*d)^5*(e*x+d)^7/e^6+5/8*b*(-a*e+b*d)^4*(e*x+d)^8/e^6-10/9*b^2* (-a*e+b*d)^3*(e*x+d)^9/e^6+b^3*(-a*e+b*d)^2*(e*x+d)^10/e^6-5/11*b^4*(-a*e+ b*d)*(e*x+d)^11/e^6+1/12*b^5*(e*x+d)^12/e^6
Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(143)=286\).
Time = 0.09 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.50 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^5 d^6 x+\frac {1}{2} a^4 d^5 (5 b d+6 a e) x^2+\frac {5}{3} a^3 d^4 \left (2 b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^3+\frac {5}{4} a^2 d^3 \left (2 b^3 d^3+12 a b^2 d^2 e+15 a^2 b d e^2+4 a^3 e^3\right ) x^4+a d^2 \left (b^4 d^4+12 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4\right ) x^5+\frac {1}{6} d \left (b^5 d^5+30 a b^4 d^4 e+150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5\right ) x^6+\frac {1}{7} e \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right ) x^7+\frac {5}{8} b e^2 \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right ) x^8+\frac {5}{9} b^2 e^3 \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right ) x^9+\frac {1}{2} b^3 e^4 \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right ) x^{10}+\frac {1}{11} b^4 e^5 (6 b d+5 a e) x^{11}+\frac {1}{12} b^5 e^6 x^{12} \] Input:
Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
a^5*d^6*x + (a^4*d^5*(5*b*d + 6*a*e)*x^2)/2 + (5*a^3*d^4*(2*b^2*d^2 + 6*a* b*d*e + 3*a^2*e^2)*x^3)/3 + (5*a^2*d^3*(2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^ 2*b*d*e^2 + 4*a^3*e^3)*x^4)/4 + a*d^2*(b^4*d^4 + 12*a*b^3*d^3*e + 30*a^2*b ^2*d^2*e^2 + 20*a^3*b*d*e^3 + 3*a^4*e^4)*x^5 + (d*(b^5*d^5 + 30*a*b^4*d^4* e + 150*a^2*b^3*d^3*e^2 + 200*a^3*b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5 )*x^6)/6 + (e*(6*b^5*d^5 + 75*a*b^4*d^4*e + 200*a^2*b^3*d^3*e^2 + 150*a^3* b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5)*x^7)/7 + (5*b*e^2*(3*b^4*d^4 + 20* a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4)*x^8)/8 + (5*b ^2*e^3*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 + 2*a^3*e^3)*x^9)/9 + (b^3*e^4*(3*b^2*d^2 + 6*a*b*d*e + 2*a^2*e^2)*x^10)/2 + (b^4*e^5*(6*b*d + 5 *a*e)*x^11)/11 + (b^5*e^6*x^12)/12
Time = 0.83 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 49, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^6 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 (d+e x)^6dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 (d+e x)^6dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{10} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^9 (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^8 (b d-a e)^3}{e^5}+\frac {5 b (d+e x)^7 (b d-a e)^4}{e^5}+\frac {(d+e x)^6 (a e-b d)^5}{e^5}+\frac {b^5 (d+e x)^{11}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 b^4 (d+e x)^{11} (b d-a e)}{11 e^6}+\frac {b^3 (d+e x)^{10} (b d-a e)^2}{e^6}-\frac {10 b^2 (d+e x)^9 (b d-a e)^3}{9 e^6}+\frac {5 b (d+e x)^8 (b d-a e)^4}{8 e^6}-\frac {(d+e x)^7 (b d-a e)^5}{7 e^6}+\frac {b^5 (d+e x)^{12}}{12 e^6}\) |
Input:
Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
-1/7*((b*d - a*e)^5*(d + e*x)^7)/e^6 + (5*b*(b*d - a*e)^4*(d + e*x)^8)/(8* e^6) - (10*b^2*(b*d - a*e)^3*(d + e*x)^9)/(9*e^6) + (b^3*(b*d - a*e)^2*(d + e*x)^10)/e^6 - (5*b^4*(b*d - a*e)*(d + e*x)^11)/(11*e^6) + (b^5*(d + e*x )^12)/(12*e^6)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(133)=266\).
Time = 1.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.57
| method | result | size |
