Integrand size = 20, antiderivative size = 120 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^4}{4 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^5}{5 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^6}{6 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4} \]
-1/4*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^4/e^4+1/5*(-a*e+b*d)*(-2*A*b*e-B*a*e+ 3*B*b*d)*(e*x+d)^5/e^4-1/6*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^6/e^4+1/7*b^ 2*B*(e*x+d)^7/e^4
Time = 0.05 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.80 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=a^2 A d^3 x+\frac {1}{2} a d^2 (2 A b d+a B d+3 a A e) x^2+\frac {1}{3} d \left (a B d (2 b d+3 a e)+A \left (b^2 d^2+6 a b d e+3 a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (6 a b d e (B d+A e)+a^2 e^2 (3 B d+A e)+b^2 d^2 (B d+3 A e)\right ) x^4+\frac {1}{5} e \left (a^2 B e^2+3 b^2 d (B d+A e)+2 a b e (3 B d+A e)\right ) x^5+\frac {1}{6} b e^2 (3 b B d+A b e+2 a B e) x^6+\frac {1}{7} b^2 B e^3 x^7 \]
a^2*A*d^3*x + (a*d^2*(2*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + (d*(a*B*d*(2*b*d + 3*a*e) + A*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2))*x^3)/3 + ((6*a*b*d*e*(B*d + A*e) + a^2*e^2*(3*B*d + A*e) + b^2*d^2*(B*d + 3*A*e))*x^4)/4 + (e*(a^2* B*e^2 + 3*b^2*d*(B*d + A*e) + 2*a*b*e*(3*B*d + A*e))*x^5)/5 + (b*e^2*(3*b* B*d + A*b*e + 2*a*B*e)*x^6)/6 + (b^2*B*e^3*x^7)/7
Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
\(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {b (d+e x)^5 (2 a B e+A b e-3 b B d)}{e^3}+\frac {(d+e x)^4 (a e-b d) (a B e+2 A b e-3 b B d)}{e^3}+\frac {(d+e x)^3 (a e-b d)^2 (A e-B d)}{e^3}+\frac {b^2 B (d+e x)^6}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b (d+e x)^6 (-2 a B e-A b e+3 b B d)}{6 e^4}+\frac {(d+e x)^5 (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {(d+e x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4}\) |
-1/4*((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^4)/e^4 + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^5)/(5*e^4) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^6)/(6*e^4) + (b^2*B*(d + e*x)^7)/(7*e^4)
3.11.23.3.1 Defintions of rubi rules used
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]))))
Leaf count of result is larger than twice the leaf count of optimal. \(236\) vs. \(2(112)=224\).
Time = 0.65 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {b^{2} B \,e^{3} x^{7}}{7}+\frac {\left (\left (b^{2} A +2 a b B \right ) e^{3}+3 b^{2} B d \,e^{2}\right ) x^{6}}{6}+\frac {\left (\left (2 a b A +a^{2} B \right ) e^{3}+3 \left (b^{2} A +2 a b B \right ) d \,e^{2}+3 b^{2} B \,d^{2} e \right ) x^{5}}{5}+\frac {\left (a^{2} A \,e^{3}+3 \left (2 a b A +a^{2} B \right ) d \,e^{2}+3 \left (b^{2} A +2 a b B \right ) d^{2} e +b^{2} B \,d^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{2} A d \,e^{2}+3 \left (2 a b A +a^{2} B \right ) d^{2} e +\left (b^{2} A +2 a b B \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{2} A \,d^{2} e +\left (2 a b A +a^{2} B \right ) d^{3}\right ) x^{2}}{2}+a^{2} A \,d^{3} x\) | \(237\) |
