Integrand size = 31, antiderivative size = 143 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^5 (a+b x)^8}{8 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e)^3 (a+b x)^{10}}{b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^{11}}{11 b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{12}}{12 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6} \] Output:
1/8*(-a*e+b*d)^5*(b*x+a)^8/b^6+5/9*e*(-a*e+b*d)^4*(b*x+a)^9/b^6+e^2*(-a*e+ b*d)^3*(b*x+a)^10/b^6+10/11*e^3*(-a*e+b*d)^2*(b*x+a)^11/b^6+5/12*e^4*(-a*e +b*d)*(b*x+a)^12/b^6+1/13*e^5*(b*x+a)^13/b^6
Leaf count is larger than twice the leaf count of optimal. \(493\) vs. \(2(143)=286\).
Time = 0.17 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.45 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x \left (1716 a^7 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+1716 a^6 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+1287 a^5 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+715 a^4 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+286 a^3 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+78 a^2 b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+13 a b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+b^7 x^7 \left (1287 d^5+5720 d^4 e x+10296 d^3 e^2 x^2+9360 d^2 e^3 x^3+4290 d e^4 x^4+792 e^5 x^5\right )\right )}{10296} \] Input:
Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
(x*(1716*a^7*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e ^4*x^4 + e^5*x^5) + 1716*a^6*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 1287*a^5*b^2*x^2*(56*d^5 + 21 0*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5 ) + 715*a^4*b^3*x^3*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3 *x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 286*a^3*b^4*x^4*(252*d^5 + 1050*d^4*e *x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 78*a^2*b^5*x^5*(462*d^5 + 1980*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x ^3 + 1386*d*e^4*x^4 + 252*e^5*x^5) + 13*a*b^6*x^6*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + b^ 7*x^7*(1287*d^5 + 5720*d^4*e*x + 10296*d^3*e^2*x^2 + 9360*d^2*e^3*x^3 + 42 90*d*e^4*x^4 + 792*e^5*x^5)))/10296
Time = 0.90 (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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^5dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 (d+e x)^5dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {5 e^4 (a+b x)^{11} (b d-a e)}{b^5}+\frac {10 e^3 (a+b x)^{10} (b d-a e)^2}{b^5}+\frac {10 e^2 (a+b x)^9 (b d-a e)^3}{b^5}+\frac {5 e (a+b x)^8 (b d-a e)^4}{b^5}+\frac {(a+b x)^7 (b d-a e)^5}{b^5}+\frac {e^5 (a+b x)^{12}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 e^4 (a+b x)^{12} (b d-a e)}{12 b^6}+\frac {10 e^3 (a+b x)^{11} (b d-a e)^2}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e)^3}{b^6}+\frac {5 e (a+b x)^9 (b d-a e)^4}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^5}{8 b^6}+\frac {e^5 (a+b x)^{13}}{13 b^6}\) |
Input:
Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
((b*d - a*e)^5*(a + b*x)^8)/(8*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^9)/(9*b ^6) + (e^2*(b*d - a*e)^3*(a + b*x)^10)/b^6 + (10*e^3*(b*d - a*e)^2*(a + b* x)^11)/(11*b^6) + (5*e^4*(b*d - a*e)*(a + b*x)^12)/(12*b^6) + (e^5*(a + b* x)^13)/(13*b^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. \(587\) vs. \(2(133)=266\).
Time = 1.22 (sec) , antiderivative size = 588, normalized size of antiderivative = 4.11
| method | result | size |
| norman | \(\frac {b^{7} e^{5} x^{13}}{13}+\left (\frac {7}{12} e^{5} a \,b^{6}+\frac {5}{12} d \,e^{4} b^{7}\right ) x^{12}+\left (\frac {21}{11} e^{5} a^{2} b^{5}+\frac {35}{11} d \,e^{4} a \,b^{6}+\frac {10}{11} d^{2} e^{3} b^{7}\right ) x^{11}+\left (\frac {7}{2} e^{5} a^{3} b^{4}+\frac {21}{2} d \,e^{4} a^{2} b^{5}+7 d^{2} e^{3} a \,b^{6}+d^{3} e^{2} b^{7}\right ) x^{10}+\left (\frac {35}{9} e^{5} a^{4} b^{3}+\frac {175}{9} d \,e^{4} a^{3} b^{4}+\frac {70}{3} d^{2} e^{3} a^{2} b^{5}+\frac {70}{9} d^{3} e^{2} a \,b^{6}+\frac {5}{9} d^{4} e \,b^{7}\right ) x^{9}+\left (\frac {21}{8} e^{5} a^{5} b^{2}+\frac {175}{8} d \,e^{4} a^{4} b^{3}+\frac {175}{4} d^{2} e^{3} a^{3} b^{4}+\frac {105}{4} d^{3} e^{2} a^{2} b^{5}+\frac {35}{8} d^{4} e a \,b^{6}+\frac {1}{8} d^{5} b^{7}\right ) x^{8}+\left (e^{5} a^{6} b +15 d \,e^{4} a^{5} b^{2}+50 d^{2} e^{3} a^{4} b^{3}+50 d^{3} e^{2} a^{3} b^{4}+15 d^{4} e \,a^{2} b^{5}+d^{5} a \,b^{6}\right ) x^{7}+\left (\frac {1}{6} e^{5} a^{7}+\frac {35}{6} d \,e^{4} a^{6} b +35 d^{2} e^{3} a^{5} b^{2}+\frac {175}{3} d^{3} e^{2} a^{4} b^{3}+\frac {175}{6} d^{4} e \,a^{3} b^{4}+\frac {7}{2} d^{5} a^{2} b^{5}\right ) x^{6}+\left (d \,e^{4} a^{7}+14 d^{2} e^{3} a^{6} b +42 d^{3} e^{2} a^{5} b^{2}+35 d^{4} e \,a^{4} b^{3}+7 d^{5} a^{3} b^{4}\right ) x^{5}+\left (\frac {5}{2} d^{2} e^{3} a^{7}+\frac {35}{2} d^{3} e^{2} a^{6} b +\frac {105}{4} d^{4} e \,a^{5} b^{2}+\frac {35}{4} d^{5} a^{4} b^{3}\right ) x^{4}+\left (\frac {10}{3} d^{3} e^{2} a^{7}+\frac {35}{3} d^{4} e \,a^{6} b +7 d^{5} a^{5} b^{2}\right ) x^{3}+\left (\frac {5}{2} d^{4} e \,a^{7}+\frac {7}{2} d^{5} a^{6} b \right ) x^{2}+d^{5} a^{7} x\) | \(588\) |
| risch | \(\frac {70}{9} x^{9} d^{3} e^{2} a \,b^{6}+\frac {175}{8} x^{8} d \,e^{4} a^{4} b^{3}+\frac {175}{4} x^{8} d^{2} e^{3} a^{3} b^{4}+\frac {105}{4} x^{8} d^{3} e^{2} a^{2} b^{5}+\frac {35}{8} x^{8} d^{4} e a \,b^{6}+42 a^{5} b^{2} d^{3} e^{2} x^{5}+35 a^{4} b^{3} d^{4} e \,x^{5}+\frac {7}{12} x^{12} e^{5} a \,b^{6}+\frac {5}{12} x^{12} d \,e^{4} b^{7}+\frac {21}{11} x^{11} e^{5} a^{2} b^{5}+\frac {10}{11} x^{11} d^{2} e^{3} b^{7}+\frac {7}{2} x^{10} e^{5} a^{3} b^{4}+x^{10} d^{3} e^{2} b^{7}+\frac {35}{9} x^{9} e^{5} a^{4} b^{3}+\frac {5}{9} x^{9} d^{4} e \,b^{7}+\frac {21}{8} x^{8} e^{5} a^{5} b^{2}+\frac {7}{2} x^{6} d^{5} a^{2} b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{7}+\frac {35}{4} x^{4} d^{5} a^{4} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{7}+7 x^{3} d^{5} a^{5} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{7}+\frac {7}{2} x^{2} d^{5} a^{6} b +\frac {70}{3} x^{9} d^{2} e^{3} a^{2} b^{5}+15 a^{2} b^{5} d^{4} e \,x^{7}+14 a^{6} b \,d^{2} e^{3} x^{5}+a^{6} b \,e^{5} x^{7}+a \,b^{6} d^{5} x^{7}+a^{7} d \,e^{4} x^{5}+7 a^{3} b^{4} d^{5} x^{5}+\frac {35}{3} x^{3} d^{4} e \,a^{6} b +50 a^{4} b^{3} d^{2} e^{3} x^{7}+50 a^{3} b^{4} d^{3} e^{2} x^{7}+\frac {35}{6} x^{6} d \,e^{4} a^{6} b +35 x^{6} d^{2} e^{3} a^{5} b^{2}+\frac {175}{3} x^{6} d^{3} e^{2} a^{4} b^{3}+\frac {175}{6} x^{6} d^{4} e \,a^{3} b^{4}+\frac {35}{2} x^{4} d^{3} e^{2} a^{6} b +\frac {105}{4} x^{4} d^{4} e \,a^{5} b^{2}+15 a^{5} b^{2} d \,e^{4} x^{7}+\frac {35}{11} x^{11} d \,e^{4} a \,b^{6}+\frac {21}{2} x^{10} d \,e^{4} a^{2} b^{5}+7 x^{10} d^{2} e^{3} a \,b^{6}+\frac {175}{9} x^{9} d \,e^{4} a^{3} b^{4}+\frac {1}{6} x^{6} e^{5} a^{7}+d^{5} a^{7} x +\frac {1}{13} b^{7} e^{5} x^{13}+\frac {1}{8} x^{8} d^{5} b^{7}\) | \(671\) |
| parallelrisch | \(\frac {70}{9} x^{9} d^{3} e^{2} a \,b^{6}+\frac {175}{8} x^{8} d \,e^{4} a^{4} b^{3}+\frac {175}{4} x^{8} d^{2} e^{3} a^{3} b^{4}+\frac {105}{4} x^{8} d^{3} e^{2} a^{2} b^{5}+\frac {35}{8} x^{8} d^{4} e a \,b^{6}+42 a^{5} b^{2} d^{3} e^{2} x^{5}+35 a^{4} b^{3} d^{4} e \,x^{5}+\frac {7}{12} x^{12} e^{5} a \,b^{6}+\frac {5}{12} x^{12} d \,e^{4} b^{7}+\frac {21}{11} x^{11} e^{5} a^{2} b^{5}+\frac {10}{11} x^{11} d^{2} e^{3} b^{7}+\frac {7}{2} x^{10} e^{5} a^{3} b^{4}+x^{10} d^{3} e^{2} b^{7}+\frac {35}{9} x^{9} e^{5} a^{4} b^{3}+\frac {5}{9} x^{9} d^{4} e \,b^{7}+\frac {21}{8} x^{8} e^{5} a^{5} b^{2}+\frac {7}{2} x^{6} d^{5} a^{2} b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{7}+\frac {35}{4} x^{4} d^{5} a^{4} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{7}+7 x^{3} d^{5} a^{5} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{7}+\frac {7}{2} x^{2} d^{5} a^{6} b +\frac {70}{3} x^{9} d^{2} e^{3} a^{2} b^{5}+15 a^{2} b^{5} d^{4} e \,x^{7}+14 a^{6} b \,d^{2} e^{3} x^{5}+a^{6} b \,e^{5} x^{7}+a \,b^{6} d^{5} x^{7}+a^{7} d \,e^{4} x^{5}+7 a^{3} b^{4} d^{5} x^{5}+\frac {35}{3} x^{3} d^{4} e \,a^{6} b +50 a^{4} b^{3} d^{2} e^{3} x^{7}+50 a^{3} b^{4} d^{3} e^{2} x^{7}+\frac {35}{6} x^{6} d \,e^{4} a^{6} b +35 x^{6} d^{2} e^{3} a^{5} b^{2}+\frac {175}{3} x^{6} d^{3} e^{2} a^{4} b^{3}+\frac {175}{6} x^{6} d^{4} e \,a^{3} b^{4}+\frac {35}{2} x^{4} d^{3} e^{2} a^{6} b +\frac {105}{4} x^{4} d^{4} e \,a^{5} b^{2}+15 a^{5} b^{2} d \,e^{4} x^{7}+\frac {35}{11} x^{11} d \,e^{4} a \,b^{6}+\frac {21}{2} x^{10} d \,e^{4} a^{2} b^{5}+7 x^{10} d^{2} e^{3} a \,b^{6}+\frac {175}{9} x^{9} d \,e^{4} a^{3} b^{4}+\frac {1}{6} x^{6} e^{5} a^{7}+d^{5} a^{7} x +\frac {1}{13} b^{7} e^{5} x^{13}+\frac {1}{8} x^{8} d^{5} b^{7}\) | \(671\) |