| norman | \(\frac {e^{6} b^{5} x^{12}}{12}+\left (\frac {5}{11} e^{6} a \,b^{4}+\frac {6}{11} d \,e^{5} b^{5}\right ) x^{11}+\left (e^{6} a^{2} b^{3}+3 d \,e^{5} a \,b^{4}+\frac {3}{2} d^{2} e^{4} b^{5}\right ) x^{10}+\left (\frac {10}{9} a^{3} b^{2} e^{6}+\frac {20}{3} a^{2} d \,e^{5} b^{3}+\frac {25}{3} a \,b^{4} d^{2} e^{4}+\frac {20}{9} d^{3} e^{3} b^{5}\right ) x^{9}+\left (\frac {5}{8} e^{6} a^{4} b +\frac {15}{2} d \,e^{5} a^{3} b^{2}+\frac {75}{4} d^{2} e^{4} a^{2} b^{3}+\frac {25}{2} e^{3} d^{3} a \,b^{4}+\frac {15}{8} d^{4} e^{2} b^{5}\right ) x^{8}+\left (\frac {1}{7} e^{6} a^{5}+\frac {30}{7} d \,e^{5} a^{4} b +\frac {150}{7} d^{2} e^{4} a^{3} b^{2}+\frac {200}{7} e^{3} d^{3} a^{2} b^{3}+\frac {75}{7} d^{4} e^{2} a \,b^{4}+\frac {6}{7} d^{5} e \,b^{5}\right ) x^{7}+\left (d \,e^{5} a^{5}+\frac {25}{2} d^{2} e^{4} a^{4} b +\frac {100}{3} e^{3} d^{3} a^{3} b^{2}+25 d^{4} e^{2} a^{2} b^{3}+5 d^{5} e a \,b^{4}+\frac {1}{6} b^{5} d^{6}\right ) x^{6}+\left (3 d^{2} e^{4} a^{5}+20 e^{3} d^{3} a^{4} b +30 d^{4} e^{2} a^{3} b^{2}+12 d^{5} e \,a^{2} b^{3}+d^{6} a \,b^{4}\right ) x^{5}+\left (5 e^{3} d^{3} a^{5}+\frac {75}{4} d^{4} e^{2} a^{4} b +15 d^{5} e \,a^{3} b^{2}+\frac {5}{2} d^{6} a^{2} b^{3}\right ) x^{4}+\left (5 d^{4} e^{2} a^{5}+10 d^{5} e \,a^{4} b +\frac {10}{3} d^{6} a^{3} b^{2}\right ) x^{3}+\left (3 d^{5} e \,a^{5}+\frac {5}{2} d^{6} a^{4} b \right ) x^{2}+d^{6} a^{5} x\) | \(510\) |
| risch | \(\frac {75}{7} x^{7} d^{4} e^{2} a \,b^{4}+\frac {25}{2} x^{6} d^{2} e^{4} a^{4} b +\frac {100}{3} x^{6} e^{3} d^{3} a^{3} b^{2}+25 x^{6} d^{4} e^{2} a^{2} b^{3}+5 x^{6} d^{5} e a \,b^{4}+\frac {75}{4} x^{4} d^{4} e^{2} a^{4} b +15 x^{4} d^{5} e \,a^{3} b^{2}+10 x^{3} d^{5} e \,a^{4} b +20 a^{4} b \,d^{3} e^{3} x^{5}+30 a^{3} b^{2} d^{4} e^{2} x^{5}+12 a^{2} b^{3} d^{5} e \,x^{5}+3 x^{10} d \,e^{5} a \,b^{4}+\frac {20}{3} x^{9} a^{2} d \,e^{5} b^{3}+\frac {25}{3} x^{9} a \,b^{4} d^{2} e^{4}+\frac {15}{2} x^{8} d \,e^{5} a^{3} b^{2}+\frac {75}{4} x^{8} d^{2} e^{4} a^{2} b^{3}+\frac {25}{2} x^{8} e^{3} d^{3} a \,b^{4}+\frac {30}{7} x^{7} d \,e^{5} a^{4} b +\frac {150}{7} x^{7} d^{2} e^{4} a^{3} b^{2}+\frac {200}{7} x^{7} e^{3} d^{3} a^{2} b^{3}+5 x^{3} d^{4} e^{2} a^{5}+\frac {10}{3} x^{3} d^{6} a^{3} b^{2}+3 x^{2} d^{5} e \,a^{5}+\frac {5}{2} x^{2} d^{6} a^{4} b +3 a^{5} d^{2} e^{4} x^{5}+a \,b^{4} d^{6} x^{5}+\frac {5}{11} x^{11} e^{6} a \,b^{4}+\frac {6}{11} x^{11} d \,e^{5} b^{5}+x^{10} e^{6} a^{2} b^{3}+\frac {3}{2} x^{10} d^{2} e^{4} b^{5}+\frac {10}{9} x^{9} a^{3} b^{2} e^{6}+\frac {20}{9} x^{9} d^{3} e^{3} b^{5}+\frac {5}{8} x^{8} e^{6} a^{4} b +\frac {15}{8} x^{8} d^{4} e^{2} b^{5}+\frac {6}{7} x^{7} d^{5} e \,b^{5}+x^{6} d \,e^{5} a^{5}+5 x^{4} e^{3} d^{3} a^{5}+\frac {5}{2} x^{4} d^{6} a^{2} b^{3}+d^{6} a^{5} x +\frac {1}{12} e^{6} b^{5} x^{12}+\frac {1}{6} x^{6} b^{5} d^{6}+\frac {1}{7} x^{7} e^{6} a^{5}\) | \(580\) |
| parallelrisch | \(\frac {75}{7} x^{7} d^{4} e^{2} a \,b^{4}+\frac {25}{2} x^{6} d^{2} e^{4} a^{4} b +\frac {100}{3} x^{6} e^{3} d^{3} a^{3} b^{2}+25 x^{6} d^{4} e^{2} a^{2} b^{3}+5 x^{6} d^{5} e a \,b^{4}+\frac {75}{4} x^{4} d^{4} e^{2} a^{4} b +15 x^{4} d^{5} e \,a^{3} b^{2}+10 x^{3} d^{5} e \,a^{4} b +20 a^{4} b \,d^{3} e^{3} x^{5}+30 a^{3} b^{2} d^{4} e^{2} x^{5}+12 a^{2} b^{3} d^{5} e \,x^{5}+3 x^{10} d \,e^{5} a \,b^{4}+\frac {20}{3} x^{9} a^{2} d \,e^{5} b^{3}+\frac {25}{3} x^{9} a \,b^{4} d^{2} e^{4}+\frac {15}{2} x^{8} d \,e^{5} a^{3} b^{2}+\frac {75}{4} x^{8} d^{2} e^{4} a^{2} b^{3}+\frac {25}{2} x^{8} e^{3} d^{3} a \,b^{4}+\frac {30}{7} x^{7} d \,e^{5} a^{4} b +\frac {150}{7} x^{7} d^{2} e^{4} a^{3} b^{2}+\frac {200}{7} x^{7} e^{3} d^{3} a^{2} b^{3}+5 x^{3} d^{4} e^{2} a^{5}+\frac {10}{3} x^{3} d^{6} a^{3} b^{2}+3 x^{2} d^{5} e \,a^{5}+\frac {5}{2} x^{2} d^{6} a^{4} b +3 a^{5} d^{2} e^{4} x^{5}+a \,b^{4} d^{6} x^{5}+\frac {5}{11} x^{11} e^{6} a \,b^{4}+\frac {6}{11} x^{11} d \,e^{5} b^{5}+x^{10} e^{6} a^{2} b^{3}+\frac {3}{2} x^{10} d^{2} e^{4} b^{5}+\frac {10}{9} x^{9} a^{3} b^{2} e^{6}+\frac {20}{9} x^{9} d^{3} e^{3} b^{5}+\frac {5}{8} x^{8} e^{6} a^{4} b +\frac {15}{8} x^{8} d^{4} e^{2} b^{5}+\frac {6}{7} x^{7} d^{5} e \,b^{5}+x^{6} d \,e^{5} a^{5}+5 x^{4} e^{3} d^{3} a^{5}+\frac {5}{2} x^{4} d^{6} a^{2} b^{3}+d^{6} a^{5} x +\frac {1}{12} e^{6} b^{5} x^{12}+\frac {1}{6} x^{6} b^{5} d^{6}+\frac {1}{7} x^{7} e^{6} a^{5}\) | \(580\) |
| gosper | \(\frac {x \left (462 e^{6} b^{5} x^{11}+2520 x^{10} e^{6} a \,b^{4}+3024 x^{10} d \,e^{5} b^{5}+5544 x^{9} e^{6} a^{2} b^{3}+16632 x^{9} d \,e^{5} a \,b^{4}+8316 x^{9} d^{2} e^{4} b^{5}+6160 x^{8} a^{3} b^{2} e^{6}+36960 x^{8} a^{2} d \,e^{5} b^{3}+46200 x^{8} a \,b^{4} d^{2} e^{4}+12320 x^{8} d^{3} e^{3} b^{5}+3465 x^{7} e^{6} a^{4} b +41580 x^{7} d \,e^{5} a^{3} b^{2}+103950 x^{7} d^{2} e^{4} a^{2} b^{3}+69300 x^{7} e^{3} d^{3} a \,b^{4}+10395 x^{7} d^{4} e^{2} b^{5}+792 x^{6} e^{6} a^{5}+23760 x^{6} d \,e^{5} a^{4} b +118800 x^{6} d^{2} e^{4} a^{3} b^{2}+158400 x^{6} e^{3} d^{3} a^{2} b^{3}+59400 x^{6} d^{4} e^{2} a \,b^{4}+4752 x^{6} d^{5} e \,b^{5}+5544 x^{5} d \,e^{5} a^{5}+69300 x^{5} d^{2} e^{4} a^{4} b +184800 x^{5} e^{3} d^{3} a^{3} b^{2}+138600 x^{5} d^{4} e^{2} a^{2} b^{3}+27720 x^{5} d^{5} e a \,b^{4}+924 x^{5} b^{5} d^{6}+16632 a^{5} d^{2} e^{4} x^{4}+110880 a^{4} b \,d^{3} e^{3} x^{4}+166320 a^{3} b^{2} d^{4} e^{2} x^{4}+66528 a^{2} b^{3} d^{5} e \,x^{4}+5544 a \,b^{4} d^{6} x^{4}+27720 x^{3} e^{3} d^{3} a^{5}+103950 x^{3} d^{4} e^{2} a^{4} b +83160 x^{3} d^{5} e \,a^{3} b^{2}+13860 x^{3} d^{6} a^{2} b^{3}+27720 x^{2} d^{4} e^{2} a^{5}+55440 x^{2} d^{5} e \,a^{4} b +18480 x^{2} d^{6} a^{3} b^{2}+16632 x \,d^{5} e \,a^{5}+13860 x \,d^{6} a^{4} b +5544 d^{6} a^{5}\right )}{5544}\) | \(582\) |
| orering | \(\frac {x \left (462 e^{6} b^{5} x^{11}+2520 x^{10} e^{6} a \,b^{4}+3024 x^{10} d \,e^{5} b^{5}+5544 x^{9} e^{6} a^{2} b^{3}+16632 x^{9} d \,e^{5} a \,b^{4}+8316 x^{9} d^{2} e^{4} b^{5}+6160 x^{8} a^{3} b^{2} e^{6}+36960 x^{8} a^{2} d \,e^{5} b^{3}+46200 x^{8} a \,b^{4} d^{2} e^{4}+12320 x^{8} d^{3} e^{3} b^{5}+3465 x^{7} e^{6} a^{4} b +41580 x^{7} d \,e^{5} a^{3} b^{2}+103950 x^{7} d^{2} e^{4} a^{2} b^{3}+69300 x^{7} e^{3} d^{3} a \,b^{4}+10395 x^{7} d^{4} e^{2} b^{5}+792 x^{6} e^{6} a^{5}+23760 x^{6} d \,e^{5} a^{4} b +118800 x^{6} d^{2} e^{4} a^{3} b^{2}+158400 x^{6} e^{3} d^{3} a^{2} b^{3}+59400 x^{6} d^{4} e^{2} a \,b^{4}+4752 x^{6} d^{5} e \,b^{5}+5544 x^{5} d \,e^{5} a^{5}+69300 x^{5} d^{2} e^{4} a^{4} b +184800 x^{5} e^{3} d^{3} a^{3} b^{2}+138600 x^{5} d^{4} e^{2} a^{2} b^{3}+27720 x^{5} d^{5} e a \,b^{4}+924 x^{5} b^{5} d^{6}+16632 a^{5} d^{2} e^{4} x^{4}+110880 a^{4} b \,d^{3} e^{3} x^{4}+166320 a^{3} b^{2} d^{4} e^{2} x^{4}+66528 a^{2} b^{3} d^{5} e \,x^{4}+5544 a \,b^{4} d^{6} x^{4}+27720 x^{3} e^{3} d^{3} a^{5}+103950 x^{3} d^{4} e^{2} a^{4} b +83160 x^{3} d^{5} e \,a^{3} b^{2}+13860 x^{3} d^{6} a^{2} b^{3}+27720 x^{2} d^{4} e^{2} a^{5}+55440 x^{2} d^{5} e \,a^{4} b +18480 x^{2} d^{6} a^{3} b^{2}+16632 x \,d^{5} e \,a^{5}+13860 x \,d^{6} a^{4} b +5544 d^{6} a^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{5544 \left (b x +a \right )^{4}}\) | \(607\) |