norman | \(\frac {b^{2} B \,e^{3} x^{7}}{7}+\left (\frac {1}{6} A \,b^{2} e^{3}+\frac {1}{3} B a b \,e^{3}+\frac {1}{2} b^{2} B d \,e^{2}\right ) x^{6}+\left (\frac {2}{5} A a b \,e^{3}+\frac {3}{5} A \,b^{2} d \,e^{2}+\frac {1}{5} B \,a^{2} e^{3}+\frac {6}{5} B a b d \,e^{2}+\frac {3}{5} b^{2} B \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} a^{2} A \,e^{3}+\frac {3}{2} A a b d \,e^{2}+\frac {3}{4} A \,b^{2} d^{2} e +\frac {3}{4} B \,a^{2} d \,e^{2}+\frac {3}{2} B a b \,d^{2} e +\frac {1}{4} b^{2} B \,d^{3}\right ) x^{4}+\left (a^{2} A d \,e^{2}+2 A a b \,d^{2} e +\frac {1}{3} A \,b^{2} d^{3}+B \,a^{2} d^{2} e +\frac {2}{3} B a b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} A \,d^{2} e +A a b \,d^{3}+\frac {1}{2} B \,a^{2} d^{3}\right ) x^{2}+a^{2} A \,d^{3} x\) | \(247\) |
gosper | \(\frac {1}{7} b^{2} B \,e^{3} x^{7}+\frac {1}{6} x^{6} A \,b^{2} e^{3}+\frac {1}{3} x^{6} B a b \,e^{3}+\frac {1}{2} x^{6} b^{2} B d \,e^{2}+\frac {2}{5} x^{5} A a b \,e^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {1}{5} x^{5} B \,a^{2} e^{3}+\frac {6}{5} x^{5} B a b d \,e^{2}+\frac {3}{5} x^{5} b^{2} B \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a b d \,e^{2}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {3}{2} x^{4} B a b \,d^{2} e +\frac {1}{4} x^{4} b^{2} B \,d^{3}+x^{3} a^{2} A d \,e^{2}+2 x^{3} A a b \,d^{2} e +\frac {1}{3} x^{3} A \,b^{2} d^{3}+x^{3} B \,a^{2} d^{2} e +\frac {2}{3} x^{3} B a b \,d^{3}+\frac {3}{2} x^{2} a^{2} A \,d^{2} e +x^{2} A a b \,d^{3}+\frac {1}{2} x^{2} B \,a^{2} d^{3}+a^{2} A \,d^{3} x\) | \(288\) |
risch | \(\frac {1}{7} b^{2} B \,e^{3} x^{7}+\frac {1}{6} x^{6} A \,b^{2} e^{3}+\frac {1}{3} x^{6} B a b \,e^{3}+\frac {1}{2} x^{6} b^{2} B d \,e^{2}+\frac {2}{5} x^{5} A a b \,e^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {1}{5} x^{5} B \,a^{2} e^{3}+\frac {6}{5} x^{5} B a b d \,e^{2}+\frac {3}{5} x^{5} b^{2} B \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a b d \,e^{2}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {3}{2} x^{4} B a b \,d^{2} e +\frac {1}{4} x^{4} b^{2} B \,d^{3}+x^{3} a^{2} A d \,e^{2}+2 x^{3} A a b \,d^{2} e +\frac {1}{3} x^{3} A \,b^{2} d^{3}+x^{3} B \,a^{2} d^{2} e +\frac {2}{3} x^{3} B a b \,d^{3}+\frac {3}{2} x^{2} a^{2} A \,d^{2} e +x^{2} A a b \,d^{3}+\frac {1}{2} x^{2} B \,a^{2} d^{3}+a^{2} A \,d^{3} x\) | \(288\) |
parallelrisch | \(\frac {1}{7} b^{2} B \,e^{3} x^{7}+\frac {1}{6} x^{6} A \,b^{2} e^{3}+\frac {1}{3} x^{6} B a b \,e^{3}+\frac {1}{2} x^{6} b^{2} B d \,e^{2}+\frac {2}{5} x^{5} A a b \,e^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {1}{5} x^{5} B \,a^{2} e^{3}+\frac {6}{5} x^{5} B a b d \,e^{2}+\frac {3}{5} x^{5} b^{2} B \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a b d \,e^{2}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {3}{2} x^{4} B a b \,d^{2} e +\frac {1}{4} x^{4} b^{2} B \,d^{3}+x^{3} a^{2} A d \,e^{2}+2 x^{3} A a b \,d^{2} e +\frac {1}{3} x^{3} A \,b^{2} d^{3}+x^{3} B \,a^{2} d^{2} e +\frac {2}{3} x^{3} B a b \,d^{3}+\frac {3}{2} x^{2} a^{2} A \,d^{2} e +x^{2} A a b \,d^{3}+\frac {1}{2} x^{2} B \,a^{2} d^{3}+a^{2} A \,d^{3} x\) | \(288\) |
