Integrand size = 31, antiderivative size = 118 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^5}{5 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^7}{7 b^4}+\frac {B e^2 (a+b x)^8}{8 b^4} \]
1/5*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^5/b^4+1/6*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B *b*d)*(b*x+a)^6/b^4+1/7*e*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^7/b^4+1/8*B*e^2* (b*x+a)^8/b^4
Leaf count is larger than twice the leaf count of optimal. \(288\) vs. \(2(118)=236\).
Time = 0.07 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 A d^2 x+\frac {1}{2} a^3 d (4 A b d+a B d+2 a A e) x^2+\frac {1}{3} a^2 \left (2 a B d (2 b d+a e)+A \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} a \left (4 A b \left (b^2 d^2+3 a b d e+a^2 e^2\right )+a B \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^4+\frac {1}{5} b \left (4 a B \left (b^2 d^2+3 a b d e+a^2 e^2\right )+A b \left (b^2 d^2+8 a b d e+6 a^2 e^2\right )\right ) x^5+\frac {1}{6} b^2 \left (6 a^2 B e^2+4 a b e (2 B d+A e)+b^2 d (B d+2 A e)\right ) x^6+\frac {1}{7} b^3 e (2 b B d+A b e+4 a B e) x^7+\frac {1}{8} b^4 B e^2 x^8 \]
a^4*A*d^2*x + (a^3*d*(4*A*b*d + a*B*d + 2*a*A*e)*x^2)/2 + (a^2*(2*a*B*d*(2 *b*d + a*e) + A*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^3)/3 + (a*(4*A*b*(b^2 *d^2 + 3*a*b*d*e + a^2*e^2) + a*B*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^4)/ 4 + (b*(4*a*B*(b^2*d^2 + 3*a*b*d*e + a^2*e^2) + A*b*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2))*x^5)/5 + (b^2*(6*a^2*B*e^2 + 4*a*b*e*(2*B*d + A*e) + b^2*d*(B *d + 2*A*e))*x^6)/6 + (b^3*e*(2*b*B*d + A*b*e + 4*a*B*e)*x^7)/7 + (b^4*B*e ^2*x^8)/8
Time = 0.42 (sec) , antiderivative size = 118, 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, 86, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^4 (A+B x) (d+e x)^2dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^4 (A+B x) (d+e x)^2dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{b^3}+\frac {(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{b^3}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^2}{b^3}+\frac {B e^2 (a+b x)^7}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac {(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac {(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac {B e^2 (a+b x)^8}{8 b^4}\) |
((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^5)/(5*b^4) + ((b*d - a*e)*(b*B*d + 2* A*b*e - 3*a*B*e)*(a + b*x)^6)/(6*b^4) + (e*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^7)/(7*b^4) + (B*e^2*(a + b*x)^8)/(8*b^4)
3.17.78.3.1 Defintions of rubi rules used
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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. \(304\) vs. \(2(110)=220\).