| gosper | \(\frac {x \left (792 b^{7} e^{5} x^{12}+6006 x^{11} e^{5} a \,b^{6}+4290 x^{11} d \,e^{4} b^{7}+19656 x^{10} e^{5} a^{2} b^{5}+32760 x^{10} d \,e^{4} a \,b^{6}+9360 x^{10} d^{2} e^{3} b^{7}+36036 x^{9} e^{5} a^{3} b^{4}+108108 x^{9} d \,e^{4} a^{2} b^{5}+72072 x^{9} d^{2} e^{3} a \,b^{6}+10296 x^{9} d^{3} e^{2} b^{7}+40040 x^{8} e^{5} a^{4} b^{3}+200200 x^{8} d \,e^{4} a^{3} b^{4}+240240 x^{8} d^{2} e^{3} a^{2} b^{5}+80080 x^{8} d^{3} e^{2} a \,b^{6}+5720 x^{8} d^{4} e \,b^{7}+27027 x^{7} e^{5} a^{5} b^{2}+225225 x^{7} d \,e^{4} a^{4} b^{3}+450450 x^{7} d^{2} e^{3} a^{3} b^{4}+270270 x^{7} d^{3} e^{2} a^{2} b^{5}+45045 x^{7} d^{4} e a \,b^{6}+1287 x^{7} d^{5} b^{7}+10296 a^{6} b \,e^{5} x^{6}+154440 a^{5} b^{2} d \,e^{4} x^{6}+514800 a^{4} b^{3} d^{2} e^{3} x^{6}+514800 a^{3} b^{4} d^{3} e^{2} x^{6}+154440 a^{2} b^{5} d^{4} e \,x^{6}+10296 a \,b^{6} d^{5} x^{6}+1716 x^{5} e^{5} a^{7}+60060 x^{5} d \,e^{4} a^{6} b +360360 x^{5} d^{2} e^{3} a^{5} b^{2}+600600 x^{5} d^{3} e^{2} a^{4} b^{3}+300300 x^{5} d^{4} e \,a^{3} b^{4}+36036 x^{5} d^{5} a^{2} b^{5}+10296 a^{7} d \,e^{4} x^{4}+144144 a^{6} b \,d^{2} e^{3} x^{4}+432432 a^{5} b^{2} d^{3} e^{2} x^{4}+360360 a^{4} b^{3} d^{4} e \,x^{4}+72072 a^{3} b^{4} d^{5} x^{4}+25740 x^{3} d^{2} e^{3} a^{7}+180180 x^{3} d^{3} e^{2} a^{6} b +270270 x^{3} d^{4} e \,a^{5} b^{2}+90090 x^{3} d^{5} a^{4} b^{3}+34320 x^{2} d^{3} e^{2} a^{7}+120120 x^{2} d^{4} e \,a^{6} b +72072 x^{2} d^{5} a^{5} b^{2}+25740 x \,d^{4} e \,a^{7}+36036 x \,d^{5} a^{6} b +10296 d^{5} a^{7}\right )}{10296}\) | \(674\) |
| orering | \(\frac {x \left (792 b^{7} e^{5} x^{12}+6006 x^{11} e^{5} a \,b^{6}+4290 x^{11} d \,e^{4} b^{7}+19656 x^{10} e^{5} a^{2} b^{5}+32760 x^{10} d \,e^{4} a \,b^{6}+9360 x^{10} d^{2} e^{3} b^{7}+36036 x^{9} e^{5} a^{3} b^{4}+108108 x^{9} d \,e^{4} a^{2} b^{5}+72072 x^{9} d^{2} e^{3} a \,b^{6}+10296 x^{9} d^{3} e^{2} b^{7}+40040 x^{8} e^{5} a^{4} b^{3}+200200 x^{8} d \,e^{4} a^{3} b^{4}+240240 x^{8} d^{2} e^{3} a^{2} b^{5}+80080 x^{8} d^{3} e^{2} a \,b^{6}+5720 x^{8} d^{4} e \,b^{7}+27027 x^{7} e^{5} a^{5} b^{2}+225225 x^{7} d \,e^{4} a^{4} b^{3}+450450 x^{7} d^{2} e^{3} a^{3} b^{4}+270270 x^{7} d^{3} e^{2} a^{2} b^{5}+45045 x^{7} d^{4} e a \,b^{6}+1287 x^{7} d^{5} b^{7}+10296 a^{6} b \,e^{5} x^{6}+154440 a^{5} b^{2} d \,e^{4} x^{6}+514800 a^{4} b^{3} d^{2} e^{3} x^{6}+514800 a^{3} b^{4} d^{3} e^{2} x^{6}+154440 a^{2} b^{5} d^{4} e \,x^{6}+10296 a \,b^{6} d^{5} x^{6}+1716 x^{5} e^{5} a^{7}+60060 x^{5} d \,e^{4} a^{6} b +360360 x^{5} d^{2} e^{3} a^{5} b^{2}+600600 x^{5} d^{3} e^{2} a^{4} b^{3}+300300 x^{5} d^{4} e \,a^{3} b^{4}+36036 x^{5} d^{5} a^{2} b^{5}+10296 a^{7} d \,e^{4} x^{4}+144144 a^{6} b \,d^{2} e^{3} x^{4}+432432 a^{5} b^{2} d^{3} e^{2} x^{4}+360360 a^{4} b^{3} d^{4} e \,x^{4}+72072 a^{3} b^{4} d^{5} x^{4}+25740 x^{3} d^{2} e^{3} a^{7}+180180 x^{3} d^{3} e^{2} a^{6} b +270270 x^{3} d^{4} e \,a^{5} b^{2}+90090 x^{3} d^{5} a^{4} b^{3}+34320 x^{2} d^{3} e^{2} a^{7}+120120 x^{2} d^{4} e \,a^{6} b +72072 x^{2} d^{5} a^{5} b^{2}+25740 x \,d^{4} e \,a^{7}+36036 x \,d^{5} a^{6} b +10296 d^{5} a^{7}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{10296 \left (b x +a \right )^{6}}\) | \(699\) |
| default | \(\frac {b^{7} e^{5} x^{13}}{13}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{6}+6 e^{5} a \,b^{6}\right ) x^{12}}{12}+\frac {\left (\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{6}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a \,b^{5}+15 e^{5} a^{2} b^{5}\right ) x^{11}}{11}+\frac {\left (\left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) b^{6}+6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a \,b^{5}+15 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2} b^{4}+20 e^{5} a^{3} b^{4}\right ) x^{10}}{10}+\frac {\left (\left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) b^{6}+6 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a \,b^{5}+15 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2} b^{4}+20 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{3} b^{3}+15 e^{5} a^{4} b^{3}\right ) x^{9}}{9}+\frac {\left (\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{6}+6 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a \,b^{5}+15 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{2} b^{4}+20 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{3} b^{3}+15 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{4} b^{2}+6 e^{5} a^{5} b^{2}\right ) x^{8}}{8}+\frac {\left (d^{5} a \,b^{6}+6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a \,b^{5}+15 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{2} b^{4}+20 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{3} b^{3}+15 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{4} b^{2}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{5} b +e^{5} a^{6} b \right ) x^{7}}{7}+\frac {\left (6 d^{5} a^{2} b^{5}+15 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2} b^{4}+20 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{3} b^{3}+15 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{4} b^{2}+6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{5} b +\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{6}\right ) x^{6}}{6}+\frac {\left (15 d^{5} a^{3} b^{4}+20 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{3} b^{3}+15 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{4} b^{2}+6 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{5} b +\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{6}\right ) x^{5}}{5}+\frac {\left (20 d^{5} a^{4} b^{3}+15 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{4} b^{2}+6 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{5} b +\left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{6}\right ) x^{4}}{4}+\frac {\left (15 d^{5} a^{5} b^{2}+6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{5} b +\left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{6}\right ) x^{3}}{3}+\frac {\left (6 d^{5} a^{6} b +\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{6}\right ) x^{2}}{2}+d^{5} a^{7} x\) | \(982\) |
Input:
int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/13*b^7*e^5*x^13+(7/12*e^5*a*b^6+5/12*d*e^4*b^7)*x^12+(21/11*e^5*a^2*b^5+ 35/11*d*e^4*a*b^6+10/11*d^2*e^3*b^7)*x^11+(7/2*e^5*a^3*b^4+21/2*d*e^4*a^2* b^5+7*d^2*e^3*a*b^6+d^3*e^2*b^7)*x^10+(35/9*e^5*a^4*b^3+175/9*d*e^4*a^3*b^ 4+70/3*d^2*e^3*a^2*b^5+70/9*d^3*e^2*a*b^6+5/9*d^4*e*b^7)*x^9+(21/8*e^5*a^5 *b^2+175/8*d*e^4*a^4*b^3+175/4*d^2*e^3*a^3*b^4+105/4*d^3*e^2*a^2*b^5+35/8* d^4*e*a*b^6+1/8*d^5*b^7)*x^8+(a^6*b*e^5+15*a^5*b^2*d*e^4+50*a^4*b^3*d^2*e^ 3+50*a^3*b^4*d^3*e^2+15*a^2*b^5*d^4*e+a*b^6*d^5)*x^7+(1/6*e^5*a^7+35/6*d*e ^4*a^6*b+35*d^2*e^3*a^5*b^2+175/3*d^3*e^2*a^4*b^3+175/6*d^4*e*a^3*b^4+7/2* d^5*a^2*b^5)*x^6+(a^7*d*e^4+14*a^6*b*d^2*e^3+42*a^5*b^2*d^3*e^2+35*a^4*b^3 *d^4*e+7*a^3*b^4*d^5)*x^5+(5/2*d^2*e^3*a^7+35/2*d^3*e^2*a^6*b+105/4*d^4*e* a^5*b^2+35/4*d^5*a^4*b^3)*x^4+(10/3*d^3*e^2*a^7+35/3*d^4*e*a^6*b+7*d^5*a^5 *b^2)*x^3+(5/2*d^4*e*a^7+7/2*d^5*a^6*b)*x^2+d^5*a^7*x
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (133) = 266\).
Time = 0.07 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{7} e^{5} x^{13} + a^{7} d^{5} x + \frac {1}{12} \, {\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} + {\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} + {\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/13*b^7*e^5*x^13 + a^7*d^5*x + 1/12*(5*b^7*d*e^4 + 7*a*b^6*e^5)*x^12 + 1/ 11*(10*b^7*d^2*e^3 + 35*a*b^6*d*e^4 + 21*a^2*b^5*e^5)*x^11 + 1/2*(2*b^7*d^ 3*e^2 + 14*a*b^6*d^2*e^3 + 21*a^2*b^5*d*e^4 + 7*a^3*b^4*e^5)*x^10 + 5/9*(b ^7*d^4*e + 14*a*b^6*d^3*e^2 + 42*a^2*b^5*d^2*e^3 + 35*a^3*b^4*d*e^4 + 7*a^ 4*b^3*e^5)*x^9 + 1/8*(b^7*d^5 + 35*a*b^6*d^4*e + 210*a^2*b^5*d^3*e^2 + 350 *a^3*b^4*d^2*e^3 + 175*a^4*b^3*d*e^4 + 21*a^5*b^2*e^5)*x^8 + (a*b^6*d^5 + 15*a^2*b^5*d^4*e + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3 + 15*a^5*b^2*d* e^4 + a^6*b*e^5)*x^7 + 1/6*(21*a^2*b^5*d^5 + 175*a^3*b^4*d^4*e + 350*a^4*b ^3*d^3*e^2 + 210*a^5*b^2*d^2*e^3 + 35*a^6*b*d*e^4 + a^7*e^5)*x^6 + (7*a^3* b^4*d^5 + 35*a^4*b^3*d^4*e + 42*a^5*b^2*d^3*e^2 + 14*a^6*b*d^2*e^3 + a^7*d *e^4)*x^5 + 5/4*(7*a^4*b^3*d^5 + 21*a^5*b^2*d^4*e + 14*a^6*b*d^3*e^2 + 2*a ^7*d^2*e^3)*x^4 + 1/3*(21*a^5*b^2*d^5 + 35*a^6*b*d^4*e + 10*a^7*d^3*e^2)*x ^3 + 1/2*(7*a^6*b*d^5 + 5*a^7*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (129) = 258\).