| default | \(\frac {e^{6} b^{5} x^{12}}{12}+\frac {\left (\left (e^{6} a +6 b d \,e^{5}\right ) b^{4}+4 e^{6} a \,b^{4}\right ) x^{11}}{11}+\frac {\left (\left (6 a \,e^{5} d +15 b \,d^{2} e^{4}\right ) b^{4}+4 \left (e^{6} a +6 b d \,e^{5}\right ) a \,b^{3}+6 e^{6} a^{2} b^{3}\right ) x^{10}}{10}+\frac {\left (\left (15 a \,e^{4} d^{2}+20 b \,e^{3} d^{3}\right ) b^{4}+4 \left (6 a \,e^{5} d +15 b \,d^{2} e^{4}\right ) a \,b^{3}+6 \left (e^{6} a +6 b d \,e^{5}\right ) a^{2} b^{2}+4 a^{3} b^{2} e^{6}\right ) x^{9}}{9}+\frac {\left (\left (20 a \,e^{3} d^{3}+15 b \,d^{4} e^{2}\right ) b^{4}+4 \left (15 a \,e^{4} d^{2}+20 b \,e^{3} d^{3}\right ) a \,b^{3}+6 \left (6 a \,e^{5} d +15 b \,d^{2} e^{4}\right ) a^{2} b^{2}+4 \left (e^{6} a +6 b d \,e^{5}\right ) a^{3} b +e^{6} a^{4} b \right ) x^{8}}{8}+\frac {\left (\left (15 a \,e^{2} d^{4}+6 b \,d^{5} e \right ) b^{4}+4 \left (20 a \,e^{3} d^{3}+15 b \,d^{4} e^{2}\right ) a \,b^{3}+6 \left (15 a \,e^{4} d^{2}+20 b \,e^{3} d^{3}\right ) a^{2} b^{2}+4 \left (6 a \,e^{5} d +15 b \,d^{2} e^{4}\right ) a^{3} b +\left (e^{6} a +6 b d \,e^{5}\right ) a^{4}\right ) x^{7}}{7}+\frac {\left (\left (6 a \,d^{5} e +b \,d^{6}\right ) b^{4}+4 \left (15 a \,e^{2} d^{4}+6 b \,d^{5} e \right ) a \,b^{3}+6 \left (20 a \,e^{3} d^{3}+15 b \,d^{4} e^{2}\right ) a^{2} b^{2}+4 \left (15 a \,e^{4} d^{2}+20 b \,e^{3} d^{3}\right ) a^{3} b +\left (6 a \,e^{5} d +15 b \,d^{2} e^{4}\right ) a^{4}\right ) x^{6}}{6}+\frac {\left (d^{6} a \,b^{4}+4 \left (6 a \,d^{5} e +b \,d^{6}\right ) a \,b^{3}+6 \left (15 a \,e^{2} d^{4}+6 b \,d^{5} e \right ) a^{2} b^{2}+4 \left (20 a \,e^{3} d^{3}+15 b \,d^{4} e^{2}\right ) a^{3} b +\left (15 a \,e^{4} d^{2}+20 b \,e^{3} d^{3}\right ) a^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{6} a^{2} b^{3}+6 \left (6 a \,d^{5} e +b \,d^{6}\right ) a^{2} b^{2}+4 \left (15 a \,e^{2} d^{4}+6 b \,d^{5} e \right ) a^{3} b +\left (20 a \,e^{3} d^{3}+15 b \,d^{4} e^{2}\right ) a^{4}\right ) x^{4}}{4}+\frac {\left (6 d^{6} a^{3} b^{2}+4 \left (6 a \,d^{5} e +b \,d^{6}\right ) a^{3} b +\left (15 a \,e^{2} d^{4}+6 b \,d^{5} e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{6} a^{4} b +\left (6 a \,d^{5} e +b \,d^{6}\right ) a^{4}\right ) x^{2}}{2}+d^{6} a^{5} x\) | \(817\) |
Input:
int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/12*e^6*b^5*x^12+(5/11*e^6*a*b^4+6/11*d*e^5*b^5)*x^11+(e^6*a^2*b^3+3*d*e^ 5*a*b^4+3/2*d^2*e^4*b^5)*x^10+(10/9*a^3*b^2*e^6+20/3*a^2*d*e^5*b^3+25/3*a* b^4*d^2*e^4+20/9*d^3*e^3*b^5)*x^9+(5/8*e^6*a^4*b+15/2*d*e^5*a^3*b^2+75/4*d ^2*e^4*a^2*b^3+25/2*e^3*d^3*a*b^4+15/8*d^4*e^2*b^5)*x^8+(1/7*e^6*a^5+30/7* d*e^5*a^4*b+150/7*d^2*e^4*a^3*b^2+200/7*e^3*d^3*a^2*b^3+75/7*d^4*e^2*a*b^4 +6/7*d^5*e*b^5)*x^7+(d*e^5*a^5+25/2*d^2*e^4*a^4*b+100/3*e^3*d^3*a^3*b^2+25 *d^4*e^2*a^2*b^3+5*d^5*e*a*b^4+1/6*b^5*d^6)*x^6+(3*a^5*d^2*e^4+20*a^4*b*d^ 3*e^3+30*a^3*b^2*d^4*e^2+12*a^2*b^3*d^5*e+a*b^4*d^6)*x^5+(5*e^3*d^3*a^5+75 /4*d^4*e^2*a^4*b+15*d^5*e*a^3*b^2+5/2*d^6*a^2*b^3)*x^4+(5*d^4*e^2*a^5+10*d ^5*e*a^4*b+10/3*d^6*a^3*b^2)*x^3+(3*d^5*e*a^5+5/2*d^6*a^4*b)*x^2+d^6*a^5*x