Time = 0.23 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(\frac {B \,e^{2} b^{4} x^{8}}{8}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) b^{4}+4 B \,e^{2} b^{3} a \right ) x^{7}}{7}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) b^{4}+4 \left (A \,e^{2}+2 B d e \right ) b^{3} a +6 B \,e^{2} b^{2} a^{2}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d^{2}+4 \left (2 A d e +B \,d^{2}\right ) b^{3} a +6 \left (A \,e^{2}+2 B d e \right ) b^{2} a^{2}+4 B \,e^{2} b \,a^{3}\right ) x^{5}}{5}+\frac {\left (4 A \,d^{2} b^{3} a +6 \left (2 A d e +B \,d^{2}\right ) b^{2} a^{2}+4 \left (A \,e^{2}+2 B d e \right ) b \,a^{3}+B \,e^{2} a^{4}\right ) x^{4}}{4}+\frac {\left (6 A \,d^{2} b^{2} a^{2}+4 \left (2 A d e +B \,d^{2}\right ) b \,a^{3}+\left (A \,e^{2}+2 B d e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 A \,d^{2} b \,a^{3}+\left (2 A d e +B \,d^{2}\right ) a^{4}\right ) x^{2}}{2}+A \,d^{2} a^{4} x\) | \(305\) |
| norman | \(\frac {B \,e^{2} b^{4} x^{8}}{8}+\left (\frac {1}{7} A \,b^{4} e^{2}+\frac {4}{7} B \,e^{2} b^{3} a +\frac {2}{7} B \,b^{4} d e \right ) x^{7}+\left (\frac {2}{3} A a \,b^{3} e^{2}+\frac {1}{3} A \,b^{4} d e +B \,e^{2} b^{2} a^{2}+\frac {4}{3} B a \,b^{3} d e +\frac {1}{6} B \,b^{4} d^{2}\right ) x^{6}+\left (\frac {6}{5} A \,a^{2} b^{2} e^{2}+\frac {8}{5} A a \,b^{3} d e +\frac {1}{5} A \,b^{4} d^{2}+\frac {4}{5} B \,e^{2} b \,a^{3}+\frac {12}{5} B \,a^{2} b^{2} d e +\frac {4}{5} B a \,b^{3} d^{2}\right ) x^{5}+\left (A \,a^{3} b \,e^{2}+3 A \,a^{2} b^{2} d e +A \,d^{2} b^{3} a +\frac {1}{4} B \,e^{2} a^{4}+2 B \,a^{3} b d e +\frac {3}{2} B \,a^{2} b^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{4} e^{2}+\frac {8}{3} A \,a^{3} b d e +2 A \,d^{2} b^{2} a^{2}+\frac {2}{3} B \,a^{4} d e +\frac {4}{3} B \,a^{3} b \,d^{2}\right ) x^{3}+\left (A \,a^{4} d e +2 A \,d^{2} b \,a^{3}+\frac {1}{2} B \,a^{4} d^{2}\right ) x^{2}+A \,d^{2} a^{4} x\) | \(321\) |
| risch | \(\frac {8}{3} x^{3} A \,a^{3} b d e +2 x^{4} B \,a^{3} b d e +3 x^{4} A \,a^{2} b^{2} d e +\frac {8}{5} x^{5} A a \,b^{3} d e +\frac {12}{5} x^{5} B \,a^{2} b^{2} d e +\frac {4}{3} x^{6} B a \,b^{3} d e +\frac {1}{7} x^{7} A \,b^{4} e^{2}+\frac {1}{6} x^{6} B \,b^{4} d^{2}+\frac {1}{5} x^{5} A \,b^{4} d^{2}+\frac {1}{4} x^{4} B \,e^{2} a^{4}+\frac {1}{3} x^{3} A \,a^{4} e^{2}+\frac {1}{2} x^{2} B \,a^{4} d^{2}+\frac {1}{8} B \,e^{2} b^{4} x^{8}+A \,d^{2} a^{4} x +\frac {4}{7} x^{7} B \,e^{2} b^{3} a +\frac {2}{7} x^{7} B \,b^{4} d e +\frac {2}{3} x^{6} A a \,b^{3} e^{2}+\frac {1}{3} x^{6} A \,b^{4} d e +x^{6} B \,e^{2} b^{2} a^{2}+\frac {6}{5} x^{5} A \,a^{2} b^{2} e^{2}+\frac {4}{5} x^{5} B \,e^{2} b \,a^{3}+\frac {4}{5} x^{5} B a \,b^{3} d^{2}+x^{4} A \,a^{3} b \,e^{2}+x^{4} A \,d^{2} b^{3} a +\frac {3}{2} x^{4} B \,a^{2} b^{2} d^{2}+2 x^{3} A \,d^{2} b^{2} a^{2}+\frac {2}{3} x^{3} B \,a^{4} d e +\frac {4}{3} x^{3} B \,a^{3} b \,d^{2}+x^{2} A \,a^{4} d e +2 x^{2} A \,d^{2} b \,a^{3}\) | \(375\) |