Time = 0.08 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.71 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{7} d^{5} x + \frac {b^{7} e^{5} x^{13}}{13} + x^{12} \cdot \left (\frac {7 a b^{6} e^{5}}{12} + \frac {5 b^{7} d e^{4}}{12}\right ) + x^{11} \cdot \left (\frac {21 a^{2} b^{5} e^{5}}{11} + \frac {35 a b^{6} d e^{4}}{11} + \frac {10 b^{7} d^{2} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {7 a^{3} b^{4} e^{5}}{2} + \frac {21 a^{2} b^{5} d e^{4}}{2} + 7 a b^{6} d^{2} e^{3} + b^{7} d^{3} e^{2}\right ) + x^{9} \cdot \left (\frac {35 a^{4} b^{3} e^{5}}{9} + \frac {175 a^{3} b^{4} d e^{4}}{9} + \frac {70 a^{2} b^{5} d^{2} e^{3}}{3} + \frac {70 a b^{6} d^{3} e^{2}}{9} + \frac {5 b^{7} d^{4} e}{9}\right ) + x^{8} \cdot \left (\frac {21 a^{5} b^{2} e^{5}}{8} + \frac {175 a^{4} b^{3} d e^{4}}{8} + \frac {175 a^{3} b^{4} d^{2} e^{3}}{4} + \frac {105 a^{2} b^{5} d^{3} e^{2}}{4} + \frac {35 a b^{6} d^{4} e}{8} + \frac {b^{7} d^{5}}{8}\right ) + x^{7} \left (a^{6} b e^{5} + 15 a^{5} b^{2} d e^{4} + 50 a^{4} b^{3} d^{2} e^{3} + 50 a^{3} b^{4} d^{3} e^{2} + 15 a^{2} b^{5} d^{4} e + a b^{6} d^{5}\right ) + x^{6} \left (\frac {a^{7} e^{5}}{6} + \frac {35 a^{6} b d e^{4}}{6} + 35 a^{5} b^{2} d^{2} e^{3} + \frac {175 a^{4} b^{3} d^{3} e^{2}}{3} + \frac {175 a^{3} b^{4} d^{4} e}{6} + \frac {7 a^{2} b^{5} d^{5}}{2}\right ) + x^{5} \left (a^{7} d e^{4} + 14 a^{6} b d^{2} e^{3} + 42 a^{5} b^{2} d^{3} e^{2} + 35 a^{4} b^{3} d^{4} e + 7 a^{3} b^{4} d^{5}\right ) + x^{4} \cdot \left (\frac {5 a^{7} d^{2} e^{3}}{2} + \frac {35 a^{6} b d^{3} e^{2}}{2} + \frac {105 a^{5} b^{2} d^{4} e}{4} + \frac {35 a^{4} b^{3} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{7} d^{3} e^{2}}{3} + \frac {35 a^{6} b d^{4} e}{3} + 7 a^{5} b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{7} d^{4} e}{2} + \frac {7 a^{6} b d^{5}}{2}\right ) \] Input:
integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
a**7*d**5*x + b**7*e**5*x**13/13 + x**12*(7*a*b**6*e**5/12 + 5*b**7*d*e**4 /12) + x**11*(21*a**2*b**5*e**5/11 + 35*a*b**6*d*e**4/11 + 10*b**7*d**2*e* *3/11) + x**10*(7*a**3*b**4*e**5/2 + 21*a**2*b**5*d*e**4/2 + 7*a*b**6*d**2 *e**3 + b**7*d**3*e**2) + x**9*(35*a**4*b**3*e**5/9 + 175*a**3*b**4*d*e**4 /9 + 70*a**2*b**5*d**2*e**3/3 + 70*a*b**6*d**3*e**2/9 + 5*b**7*d**4*e/9) + x**8*(21*a**5*b**2*e**5/8 + 175*a**4*b**3*d*e**4/8 + 175*a**3*b**4*d**2*e **3/4 + 105*a**2*b**5*d**3*e**2/4 + 35*a*b**6*d**4*e/8 + b**7*d**5/8) + x* *7*(a**6*b*e**5 + 15*a**5*b**2*d*e**4 + 50*a**4*b**3*d**2*e**3 + 50*a**3*b **4*d**3*e**2 + 15*a**2*b**5*d**4*e + a*b**6*d**5) + x**6*(a**7*e**5/6 + 3 5*a**6*b*d*e**4/6 + 35*a**5*b**2*d**2*e**3 + 175*a**4*b**3*d**3*e**2/3 + 1 75*a**3*b**4*d**4*e/6 + 7*a**2*b**5*d**5/2) + x**5*(a**7*d*e**4 + 14*a**6* b*d**2*e**3 + 42*a**5*b**2*d**3*e**2 + 35*a**4*b**3*d**4*e + 7*a**3*b**4*d **5) + x**4*(5*a**7*d**2*e**3/2 + 35*a**6*b*d**3*e**2/2 + 105*a**5*b**2*d* *4*e/4 + 35*a**4*b**3*d**5/4) + x**3*(10*a**7*d**3*e**2/3 + 35*a**6*b*d**4 *e/3 + 7*a**5*b**2*d**5) + x**2*(5*a**7*d**4*e/2 + 7*a**6*b*d**5/2)
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (133) = 266\).