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.08 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{12} \, b^{5} e^{6} x^{12} + a^{5} d^{6} x + \frac {1}{11} \, {\left (6 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{11} + \frac {1}{2} \, {\left (3 \, b^{5} d^{2} e^{4} + 6 \, a b^{4} d e^{5} + 2 \, a^{2} b^{3} e^{6}\right )} x^{10} + \frac {5}{9} \, {\left (4 \, b^{5} d^{3} e^{3} + 15 \, a b^{4} d^{2} e^{4} + 12 \, a^{2} b^{3} d e^{5} + 2 \, a^{3} b^{2} e^{6}\right )} x^{9} + \frac {5}{8} \, {\left (3 \, b^{5} d^{4} e^{2} + 20 \, a b^{4} d^{3} e^{3} + 30 \, a^{2} b^{3} d^{2} e^{4} + 12 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, b^{5} d^{5} e + 75 \, a b^{4} d^{4} e^{2} + 200 \, a^{2} b^{3} d^{3} e^{3} + 150 \, a^{3} b^{2} d^{2} e^{4} + 30 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{6} + 30 \, a b^{4} d^{5} e + 150 \, a^{2} b^{3} d^{4} e^{2} + 200 \, a^{3} b^{2} d^{3} e^{3} + 75 \, a^{4} b d^{2} e^{4} + 6 \, a^{5} d e^{5}\right )} x^{6} + {\left (a b^{4} d^{6} + 12 \, a^{2} b^{3} d^{5} e + 30 \, a^{3} b^{2} d^{4} e^{2} + 20 \, a^{4} b d^{3} e^{3} + 3 \, a^{5} d^{2} e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{6} + 12 \, a^{3} b^{2} d^{5} e + 15 \, a^{4} b d^{4} e^{2} + 4 \, a^{5} d^{3} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{6} + 6 \, a^{4} b d^{5} e + 3 \, a^{5} d^{4} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{6} + 6 \, a^{5} d^{5} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
1/12*b^5*e^6*x^12 + a^5*d^6*x + 1/11*(6*b^5*d*e^5 + 5*a*b^4*e^6)*x^11 + 1/ 2*(3*b^5*d^2*e^4 + 6*a*b^4*d*e^5 + 2*a^2*b^3*e^6)*x^10 + 5/9*(4*b^5*d^3*e^ 3 + 15*a*b^4*d^2*e^4 + 12*a^2*b^3*d*e^5 + 2*a^3*b^2*e^6)*x^9 + 5/8*(3*b^5* d^4*e^2 + 20*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 + 12*a^3*b^2*d*e^5 + a^4*b *e^6)*x^8 + 1/7*(6*b^5*d^5*e + 75*a*b^4*d^4*e^2 + 200*a^2*b^3*d^3*e^3 + 15 0*a^3*b^2*d^2*e^4 + 30*a^4*b*d*e^5 + a^5*e^6)*x^7 + 1/6*(b^5*d^6 + 30*a*b^ 4*d^5*e + 150*a^2*b^3*d^4*e^2 + 200*a^3*b^2*d^3*e^3 + 75*a^4*b*d^2*e^4 + 6 *a^5*d*e^5)*x^6 + (a*b^4*d^6 + 12*a^2*b^3*d^5*e + 30*a^3*b^2*d^4*e^2 + 20* a^4*b*d^3*e^3 + 3*a^5*d^2*e^4)*x^5 + 5/4*(2*a^2*b^3*d^6 + 12*a^3*b^2*d^5*e + 15*a^4*b*d^4*e^2 + 4*a^5*d^3*e^3)*x^4 + 5/3*(2*a^3*b^2*d^6 + 6*a^4*b*d^ 5*e + 3*a^5*d^4*e^2)*x^3 + 1/2*(5*a^4*b*d^6 + 6*a^5*d^5*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (129) = 258\).
Time = 0.09 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.06 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{5} d^{6} x + \frac {b^{5} e^{6} x^{12}}{12} + x^{11} \cdot \left (\frac {5 a b^{4} e^{6}}{11} + \frac {6 b^{5} d e^{5}}{11}\right ) + x^{10} \left (a^{2} b^{3} e^{6} + 3 a b^{4} d e^{5} + \frac {3 b^{5} d^{2} e^{4}}{2}\right ) + x^{9} \cdot \left (\frac {10 a^{3} b^{2} e^{6}}{9} + \frac {20 a^{2} b^{3} d e^{5}}{3} + \frac {25 a b^{4} d^{2} e^{4}}{3} + \frac {20 b^{5} d^{3} e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 a^{4} b e^{6}}{8} + \frac {15 a^{3} b^{2} d e^{5}}{2} + \frac {75 a^{2} b^{3} d^{2} e^{4}}{4} + \frac {25 a b^{4} d^{3} e^{3}}{2} + \frac {15 b^{5} d^{4} e^{2}}{8}\right ) + x^{7} \left (\frac {a^{5} e^{6}}{7} + \frac {30 a^{4} b d e^{5}}{7} + \frac {150 a^{3} b^{2} d^{2} e^{4}}{7} + \frac {200 a^{2} b^{3} d^{3} e^{3}}{7} + \frac {75 a b^{4} d^{4} e^{2}}{7} + \frac {6 b^{5} d^{5} e}{7}\right ) + x^{6} \left (a^{5} d e^{5} + \frac {25 a^{4} b d^{2} e^{4}}{2} + \frac {100 a^{3} b^{2} d^{3} e^{3}}{3} + 25 a^{2} b^{3} d^{4} e^{2} + 5 a b^{4} d^{5} e + \frac {b^{5} d^{6}}{6}\right ) + x^{5} \cdot \left (3 a^{5} d^{2} e^{4} + 20 a^{4} b d^{3} e^{3} + 30 a^{3} b^{2} d^{4} e^{2} + 12 a^{2} b^{3} d^{5} e + a b^{4} d^{6}\right ) + x^{4} \cdot \left (5 a^{5} d^{3} e^{3} + \frac {75 a^{4} b d^{4} e^{2}}{4} + 15 a^{3} b^{2} d^{5} e + \frac {5 a^{2} b^{3} d^{6}}{2}\right ) + x^{3} \cdot \left (5 a^{5} d^{4} e^{2} + 10 a^{4} b d^{5} e + \frac {10 a^{3} b^{2} d^{6}}{3}\right ) + x^{2} \cdot \left (3 a^{5} d^{5} e + \frac {5 a^{4} b d^{6}}{2}\right ) \] Input:
integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
a**5*d**6*x + b**5*e**6*x**12/12 + x**11*(5*a*b**4*e**6/11 + 6*b**5*d*e**5 /11) + x**10*(a**2*b**3*e**6 + 3*a*b**4*d*e**5 + 3*b**5*d**2*e**4/2) + x** 9*(10*a**3*b**2*e**6/9 + 20*a**2*b**3*d*e**5/3 + 25*a*b**4*d**2*e**4/3 + 2 0*b**5*d**3*e**3/9) + x**8*(5*a**4*b*e**6/8 + 15*a**3*b**2*d*e**5/2 + 75*a **2*b**3*d**2*e**4/4 + 25*a*b**4*d**3*e**3/2 + 15*b**5*d**4*e**2/8) + x**7 *(a**5*e**6/7 + 30*a**4*b*d*e**5/7 + 150*a**3*b**2*d**2*e**4/7 + 200*a**2* b**3*d**3*e**3/7 + 75*a*b**4*d**4*e**2/7 + 6*b**5*d**5*e/7) + x**6*(a**5*d *e**5 + 25*a**4*b*d**2*e**4/2 + 100*a**3*b**2*d**3*e**3/3 + 25*a**2*b**3*d **4*e**2 + 5*a*b**4*d**5*e + b**5*d**6/6) + x**5*(3*a**5*d**2*e**4 + 20*a* *4*b*d**3*e**3 + 30*a**3*b**2*d**4*e**2 + 12*a**2*b**3*d**5*e + a*b**4*d** 6) + x**4*(5*a**5*d**3*e**3 + 75*a**4*b*d**4*e**2/4 + 15*a**3*b**2*d**5*e + 5*a**2*b**3*d**6/2) + x**3*(5*a**5*d**4*e**2 + 10*a**4*b*d**5*e + 10*a** 3*b**2*d**6/3) + x**2*(3*a**5*d**5*e + 5*a**4*b*d**6/2)
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.04 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{12} \, b^{5} e^{6} x^{12} + a^{5} d^{6} x + \frac {1}{11} \, {\left (6 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{11} + \frac {1}{2} \, {\left (3 \, b^{5} d^{2} e^{4} + 6 \, a b^{4} d e^{5} + 2 \, a^{2} b^{3} e^{6}\right )} x^{10} + \frac {5}{9} \, {\left (4 \, b^{5} d^{3} e^{3} + 15 \, a b^{4} d^{2} e^{4} + 12 \, a^{2} b^{3} d e^{5} + 2 \, a^{3} b^{2} e^{6}\right )} x^{9} + \frac {5}{8} \, {\left (3 \, b^{5} d^{4} e^{2} + 20 \, a b^{4} d^{3} e^{3} + 30 \, a^{2} b^{3} d^{2} e^{4} + 12 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, b^{5} d^{5} e + 75 \, a b^{4} d^{4} e^{2} + 200 \, a^{2} b^{3} d^{3} e^{3} + 150 \, a^{3} b^{2} d^{2} e^{4} + 30 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{6} + 30 \, a b^{4} d^{5} e + 150 \, a^{2} b^{3} d^{4} e^{2} + 200 \, a^{3} b^{2} d^{3} e^{3} + 75 \, a^{4} b d^{2} e^{4} + 6 \, a^{5} d e^{5}\right )} x^{6} + {\left (a b^{4} d^{6} + 12 \, a^{2} b^{3} d^{5} e + 30 \, a^{3} b^{2} d^{4} e^{2} + 20 \, a^{4} b d^{3} e^{3} + 3 \, a^{5} d^{2} e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{6} + 12 \, a^{3} b^{2} d^{5} e + 15 \, a^{4} b d^{4} e^{2} + 4 \, a^{5} d^{3} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{6} + 6 \, a^{4} b d^{5} e + 3 \, a^{5} d^{4} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{6} + 6 \, a^{5} d^{5} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
1/12*b^5*e^6*x^12 + a^5*d^6*x + 1/11*(6*b^5*d*e^5 + 5*a*b^4*e^6)*x^11 + 1/ 2*(3*b^5*d^2*e^4 + 6*a*b^4*d*e^5 + 2*a^2*b^3*e^6)*x^10 + 5/9*(4*b^5*d^3*e^ 3 + 15*a*b^4*d^2*e^4 + 12*a^2*b^3*d*e^5 + 2*a^3*b^2*e^6)*x^9 + 5/8*(3*b^5* d^4*e^2 + 20*a*b^4*d^3*e^3 + 30*a^2*b^3*d^2*e^4 + 12*a^3*b^2*d*e^5 + a^4*b *e^6)*x^8 + 1/7*(6*b^5*d^5*e + 75*a*b^4*d^4*e^2 + 200*a^2*b^3*d^3*e^3 + 15 0*a^3*b^2*d^2*e^4 + 30*a^4*b*d*e^5 + a^5*e^6)*x^7 + 1/6*(b^5*d^6 + 30*a*b^ 4*d^5*e + 150*a^2*b^3*d^4*e^2 + 200*a^3*b^2*d^3*e^3 + 75*a^4*b*d^2*e^4 + 6 *a^5*d*e^5)*x^6 + (a*b^4*d^6 + 12*a^2*b^3*d^5*e + 30*a^3*b^2*d^4*e^2 + 20* a^4*b*d^3*e^3 + 3*a^5*d^2*e^4)*x^5 + 5/4*(2*a^2*b^3*d^6 + 12*a^3*b^2*d^5*e + 15*a^4*b*d^4*e^2 + 4*a^5*d^3*e^3)*x^4 + 5/3*(2*a^3*b^2*d^6 + 6*a^4*b*d^ 5*e + 3*a^5*d^4*e^2)*x^3 + 1/2*(5*a^4*b*d^6 + 6*a^5*d^5*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (133) = 266\).