| parallelrisch | \(\frac {8}{3} x^{3} A \,a^{3} b d e +2 x^{4} B \,a^{3} b d e +3 x^{4} A \,a^{2} b^{2} d e +\frac {8}{5} x^{5} A a \,b^{3} d e +\frac {12}{5} x^{5} B \,a^{2} b^{2} d e +\frac {4}{3} x^{6} B a \,b^{3} d e +\frac {1}{7} x^{7} A \,b^{4} e^{2}+\frac {1}{6} x^{6} B \,b^{4} d^{2}+\frac {1}{5} x^{5} A \,b^{4} d^{2}+\frac {1}{4} x^{4} B \,e^{2} a^{4}+\frac {1}{3} x^{3} A \,a^{4} e^{2}+\frac {1}{2} x^{2} B \,a^{4} d^{2}+\frac {1}{8} B \,e^{2} b^{4} x^{8}+A \,d^{2} a^{4} x +\frac {4}{7} x^{7} B \,e^{2} b^{3} a +\frac {2}{7} x^{7} B \,b^{4} d e +\frac {2}{3} x^{6} A a \,b^{3} e^{2}+\frac {1}{3} x^{6} A \,b^{4} d e +x^{6} B \,e^{2} b^{2} a^{2}+\frac {6}{5} x^{5} A \,a^{2} b^{2} e^{2}+\frac {4}{5} x^{5} B \,e^{2} b \,a^{3}+\frac {4}{5} x^{5} B a \,b^{3} d^{2}+x^{4} A \,a^{3} b \,e^{2}+x^{4} A \,d^{2} b^{3} a +\frac {3}{2} x^{4} B \,a^{2} b^{2} d^{2}+2 x^{3} A \,d^{2} b^{2} a^{2}+\frac {2}{3} x^{3} B \,a^{4} d e +\frac {4}{3} x^{3} B \,a^{3} b \,d^{2}+x^{2} A \,a^{4} d e +2 x^{2} A \,d^{2} b \,a^{3}\) | \(375\) |
| gosper | \(\frac {x \left (105 B \,e^{2} b^{4} x^{7}+120 x^{6} A \,b^{4} e^{2}+480 x^{6} B \,e^{2} b^{3} a +240 x^{6} B \,b^{4} d e +560 x^{5} A a \,b^{3} e^{2}+280 x^{5} A \,b^{4} d e +840 x^{5} B \,e^{2} b^{2} a^{2}+1120 x^{5} B a \,b^{3} d e +140 x^{5} B \,b^{4} d^{2}+1008 x^{4} A \,a^{2} b^{2} e^{2}+1344 x^{4} A a \,b^{3} d e +168 x^{4} A \,b^{4} d^{2}+672 x^{4} B \,e^{2} b \,a^{3}+2016 x^{4} B \,a^{2} b^{2} d e +672 x^{4} B a \,b^{3} d^{2}+840 x^{3} A \,a^{3} b \,e^{2}+2520 x^{3} A \,a^{2} b^{2} d e +840 x^{3} A \,d^{2} b^{3} a +210 x^{3} B \,e^{2} a^{4}+1680 x^{3} B \,a^{3} b d e +1260 x^{3} B \,a^{2} b^{2} d^{2}+280 x^{2} A \,a^{4} e^{2}+2240 x^{2} A \,a^{3} b d e +1680 x^{2} A \,d^{2} b^{2} a^{2}+560 x^{2} B \,a^{4} d e +1120 x^{2} B \,a^{3} b \,d^{2}+840 x A \,a^{4} d e +1680 x A \,d^{2} b \,a^{3}+420 x B \,a^{4} d^{2}+840 A \,d^{2} a^{4}\right )}{840}\) | \(376\) |
1/8*B*e^2*b^4*x^8+1/7*((A*e^2+2*B*d*e)*b^4+4*B*e^2*b^3*a)*x^7+1/6*((2*A*d* e+B*d^2)*b^4+4*(A*e^2+2*B*d*e)*b^3*a+6*B*e^2*b^2*a^2)*x^6+1/5*(A*b^4*d^2+4 *(2*A*d*e+B*d^2)*b^3*a+6*(A*e^2+2*B*d*e)*b^2*a^2+4*B*e^2*b*a^3)*x^5+1/4*(4 *A*d^2*b^3*a+6*(2*A*d*e+B*d^2)*b^2*a^2+4*(A*e^2+2*B*d*e)*b*a^3+B*e^2*a^4)* x^4+1/3*(6*A*d^2*b^2*a^2+4*(2*A*d*e+B*d^2)*b*a^3+(A*e^2+2*B*d*e)*a^4)*x^3+ 1/2*(4*A*d^2*b*a^3+(2*A*d*e+B*d^2)*a^4)*x^2+A*d^2*a^4*x