Time = 0.04 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{7} e^{5} x^{13} + a^{7} d^{5} x + \frac {1}{12} \, {\left (5 \, b^{7} d e^{4} + 7 \, a b^{6} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, b^{7} d^{2} e^{3} + 35 \, a b^{6} d e^{4} + 21 \, a^{2} b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{7} d^{3} e^{2} + 14 \, a b^{6} d^{2} e^{3} + 21 \, a^{2} b^{5} d e^{4} + 7 \, a^{3} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (b^{7} d^{4} e + 14 \, a b^{6} d^{3} e^{2} + 42 \, a^{2} b^{5} d^{2} e^{3} + 35 \, a^{3} b^{4} d e^{4} + 7 \, a^{4} b^{3} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{5} + 35 \, a b^{6} d^{4} e + 210 \, a^{2} b^{5} d^{3} e^{2} + 350 \, a^{3} b^{4} d^{2} e^{3} + 175 \, a^{4} b^{3} d e^{4} + 21 \, a^{5} b^{2} e^{5}\right )} x^{8} + {\left (a b^{6} d^{5} + 15 \, a^{2} b^{5} d^{4} e + 50 \, a^{3} b^{4} d^{3} e^{2} + 50 \, a^{4} b^{3} d^{2} e^{3} + 15 \, a^{5} b^{2} d e^{4} + a^{6} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (21 \, a^{2} b^{5} d^{5} + 175 \, a^{3} b^{4} d^{4} e + 350 \, a^{4} b^{3} d^{3} e^{2} + 210 \, a^{5} b^{2} d^{2} e^{3} + 35 \, a^{6} b d e^{4} + a^{7} e^{5}\right )} x^{6} + {\left (7 \, a^{3} b^{4} d^{5} + 35 \, a^{4} b^{3} d^{4} e + 42 \, a^{5} b^{2} d^{3} e^{2} + 14 \, a^{6} b d^{2} e^{3} + a^{7} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (7 \, a^{4} b^{3} d^{5} + 21 \, a^{5} b^{2} d^{4} e + 14 \, a^{6} b d^{3} e^{2} + 2 \, a^{7} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (21 \, a^{5} b^{2} d^{5} + 35 \, a^{6} b d^{4} e + 10 \, a^{7} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{5} + 5 \, a^{7} d^{4} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
1/13*b^7*e^5*x^13 + a^7*d^5*x + 1/12*(5*b^7*d*e^4 + 7*a*b^6*e^5)*x^12 + 1/ 11*(10*b^7*d^2*e^3 + 35*a*b^6*d*e^4 + 21*a^2*b^5*e^5)*x^11 + 1/2*(2*b^7*d^ 3*e^2 + 14*a*b^6*d^2*e^3 + 21*a^2*b^5*d*e^4 + 7*a^3*b^4*e^5)*x^10 + 5/9*(b ^7*d^4*e + 14*a*b^6*d^3*e^2 + 42*a^2*b^5*d^2*e^3 + 35*a^3*b^4*d*e^4 + 7*a^ 4*b^3*e^5)*x^9 + 1/8*(b^7*d^5 + 35*a*b^6*d^4*e + 210*a^2*b^5*d^3*e^2 + 350 *a^3*b^4*d^2*e^3 + 175*a^4*b^3*d*e^4 + 21*a^5*b^2*e^5)*x^8 + (a*b^6*d^5 + 15*a^2*b^5*d^4*e + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3 + 15*a^5*b^2*d* e^4 + a^6*b*e^5)*x^7 + 1/6*(21*a^2*b^5*d^5 + 175*a^3*b^4*d^4*e + 350*a^4*b ^3*d^3*e^2 + 210*a^5*b^2*d^2*e^3 + 35*a^6*b*d*e^4 + a^7*e^5)*x^6 + (7*a^3* b^4*d^5 + 35*a^4*b^3*d^4*e + 42*a^5*b^2*d^3*e^2 + 14*a^6*b*d^2*e^3 + a^7*d *e^4)*x^5 + 5/4*(7*a^4*b^3*d^5 + 21*a^5*b^2*d^4*e + 14*a^6*b*d^3*e^2 + 2*a ^7*d^2*e^3)*x^4 + 1/3*(21*a^5*b^2*d^5 + 35*a^6*b*d^4*e + 10*a^7*d^3*e^2)*x ^3 + 1/2*(7*a^6*b*d^5 + 5*a^7*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (133) = 266\).
Time = 0.20 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.69 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{7} e^{5} x^{13} + \frac {5}{12} \, b^{7} d e^{4} x^{12} + \frac {7}{12} \, a b^{6} e^{5} x^{12} + \frac {10}{11} \, b^{7} d^{2} e^{3} x^{11} + \frac {35}{11} \, a b^{6} d e^{4} x^{11} + \frac {21}{11} \, a^{2} b^{5} e^{5} x^{11} + b^{7} d^{3} e^{2} x^{10} + 7 \, a b^{6} d^{2} e^{3} x^{10} + \frac {21}{2} \, a^{2} b^{5} d e^{4} x^{10} + \frac {7}{2} \, a^{3} b^{4} e^{5} x^{10} + \frac {5}{9} \, b^{7} d^{4} e x^{9} + \frac {70}{9} \, a b^{6} d^{3} e^{2} x^{9} + \frac {70}{3} \, a^{2} b^{5} d^{2} e^{3} x^{9} + \frac {175}{9} \, a^{3} b^{4} d e^{4} x^{9} + \frac {35}{9} \, a^{4} b^{3} e^{5} x^{9} + \frac {1}{8} \, b^{7} d^{5} x^{8} + \frac {35}{8} \, a b^{6} d^{4} e x^{8} + \frac {105}{4} \, a^{2} b^{5} d^{3} e^{2} x^{8} + \frac {175}{4} \, a^{3} b^{4} d^{2} e^{3} x^{8} + \frac {175}{8} \, a^{4} b^{3} d e^{4} x^{8} + \frac {21}{8} \, a^{5} b^{2} e^{5} x^{8} + a b^{6} d^{5} x^{7} + 15 \, a^{2} b^{5} d^{4} e x^{7} + 50 \, a^{3} b^{4} d^{3} e^{2} x^{7} + 50 \, a^{4} b^{3} d^{2} e^{3} x^{7} + 