Time = 0.11 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.05 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{12} \, b^{5} e^{6} x^{12} + \frac {6}{11} \, b^{5} d e^{5} x^{11} + \frac {5}{11} \, a b^{4} e^{6} x^{11} + \frac {3}{2} \, b^{5} d^{2} e^{4} x^{10} + 3 \, a b^{4} d e^{5} x^{10} + a^{2} b^{3} e^{6} x^{10} + \frac {20}{9} \, b^{5} d^{3} e^{3} x^{9} + \frac {25}{3} \, a b^{4} d^{2} e^{4} x^{9} + \frac {20}{3} \, a^{2} b^{3} d e^{5} x^{9} + \frac {10}{9} \, a^{3} b^{2} e^{6} x^{9} + \frac {15}{8} \, b^{5} d^{4} e^{2} x^{8} + \frac {25}{2} \, a b^{4} d^{3} e^{3} x^{8} + \frac {75}{4} \, a^{2} b^{3} d^{2} e^{4} x^{8} + \frac {15}{2} \, a^{3} b^{2} d e^{5} x^{8} + \frac {5}{8} \, a^{4} b e^{6} x^{8} + \frac {6}{7} \, b^{5} d^{5} e x^{7} + \frac {75}{7} \, a b^{4} d^{4} e^{2} x^{7} + \frac {200}{7} \, a^{2} b^{3} d^{3} e^{3} x^{7} + \frac {150}{7} \, a^{3} b^{2} d^{2} e^{4} x^{7} + \frac {30}{7} \, a^{4} b d e^{5} x^{7} + \frac {1}{7} \, a^{5} e^{6} x^{7} + \frac {1}{6} \, b^{5} d^{6} x^{6} + 5 \, a b^{4} d^{5} e x^{6} + 25 \, a^{2} b^{3} d^{4} e^{2} x^{6} + \frac {100}{3} \, a^{3} b^{2} d^{3} e^{3} x^{6} + \frac {25}{2} \, a^{4} b d^{2} e^{4} x^{6} + a^{5} d e^{5} x^{6} + a b^{4} d^{6} x^{5} + 12 \, a^{2} b^{3} d^{5} e x^{5} + 30 \, a^{3} b^{2} d^{4} e^{2} x^{5} + 20 \, a^{4} b d^{3} e^{3} x^{5} + 3 \, a^{5} d^{2} e^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{6} x^{4} + 15 \, a^{3} b^{2} d^{5} e x^{4} + \frac {75}{4} \, a^{4} b d^{4} e^{2} x^{4} + 5 \, a^{5} d^{3} e^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{6} x^{3} + 10 \, a^{4} b d^{5} e x^{3} + 5 \, a^{5} d^{4} e^{2} x^{3} + \frac {5}{2} \, a^{4} b d^{6} x^{2} + 3 \, a^{5} d^{5} e x^{2} + a^{5} d^{6} x \] Input:
integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
1/12*b^5*e^6*x^12 + 6/11*b^5*d*e^5*x^11 + 5/11*a*b^4*e^6*x^11 + 3/2*b^5*d^ 2*e^4*x^10 + 3*a*b^4*d*e^5*x^10 + a^2*b^3*e^6*x^10 + 20/9*b^5*d^3*e^3*x^9 + 25/3*a*b^4*d^2*e^4*x^9 + 20/3*a^2*b^3*d*e^5*x^9 + 10/9*a^3*b^2*e^6*x^9 + 15/8*b^5*d^4*e^2*x^8 + 25/2*a*b^4*d^3*e^3*x^8 + 75/4*a^2*b^3*d^2*e^4*x^8 + 15/2*a^3*b^2*d*e^5*x^8 + 5/8*a^4*b*e^6*x^8 + 6/7*b^5*d^5*e*x^7 + 75/7*a* b^4*d^4*e^2*x^7 + 200/7*a^2*b^3*d^3*e^3*x^7 + 150/7*a^3*b^2*d^2*e^4*x^7 + 30/7*a^4*b*d*e^5*x^7 + 1/7*a^5*e^6*x^7 + 1/6*b^5*d^6*x^6 + 5*a*b^4*d^5*e*x ^6 + 25*a^2*b^3*d^4*e^2*x^6 + 100/3*a^3*b^2*d^3*e^3*x^6 + 25/2*a^4*b*d^2*e ^4*x^6 + a^5*d*e^5*x^6 + a*b^4*d^6*x^5 + 12*a^2*b^3*d^5*e*x^5 + 30*a^3*b^2 *d^4*e^2*x^5 + 20*a^4*b*d^3*e^3*x^5 + 3*a^5*d^2*e^4*x^5 + 5/2*a^2*b^3*d^6* x^4 + 15*a^3*b^2*d^5*e*x^4 + 75/4*a^4*b*d^4*e^2*x^4 + 5*a^5*d^3*e^3*x^4 + 10/3*a^3*b^2*d^6*x^3 + 10*a^4*b*d^5*e*x^3 + 5*a^5*d^4*e^2*x^3 + 5/2*a^4*b* d^6*x^2 + 3*a^5*d^5*e*x^2 + a^5*d^6*x
Time = 0.20 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.44 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^5\,\left (3\,a^5\,d^2\,e^4+20\,a^4\,b\,d^3\,e^3+30\,a^3\,b^2\,d^4\,e^2+12\,a^2\,b^3\,d^5\,e+a\,b^4\,d^6\right )+x^8\,\left (\frac {5\,a^4\,b\,e^6}{8}+\frac {15\,a^3\,b^2\,d\,e^5}{2}+\frac {75\,a^2\,b^3\,d^2\,e^4}{4}+\frac {25\,a\,b^4\,d^3\,e^3}{2}+\frac {15\,b^5\,d^4\,e^2}{8}\right )+x^6\,\left (a^5\,d\,e^5+\frac {25\,a^4\,b\,d^2\,e^4}{2}+\frac {100\,a^3\,b^2\,d^3\,e^3}{3}+25\,a^2\,b^3\,d^4\,e^2+5\,a\,b^4\,d^5\,e+\frac {b^5\,d^6}{6}\right )+x^7\,\left (\frac {a^5\,e^6}{7}+\frac {30\,a^4\,b\,d\,e^5}{7}+\frac {150\,a^3\,b^2\,d^2\,e^4}{7}+\frac {200\,a^2\,b^3\,d^3\,e^3}{7}+\frac {75\,a\,b^4\,d^4\,e^2}{7}+\frac {6\,b^5\,d^5\,e}{7}\right )+a^5\,d^6\,x+\frac {b^5\,e^6\,x^{12}}{12}+\frac {5\,a^2\,d^3\,x^4\,\left (4\,a^3\,e^3+15\,a^2\,b\,d\,e^2+12\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{4}+\frac {5\,b^2\,e^3\,x^9\,\left (2\,a^3\,e^3+12\,a^2\,b\,d\,e^2+15\,a\,b^2\,d^2\,e+4\,b^3\,d^3\right )}{9}+\frac {a^4\,d^5\,x^2\,\left (6\,a\,e+5\,b\,d\right )}{2}+\frac {b^4\,e^5\,x^{11}\,\left (5\,a\,e+6\,b\,d\right )}{11}+\frac {5\,a^3\,d^4\,x^3\,\left (3\,a^2\,e^2+6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3}+\frac {b^3\,e^4\,x^{10}\,\left (2\,a^2\,e^2+6\,a\,b\,d\,e+3\,b^2\,d^2\right )}{2} \] Input:
int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
x^5*(a*b^4*d^6 + 3*a^5*d^2*e^4 + 12*a^2*b^3*d^5*e + 20*a^4*b*d^3*e^3 + 30* a^3*b^2*d^4*e^2) + x^8*((5*a^4*b*e^6)/8 + (15*b^5*d^4*e^2)/8 + (25*a*b^4*d ^3*e^3)/2 + (15*a^3*b^2*d*e^5)/2 + (75*a^2*b^3*d^2*e^4)/4) + x^6*((b^5*d^6 )/6 + a^5*d*e^5 + (25*a^4*b*d^2*e^4)/2 + 25*a^2*b^3*d^4*e^2 + (100*a^3*b^2 *d^3*e^3)/3 + 5*a*b^4*d^5*e) + x^7*((a^5*e^6)/7 + (6*b^5*d^5*e)/7 + (75*a* b^4*d^4*e^2)/7 + (200*a^2*b^3*d^3*e^3)/7 + (150*a^3*b^2*d^2*e^4)/7 + (30*a ^4*b*d*e^5)/7) + a^5*d^6*x + (b^5*e^6*x^12)/12 + (5*a^2*d^3*x^4*(4*a^3*e^3 + 2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*e^2))/4 + (5*b^2*e^3*x^9*(2*a^3 *e^3 + 4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2))/9 + (a^4*d^5*x^2*(6*a *e + 5*b*d))/2 + (b^4*e^5*x^11*(5*a*e + 6*b*d))/11 + (5*a^3*d^4*x^3*(3*a^2 *e^2 + 2*b^2*d^2 + 6*a*b*d*e))/3 + (b^3*e^4*x^10*(2*a^2*e^2 + 3*b^2*d^2 + 6*a*b*d*e))/2
Time = 0.26 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.06 \[ \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {x \left (462 b^{5} e^{6} x^{11}+2520 a \,b^{4} e^{6} x^{10}+3024 b^{5} d \,e^{5} x^{10}+5544 a^{2} b^{3} e^{6} x^{9}+16632 a \,b^{4} d \,e^{5} x^{9}+8316 b^{5} d^{2} e^{4} x^{9}+6160 a^{3} b^{2} e^{6} x^{8}+36960 a^{2} b^{3} d \,e^{5} x^{8}+46200 a \,b^{4} d^{2} e^{4} x^{8}+12320 b^{5} d^{3} e^{3} x^{8}+3465 a^{4} b \,e^{6} x^{7}+41580 a^{3} b^{2} d \,e^{5} x^{7}+103950 a^{2} b^{3} d^{2} e^{4} x^{7}+69300 a \,b^{4} d^{3} e^{3} x^{7}+10395 b^{5} d^{4} e^{2} x^{7}+792 a^{5} e^{6} x^{6}+23760 a^{4} b d \,e^{5} x^{6}+118800 a^{3} b^{2} d^{2} e^{4} x^{6}+158400 a^{2} b^{3} d^{3} e^{3} x^{6}+59400 a \,b^{4} d^{4} e^{2} x^{6}+4752 b^{5} d^{5} e \,x^{6}+5544 a^{5} d \,e^{5} x^{5}+69300 a^{4} b \,d^{2} e^{4} x^{5}+184800 a^{3} b^{2} d^{3} e^{3} x^{5}+138600 a^{2} b^{3} d^{4} e^{2} x^{5}+27720 a \,b^{4} d^{5} e \,x^{5}+924 b^{5} d^{6} x^{5}+16632 a^{5} d^{2} e^{4} x^{4}+110880 a^{4} b \,d^{3} e^{3} x^{4}+166320 a^{3} b^{2} d^{4} e^{2} x^{4}+66528 a^{2} b^{3} d^{5} e \,x^{4}+5544 a \,b^{4} d^{6} x^{4}+27720 a^{5} d^{3} e^{3} x^{3}+103950 a^{4} b \,d^{4} e^{2} x^{3}+83160 a^{3} b^{2} d^{5} e \,x^{3}+13860 a^{2} b^{3} d^{6} x^{3}+27720 a^{5} d^{4} e^{2} x^{2}+55440 a^{4} b \,d^{5} e \,x^{2}+18480 a^{3} b^{2} d^{6} x^{2}+16632 a^{5} d^{5} e x +13860 a^{4} b \,d^{6} x +5544 a^{5} d^{6}\right )}{5544} \] Input:
int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(x*(5544*a**5*d**6 + 16632*a**5*d**5*e*x + 27720*a**5*d**4*e**2*x**2 + 277 20*a**5*d**3*e**3*x**3 + 16632*a**5*d**2*e**4*x**4 + 5544*a**5*d*e**5*x**5 + 792*a**5*e**6*x**6 + 13860*a**4*b*d**6*x + 55440*a**4*b*d**5*e*x**2 + 1 03950*a**4*b*d**4*e**2*x**3 + 110880*a**4*b*d**3*e**3*x**4 + 69300*a**4*b* d**2*e**4*x**5 + 23760*a**4*b*d*e**5*x**6 + 3465*a**4*b*e**6*x**7 + 18480* a**3*b**2*d**6*x**2 + 83160*a**3*b**2*d**5*e*x**3 + 166320*a**3*b**2*d**4* e**2*x**4 + 184800*a**3*b**2*d**3*e**3*x**5 + 118800*a**3*b**2*d**2*e**4*x **6 + 41580*a**3*b**2*d*e**5*x**7 + 6160*a**3*b**2*e**6*x**8 + 13860*a**2* b**3*d**6*x**3 + 66528*a**2*b**3*d**5*e*x**4 + 138600*a**2*b**3*d**4*e**2* x**5 + 158400*a**2*b**3*d**3*e**3*x**6 + 103950*a**2*b**3*d**2*e**4*x**7 + 36960*a**2*b**3*d*e**5*x**8 + 5544*a**2*b**3*e**6*x**9 + 5544*a*b**4*d**6 *x**4 + 27720*a*b**4*d**5*e*x**5 + 59400*a*b**4*d**4*e**2*x**6 + 69300*a*b **4*d**3*e**3*x**7 + 46200*a*b**4*d**2*e**4*x**8 + 16632*a*b**4*d*e**5*x** 9 + 2520*a*b**4*e**6*x**10 + 924*b**5*d**6*x**5 + 4752*b**5*d**5*e*x**6 + 10395*b**5*d**4*e**2*x**7 + 12320*b**5*d**3*e**3*x**8 + 8316*b**5*d**2*e** 4*x**9 + 3024*b**5*d*e**5*x**10 + 462*b**5*e**6*x**11))/5544