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.73 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} e^{2} x^{8} + A a^{4} d^{2} x + \frac {1}{7} \, {\left (2 \, B b^{4} d e + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{4} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{4} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} x^{2} \]
1/8*B*b^4*e^2*x^8 + A*a^4*d^2*x + 1/7*(2*B*b^4*d*e + (4*B*a*b^3 + A*b^4)*e ^2)*x^7 + 1/6*(B*b^4*d^2 + 2*(4*B*a*b^3 + A*b^4)*d*e + 2*(3*B*a^2*b^2 + 2* A*a*b^3)*e^2)*x^6 + 1/5*((4*B*a*b^3 + A*b^4)*d^2 + 4*(3*B*a^2*b^2 + 2*A*a* b^3)*d*e + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^2)*x^5 + 1/4*(2*(3*B*a^2*b^2 + 2* A*a*b^3)*d^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e + (B*a^4 + 4*A*a^3*b)*e^2)* x^4 + 1/3*(A*a^4*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2 + 2*(B*a^4 + 4*A*a^ 3*b)*d*e)*x^3 + 1/2*(2*A*a^4*d*e + (B*a^4 + 4*A*a^3*b)*d^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (116) = 232\).
Time = 0.04 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.25 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=A a^{4} d^{2} x + \frac {B b^{4} e^{2} x^{8}}{8} + x^{7} \left (\frac {A b^{4} e^{2}}{7} + \frac {4 B a b^{3} e^{2}}{7} + \frac {2 B b^{4} d e}{7}\right ) + x^{6} \cdot \left (\frac {2 A a b^{3} e^{2}}{3} + \frac {A b^{4} d e}{3} + B a^{2} b^{2} e^{2} + \frac {4 B a b^{3} d e}{3} + \frac {B b^{4} d^{2}}{6}\right ) + x^{5} \cdot \left (\frac {6 A a^{2} b^{2} e^{2}}{5} + \frac {8 A a b^{3} d e}{5} + \frac {A b^{4} d^{2}}{5} + \frac {4 B a^{3} b e^{2}}{5} + \frac {12 B a^{2} b^{2} d e}{5} + \frac {4 B a b^{3} d^{2}}{5}\right ) + x^{4} \left (A a^{3} b e^{2} + 3 A a^{2} b^{2} d e + A a b^{3} d^{2} + \frac {B a^{4} e^{2}}{4} + 2 B a^{3} b d e + \frac {3 B a^{2} b^{2} d^{2}}{2}\right ) + x^{3} \left (\frac {A a^{4} e^{2}}{3} + \frac {8 A a^{3} b d e}{3} + 2 A a^{2} b^{2} d^{2} + \frac {2 B a^{4} d e}{3} + \frac {4 B a^{3} b d^{2}}{3}\right ) + x^{2} \left (A a^{4} d e + 2 A a^{3} b d^{2} + \frac {B a^{4} d^{2}}{2}\right ) \]
A*a**4*d**2*x + B*b**4*e**2*x**8/8 + x**7*(A*b**4*e**2/7 + 4*B*a*b**3*e**2 /7 + 2*B*b**4*d*e/7) + x**6*(2*A*a*b**3*e**2/3 + A*b**4*d*e/3 + B*a**2*b** 2*e**2 + 4*B*a*b**3*d*e/3 + B*b**4*d**2/6) + x**5*(6*A*a**2*b**2*e**2/5 + 8*A*a*b**3*d*e/5 + A*b**4*d**2/5 + 4*B*a**3*b*e**2/5 + 12*B*a**2*b**2*d*e/ 5 + 4*B*a*b**3*d**2/5) + x**4*(A*a**3*b*e**2 + 3*A*a**2*b**2*d*e + A*a*b** 3*d**2 + B*a**4*e**2/4 + 2*B*a**3*b*d*e + 3*B*a**2*b**2*d**2/2) + x**3*(A* a**4*e**2/3 + 8*A*a**3*b*d*e/3 + 2*A*a**2*b**2*d**2 + 2*B*a**4*d*e/3 + 4*B *a**3*b*d**2/3) + x**2*(A*a**4*d*e + 2*A*a**3*b*d**2 + B*a**4*d**2/2)
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (110) = 220\).