15 \, a^{5} b^{2} d e^{4} x^{7} + a^{6} b e^{5} x^{7} + \frac {7}{2} \, a^{2} b^{5} d^{5} x^{6} + \frac {175}{6} \, a^{3} b^{4} d^{4} e x^{6} + \frac {175}{3} \, a^{4} b^{3} d^{3} e^{2} x^{6} + 35 \, a^{5} b^{2} d^{2} e^{3} x^{6} + \frac {35}{6} \, a^{6} b d e^{4} x^{6} + \frac {1}{6} \, a^{7} e^{5} x^{6} + 7 \, a^{3} b^{4} d^{5} x^{5} + 35 \, a^{4} b^{3} d^{4} e x^{5} + 42 \, a^{5} b^{2} d^{3} e^{2} x^{5} + 14 \, a^{6} b d^{2} e^{3} x^{5} + a^{7} d e^{4} x^{5} + \frac {35}{4} \, a^{4} b^{3} d^{5} x^{4} + \frac {105}{4} \, a^{5} b^{2} d^{4} e x^{4} + \frac {35}{2} \, a^{6} b d^{3} e^{2} x^{4} + \frac {5}{2} \, a^{7} d^{2} e^{3} x^{4} + 7 \, a^{5} b^{2} d^{5} x^{3} + \frac {35}{3} \, a^{6} b d^{4} e x^{3} + \frac {10}{3} \, a^{7} d^{3} e^{2} x^{3} + \frac {7}{2} \, a^{6} b d^{5} x^{2} + \frac {5}{2} \, a^{7} d^{4} e x^{2} + a^{7} d^{5} x \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
1/13*b^7*e^5*x^13 + 5/12*b^7*d*e^4*x^12 + 7/12*a*b^6*e^5*x^12 + 10/11*b^7* d^2*e^3*x^11 + 35/11*a*b^6*d*e^4*x^11 + 21/11*a^2*b^5*e^5*x^11 + b^7*d^3*e ^2*x^10 + 7*a*b^6*d^2*e^3*x^10 + 21/2*a^2*b^5*d*e^4*x^10 + 7/2*a^3*b^4*e^5 *x^10 + 5/9*b^7*d^4*e*x^9 + 70/9*a*b^6*d^3*e^2*x^9 + 70/3*a^2*b^5*d^2*e^3* x^9 + 175/9*a^3*b^4*d*e^4*x^9 + 35/9*a^4*b^3*e^5*x^9 + 1/8*b^7*d^5*x^8 + 3 5/8*a*b^6*d^4*e*x^8 + 105/4*a^2*b^5*d^3*e^2*x^8 + 175/4*a^3*b^4*d^2*e^3*x^ 8 + 175/8*a^4*b^3*d*e^4*x^8 + 21/8*a^5*b^2*e^5*x^8 + a*b^6*d^5*x^7 + 15*a^ 2*b^5*d^4*e*x^7 + 50*a^3*b^4*d^3*e^2*x^7 + 50*a^4*b^3*d^2*e^3*x^7 + 15*a^5 *b^2*d*e^4*x^7 + a^6*b*e^5*x^7 + 7/2*a^2*b^5*d^5*x^6 + 175/6*a^3*b^4*d^4*e *x^6 + 175/3*a^4*b^3*d^3*e^2*x^6 + 35*a^5*b^2*d^2*e^3*x^6 + 35/6*a^6*b*d*e ^4*x^6 + 1/6*a^7*e^5*x^6 + 7*a^3*b^4*d^5*x^5 + 35*a^4*b^3*d^4*e*x^5 + 42*a ^5*b^2*d^3*e^2*x^5 + 14*a^6*b*d^2*e^3*x^5 + a^7*d*e^4*x^5 + 35/4*a^4*b^3*d ^5*x^4 + 105/4*a^5*b^2*d^4*e*x^4 + 35/2*a^6*b*d^3*e^2*x^4 + 5/2*a^7*d^2*e^ 3*x^4 + 7*a^5*b^2*d^5*x^3 + 35/3*a^6*b*d^4*e*x^3 + 10/3*a^7*d^3*e^2*x^3 + 7/2*a^6*b*d^5*x^2 + 5/2*a^7*d^4*e*x^2 + a^7*d^5*x
Time = 0.24 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.99 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (a^7\,d\,e^4+14\,a^6\,b\,d^2\,e^3+42\,a^5\,b^2\,d^3\,e^2+35\,a^4\,b^3\,d^4\,e+7\,a^3\,b^4\,d^5\right )+x^9\,\left (\frac {35\,a^4\,b^3\,e^5}{9}+\frac {175\,a^3\,b^4\,d\,e^4}{9}+\frac {70\,a^2\,b^5\,d^2\,e^3}{3}+\frac {70\,a\,b^6\,d^3\,e^2}{9}+\frac {5\,b^7\,d^4\,e}{9}\right )+x^7\,\left (a^6\,b\,e^5+15\,a^5\,b^2\,d\,e^4+50\,a^4\,b^3\,d^2\,e^3+50\,a^3\,b^4\,d^3\,e^2+15\,a^2\,b^5\,d^4\,e+a\,b^6\,d^5\right )+x^6\,\left (\frac {a^7\,e^5}{6}+\frac {35\,a^6\,b\,d\,e^4}{6}+35\,a^5\,b^2\,d^2\,e^3+\frac {175\,a^4\,b^3\,d^3\,e^2}{3}+\frac {175\,a^3\,b^4\,d^4\,e}{6}+\frac {7\,a^2\,b^5\,d^5}{2}\right )+x^8\,\left (\frac {21\,a^5\,b^2\,e^5}{8}+\frac {175\,a^4\,b^3\,d\,e^4}{8}+\frac {175\,a^3\,b^4\,d^2\,e^3}{4}+\frac {105\,a^2\,b^5\,d^3\,e^2}{4}+\frac {35\,a\,b^6\,d^4\,e}{8}+\frac {b^7\,d^5}{8}\right )+a^7\,d^5\,x+\frac {b^7\,e^5\,x^{13}}{13}+\frac {5\,a^4\,d^2\,x^4\,\left (2\,a^3\,e^3+14\,a^2\,b\,d\,e^2+21\,a\,b^2\,d^2\,e+7\,b^3\,d^3\right )}{4}+\frac {b^4\,e^2\,x^{10}\,\left (7\,a^3\,e^3+21\,a^2\,b\,d\,e^2+14\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+\frac {a^6\,d^4\,x^2\,\left (5\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^4\,x^{12}\,\left (7\,a\,e+5\,b\,d\right )}{12}+\frac {a^5\,d^3\,x^3\,\left (10\,a^2\,e^2+35\,a\,b\,d\,e+21\,b^2\,d^2\right )}{3}+\frac {b^5\,e^3\,x^{11}\,\left (21\,a^2\,e^2+35\,a\,b\,d\,e+10\,b^2\,d^2\right )}{11} \] Input:
int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