Time = 0.20 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.73 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} e^{2} x^{8} + A a^{4} d^{2} x + \frac {1}{7} \, {\left (2 \, B b^{4} d e + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{4} d^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} + 4 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{4} e^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{4} d e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} x^{2} \]
1/8*B*b^4*e^2*x^8 + A*a^4*d^2*x + 1/7*(2*B*b^4*d*e + (4*B*a*b^3 + A*b^4)*e ^2)*x^7 + 1/6*(B*b^4*d^2 + 2*(4*B*a*b^3 + A*b^4)*d*e + 2*(3*B*a^2*b^2 + 2* A*a*b^3)*e^2)*x^6 + 1/5*((4*B*a*b^3 + A*b^4)*d^2 + 4*(3*B*a^2*b^2 + 2*A*a* b^3)*d*e + 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^2)*x^5 + 1/4*(2*(3*B*a^2*b^2 + 2* A*a*b^3)*d^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e + (B*a^4 + 4*A*a^3*b)*e^2)* x^4 + 1/3*(A*a^4*e^2 + 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2 + 2*(B*a^4 + 4*A*a^ 3*b)*d*e)*x^3 + 1/2*(2*A*a^4*d*e + (B*a^4 + 4*A*a^3*b)*d^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (110) = 220\).
Time = 0.29 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.17 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{8} \, B b^{4} e^{2} x^{8} + \frac {2}{7} \, B b^{4} d e x^{7} + \frac {4}{7} \, B a b^{3} e^{2} x^{7} + \frac {1}{7} \, A b^{4} e^{2} x^{7} + \frac {1}{6} \, B b^{4} d^{2} x^{6} + \frac {4}{3} \, B a b^{3} d e x^{6} + \frac {1}{3} \, A b^{4} d e x^{6} + B a^{2} b^{2} e^{2} x^{6} + \frac {2}{3} \, A a b^{3} e^{2} x^{6} + \frac {4}{5} \, B a b^{3} d^{2} x^{5} + \frac {1}{5} \, A b^{4} d^{2} x^{5} + \frac {12}{5} \, B a^{2} b^{2} d e x^{5} + \frac {8}{5} \, A a b^{3} d e x^{5} + \frac {4}{5} \, B a^{3} b e^{2} x^{5} + \frac {6}{5} \, A a^{2} b^{2} e^{2} x^{5} + \frac {3}{2} \, B a^{2} b^{2} d^{2} x^{4} + A a b^{3} d^{2} x^{4} + 2 \, B a^{3} b d e x^{4} + 3 \, A a^{2} b^{2} d e x^{4} + \frac {1}{4} \, B a^{4} e^{2} x^{4} + A a^{3} b e^{2} x^{4} + \frac {4}{3} \, B a^{3} b d^{2} x^{3} + 2 \, A a^{2} b^{2} d^{2} x^{3} + \frac {2}{3} \, B a^{4} d e x^{3} + \frac {8}{3} \, A a^{3} b d e x^{3} + \frac {1}{3} \, A a^{4} e^{2} x^{3} + \frac {1}{2} \, B a^{4} d^{2} x^{2} + 2 \, A a^{3} b d^{2} x^{2} + A a^{4} d e x^{2} + A a^{4} d^{2} x \]