x^5*(a^7*d*e^4 + 7*a^3*b^4*d^5 + 35*a^4*b^3*d^4*e + 14*a^6*b*d^2*e^3 + 42* a^5*b^2*d^3*e^2) + x^9*((5*b^7*d^4*e)/9 + (35*a^4*b^3*e^5)/9 + (70*a*b^6*d ^3*e^2)/9 + (175*a^3*b^4*d*e^4)/9 + (70*a^2*b^5*d^2*e^3)/3) + x^7*(a*b^6*d ^5 + a^6*b*e^5 + 15*a^2*b^5*d^4*e + 15*a^5*b^2*d*e^4 + 50*a^3*b^4*d^3*e^2 + 50*a^4*b^3*d^2*e^3) + x^6*((a^7*e^5)/6 + (7*a^2*b^5*d^5)/2 + (175*a^3*b^ 4*d^4*e)/6 + (175*a^4*b^3*d^3*e^2)/3 + 35*a^5*b^2*d^2*e^3 + (35*a^6*b*d*e^ 4)/6) + x^8*((b^7*d^5)/8 + (21*a^5*b^2*e^5)/8 + (175*a^4*b^3*d*e^4)/8 + (1 05*a^2*b^5*d^3*e^2)/4 + (175*a^3*b^4*d^2*e^3)/4 + (35*a*b^6*d^4*e)/8) + a^ 7*d^5*x + (b^7*e^5*x^13)/13 + (5*a^4*d^2*x^4*(2*a^3*e^3 + 7*b^3*d^3 + 21*a *b^2*d^2*e + 14*a^2*b*d*e^2))/4 + (b^4*e^2*x^10*(7*a^3*e^3 + 2*b^3*d^3 + 1 4*a*b^2*d^2*e + 21*a^2*b*d*e^2))/2 + (a^6*d^4*x^2*(5*a*e + 7*b*d))/2 + (b^ 6*e^4*x^12*(7*a*e + 5*b*d))/12 + (a^5*d^3*x^3*(10*a^2*e^2 + 21*b^2*d^2 + 3 5*a*b*d*e))/3 + (b^5*e^3*x^11*(21*a^2*e^2 + 10*b^2*d^2 + 35*a*b*d*e))/11
Time = 0.28 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.71 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x \left (792 b^{7} e^{5} x^{12}+6006 a \,b^{6} e^{5} x^{11}+4290 b^{7} d \,e^{4} x^{11}+19656 a^{2} b^{5} e^{5} x^{10}+32760 a \,b^{6} d \,e^{4} x^{10}+9360 b^{7} d^{2} e^{3} x^{10}+36036 a^{3} b^{4} e^{5} x^{9}+108108 a^{2} b^{5} d \,e^{4} x^{9}+72072 a \,b^{6} d^{2} e^{3} x^{9}+10296 b^{7} d^{3} e^{2} x^{9}+40040 a^{4} b^{3} e^{5} x^{8}+200200 a^{3} b^{4} d \,e^{4} x^{8}+240240 a^{2} b^{5} d^{2} e^{3} x^{8}+80080 a \,b^{6} d^{3} e^{2} x^{8}+5720 b^{7} d^{4} e \,x^{8}+27027 a^{5} b^{2} e^{5} x^{7}+225225 a^{4} b^{3} d \,e^{4} x^{7}+450450 a^{3} b^{4} d^{2} e^{3} x^{7}+270270 a^{2} b^{5} d^{3} e^{2} x^{7}+45045 a \,b^{6} d^{4} e \,x^{7}+1287 b^{7} d^{5} x^{7}+10296 a^{6} b \,e^{5} x^{6}+154440 a^{5} b^{2} d \,e^{4} x^{6}+514800 a^{4} b^{3} d^{2} e^{3} x^{6}+514800 a^{3} b^{4} d^{3} e^{2} x^{6}+154440 a^{2} b^{5} d^{4} e \,x^{6}+10296 a \,b^{6} d^{5} x^{6}+1716 a^{7} e^{5} x^{5}+60060 a^{6} b d \,e^{4} x^{5}+360360 a^{5} b^{2} d^{2} e^{3} x^{5}+600600 a^{4} b^{3} d^{3} e^{2} x^{5}+300300 a^{3} b^{4} d^{4} e \,x^{5}+36036 a^{2} b^{5} d^{5} x^{5}+10296 a^{7} d \,e^{4} x^{4}+144144 a^{6} b \,d^{2} e^{3} x^{4}+432432 a^{5} b^{2} d^{3} e^{2} x^{4}+360360 a^{4} b^{3} d^{4} e \,x^{4}+72072 a^{3} b^{4} d^{5} x^{4}+25740 a^{7} d^{2} e^{3} x^{3}+180180 a^{6} b \,d^{3} e^{2} x^{3}+270270 a^{5} b^{2} d^{4} e \,x^{3}+90090 a^{4} b^{3} d^{5} x^{3}+34320 a^{7} d^{3} e^{2} x^{2}+120120 a^{6} b \,d^{4} e \,x^{2}+72072 a^{5} b^{2} d^{5} x^{2}+25740 a^{7} d^{4} e x +36036 a^{6} b \,d^{5} x +10296 a^{7} d^{5}\right )}{10296} \] Input:
int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
(x*(10296*a**7*d**5 + 25740*a**7*d**4*e*x + 34320*a**7*d**3*e**2*x**2 + 25 740*a**7*d**2*e**3*x**3 + 10296*a**7*d*e**4*x**4 + 1716*a**7*e**5*x**5 + 3 6036*a**6*b*d**5*x + 120120*a**6*b*d**4*e*x**2 + 180180*a**6*b*d**3*e**2*x **3 + 144144*a**6*b*d**2*e**3*x**4 + 60060*a**6*b*d*e**4*x**5 + 10296*a**6 *b*e**5*x**6 + 72072*a**5*b**2*d**5*x**2 + 270270*a**5*b**2*d**4*e*x**3 + 432432*a**5*b**2*d**3*e**2*x**4 + 360360*a**5*b**2*d**2*e**3*x**5 + 154440 *a**5*b**2*d*e**4*x**6 + 27027*a**5*b**2*e**5*x**7 + 90090*a**4*b**3*d**5* x**3 + 360360*a**4*b**3*d**4*e*x**4 + 600600*a**4*b**3*d**3*e**2*x**5 + 51 4800*a**4*b**3*d**2*e**3*x**6 + 225225*a**4*b**3*d*e**4*x**7 + 40040*a**4* b**3*e**5*x**8 + 72072*a**3*b**4*d**5*x**4 + 300300*a**3*b**4*d**4*e*x**5 + 514800*a**3*b**4*d**3*e**2*x**6 + 450450*a**3*b**4*d**2*e**3*x**7 + 2002 00*a**3*b**4*d*e**4*x**8 + 36036*a**3*b**4*e**5*x**9 + 36036*a**2*b**5*d** 5*x**5 + 154440*a**2*b**5*d**4*e*x**6 + 270270*a**2*b**5*d**3*e**2*x**7 + 240240*a**2*b**5*d**2*e**3*x**8 + 108108*a**2*b**5*d*e**4*x**9 + 19656*a** 2*b**5*e**5*x**10 + 10296*a*b**6*d**5*x**6 + 45045*a*b**6*d**4*e*x**7 + 80 080*a*b**6*d**3*e**2*x**8 + 72072*a*b**6*d**2*e**3*x**9 + 32760*a*b**6*d*e **4*x**10 + 6006*a*b**6*e**5*x**11 + 1287*b**7*d**5*x**7 + 5720*b**7*d**4* e*x**8 + 10296*b**7*d**3*e**2*x**9 + 9360*b**7*d**2*e**3*x**10 + 4290*b**7 *d*e**4*x**11 + 792*b**7*e**5*x**12))/10296