1/8*B*b^4*e^2*x^8 + 2/7*B*b^4*d*e*x^7 + 4/7*B*a*b^3*e^2*x^7 + 1/7*A*b^4*e^ 2*x^7 + 1/6*B*b^4*d^2*x^6 + 4/3*B*a*b^3*d*e*x^6 + 1/3*A*b^4*d*e*x^6 + B*a^ 2*b^2*e^2*x^6 + 2/3*A*a*b^3*e^2*x^6 + 4/5*B*a*b^3*d^2*x^5 + 1/5*A*b^4*d^2* x^5 + 12/5*B*a^2*b^2*d*e*x^5 + 8/5*A*a*b^3*d*e*x^5 + 4/5*B*a^3*b*e^2*x^5 + 6/5*A*a^2*b^2*e^2*x^5 + 3/2*B*a^2*b^2*d^2*x^4 + A*a*b^3*d^2*x^4 + 2*B*a^3 *b*d*e*x^4 + 3*A*a^2*b^2*d*e*x^4 + 1/4*B*a^4*e^2*x^4 + A*a^3*b*e^2*x^4 + 4 /3*B*a^3*b*d^2*x^3 + 2*A*a^2*b^2*d^2*x^3 + 2/3*B*a^4*d*e*x^3 + 8/3*A*a^3*b *d*e*x^3 + 1/3*A*a^4*e^2*x^3 + 1/2*B*a^4*d^2*x^2 + 2*A*a^3*b*d^2*x^2 + A*a ^4*d*e*x^2 + A*a^4*d^2*x
Time = 10.82 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.58 \[ \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^4\,\left (\frac {B\,a^4\,e^2}{4}+2\,B\,a^3\,b\,d\,e+A\,a^3\,b\,e^2+\frac {3\,B\,a^2\,b^2\,d^2}{2}+3\,A\,a^2\,b^2\,d\,e+A\,a\,b^3\,d^2\right )+x^5\,\left (\frac {4\,B\,a^3\,b\,e^2}{5}+\frac {12\,B\,a^2\,b^2\,d\,e}{5}+\frac {6\,A\,a^2\,b^2\,e^2}{5}+\frac {4\,B\,a\,b^3\,d^2}{5}+\frac {8\,A\,a\,b^3\,d\,e}{5}+\frac {A\,b^4\,d^2}{5}\right )+x^3\,\left (\frac {2\,B\,a^4\,d\,e}{3}+\frac {A\,a^4\,e^2}{3}+\frac {4\,B\,a^3\,b\,d^2}{3}+\frac {8\,A\,a^3\,b\,d\,e}{3}+2\,A\,a^2\,b^2\,d^2\right )+x^6\,\left (B\,a^2\,b^2\,e^2+\frac {4\,B\,a\,b^3\,d\,e}{3}+\frac {2\,A\,a\,b^3\,e^2}{3}+\frac {B\,b^4\,d^2}{6}+\frac {A\,b^4\,d\,e}{3}\right )+A\,a^4\,d^2\,x+\frac {a^3\,d\,x^2\,\left (2\,A\,a\,e+4\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^3\,e\,x^7\,\left (A\,b\,e+4\,B\,a\,e+2\,B\,b\,d\right )}{7}+\frac {B\,b^4\,e^2\,x^8}{8} \]
x^4*((B*a^4*e^2)/4 + A*a*b^3*d^2 + A*a^3*b*e^2 + (3*B*a^2*b^2*d^2)/2 + 2*B *a^3*b*d*e + 3*A*a^2*b^2*d*e) + x^5*((A*b^4*d^2)/5 + (4*B*a*b^3*d^2)/5 + ( 4*B*a^3*b*e^2)/5 + (6*A*a^2*b^2*e^2)/5 + (8*A*a*b^3*d*e)/5 + (12*B*a^2*b^2 *d*e)/5) + x^3*((A*a^4*e^2)/3 + (2*B*a^4*d*e)/3 + (4*B*a^3*b*d^2)/3 + 2*A* a^2*b^2*d^2 + (8*A*a^3*b*d*e)/3) + x^6*((B*b^4*d^2)/6 + (A*b^4*d*e)/3 + (2 *A*a*b^3*e^2)/3 + B*a^2*b^2*e^2 + (4*B*a*b^3*d*e)/3) + A*a^4*d^2*x + (a^3* d*x^2*(2*A*a*e + 4*A*b*d + B*a*d))/2 + (b^3*e*x^7*(A*b*e + 4*B*a*e + 2*B*b *d))/7 + (B*b^4*e^2*x^8)/8