1/7*b^2*B*e^3*x^7+1/6*((A*b^2+2*B*a*b)*e^3+3*b^2*B*d*e^2)*x^6+1/5*((2*A*a* b+B*a^2)*e^3+3*(A*b^2+2*B*a*b)*d*e^2+3*b^2*B*d^2*e)*x^5+1/4*(a^2*A*e^3+3*( 2*A*a*b+B*a^2)*d*e^2+3*(A*b^2+2*B*a*b)*d^2*e+b^2*B*d^3)*x^4+1/3*(3*a^2*A*d *e^2+3*(2*A*a*b+B*a^2)*d^2*e+(A*b^2+2*B*a*b)*d^3)*x^3+1/2*(3*a^2*A*d^2*e+( 2*A*a*b+B*a^2)*d^3)*x^2+a^2*A*d^3*x
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (112) = 224\).
Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.97 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=\frac {1}{7} \, B b^{2} e^{3} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B b^{2} d^{2} e + 3 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{3} + A a^{2} e^{3} + 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \]
1/7*B*b^2*e^3*x^7 + A*a^2*d^3*x + 1/6*(3*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e ^3)*x^6 + 1/5*(3*B*b^2*d^2*e + 3*(2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a* b)*e^3)*x^5 + 1/4*(B*b^2*d^3 + A*a^2*e^3 + 3*(2*B*a*b + A*b^2)*d^2*e + 3*( B*a^2 + 2*A*a*b)*d*e^2)*x^4 + 1/3*(3*A*a^2*d*e^2 + (2*B*a*b + A*b^2)*d^3 + 3*(B*a^2 + 2*A*a*b)*d^2*e)*x^3 + 1/2*(3*A*a^2*d^2*e + (B*a^2 + 2*A*a*b)*d ^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (114) = 228\).
Time = 0.04 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.47 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=A a^{2} d^{3} x + \frac {B b^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {A b^{2} e^{3}}{6} + \frac {B a b e^{3}}{3} + \frac {B b^{2} d e^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 A a b e^{3}}{5} + \frac {3 A b^{2} d e^{2}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a b d e^{2}}{5} + \frac {3 B b^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a b d e^{2}}{2} + \frac {3 A b^{2} d^{2} e}{4} + \frac {3 B a^{2} d e^{2}}{4} + \frac {3 B a b d^{2} e}{2} + \frac {B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac {A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac {2 B a b d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac {B a^{2} d^{3}}{2}\right ) \]
A*a**2*d**3*x + B*b**2*e**3*x**7/7 + x**6*(A*b**2*e**3/6 + B*a*b*e**3/3 + B*b**2*d*e**2/2) + x**5*(2*A*a*b*e**3/5 + 3*A*b**2*d*e**2/5 + B*a**2*e**3/ 5 + 6*B*a*b*d*e**2/5 + 3*B*b**2*d**2*e/5) + x**4*(A*a**2*e**3/4 + 3*A*a*b* d*e**2/2 + 3*A*b**2*d**2*e/4 + 3*B*a**2*d*e**2/4 + 3*B*a*b*d**2*e/2 + B*b* *2*d**3/4) + x**3*(A*a**2*d*e**2 + 2*A*a*b*d**2*e + A*b**2*d**3/3 + B*a**2 *d**2*e + 2*B*a*b*d**3/3) + x**2*(3*A*a**2*d**2*e/2 + A*a*b*d**3 + B*a**2* d**3/2)
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (112) = 224\).
Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.97 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=\frac {1}{7} \, B b^{2} e^{3} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B b^{2} d^{2} e + 3 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{3} + A a^{2} e^{3} + 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \]
1/7*B*b^2*e^3*x^7 + A*a^2*d^3*x + 1/6*(3*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e ^3)*x^6 + 1/5*(3*B*b^2*d^2*e + 3*(2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a* b)*e^3)*x^5 + 1/4*(B*b^2*d^3 + A*a^2*e^3 + 3*(2*B*a*b + A*b^2)*d^2*e + 3*( B*a^2 + 2*A*a*b)*d*e^2)*x^4 + 1/3*(3*A*a^2*d*e^2 + (2*B*a*b + A*b^2)*d^3 + 3*(B*a^2 + 2*A*a*b)*d^2*e)*x^3 + 1/2*(3*A*a^2*d^2*e + (B*a^2 + 2*A*a*b)*d ^3)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (112) = 224\).
Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.39 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=\frac {1}{7} \, B b^{2} e^{3} x^{7} + \frac {1}{2} \, B b^{2} d e^{2} x^{6} + \frac {1}{3} \, B a b e^{3} x^{6} + \frac {1}{6} \, A b^{2} e^{3} x^{6} + \frac {3}{5} \, B b^{2} d^{2} e x^{5} + \frac {6}{5} \, B a b d e^{2} x^{5} + \frac {3}{5} \, A b^{2} d e^{2} x^{5} + \frac {1}{5} \, B a^{2} e^{3} x^{5} + \frac {2}{5} \, A a b e^{3} x^{5} + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {3}{2} \, B a b d^{2} e x^{4} + \frac {3}{4} \, A b^{2} d^{2} e x^{4} + \frac {3}{4} \, B a^{2} d e^{2} x^{4} + \frac {3}{2} \, A a b d e^{2} x^{4} + \frac {1}{4} \, A a^{2} e^{3} x^{4} + \frac {2}{3} \, B a b d^{3} x^{3} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + B a^{2} d^{2} e x^{3} + 2 \, A a b d^{2} e x^{3} + A a^{2} d e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a b d^{3} x^{2} + \frac {3}{2} \, A a^{2} d^{2} e x^{2} + A a^{2} d^{3} x \]
1/7*B*b^2*e^3*x^7 + 1/2*B*b^2*d*e^2*x^6 + 1/3*B*a*b*e^3*x^6 + 1/6*A*b^2*e^ 3*x^6 + 3/5*B*b^2*d^2*e*x^5 + 6/5*B*a*b*d*e^2*x^5 + 3/5*A*b^2*d*e^2*x^5 + 1/5*B*a^2*e^3*x^5 + 2/5*A*a*b*e^3*x^5 + 1/4*B*b^2*d^3*x^4 + 3/2*B*a*b*d^2* e*x^4 + 3/4*A*b^2*d^2*e*x^4 + 3/4*B*a^2*d*e^2*x^4 + 3/2*A*a*b*d*e^2*x^4 + 1/4*A*a^2*e^3*x^4 + 2/3*B*a*b*d^3*x^3 + 1/3*A*b^2*d^3*x^3 + B*a^2*d^2*e*x^ 3 + 2*A*a*b*d^2*e*x^3 + A*a^2*d*e^2*x^3 + 1/2*B*a^2*d^3*x^2 + A*a*b*d^3*x^ 2 + 3/2*A*a^2*d^2*e*x^2 + A*a^2*d^3*x
Time = 0.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.92 \[ \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx=x^4\,\left (\frac {3\,B\,a^2\,d\,e^2}{4}+\frac {A\,a^2\,e^3}{4}+\frac {3\,B\,a\,b\,d^2\,e}{2}+\frac {3\,A\,a\,b\,d\,e^2}{2}+\frac {B\,b^2\,d^3}{4}+\frac {3\,A\,b^2\,d^2\,e}{4}\right )+x^3\,\left (B\,a^2\,d^2\,e+A\,a^2\,d\,e^2+\frac {2\,B\,a\,b\,d^3}{3}+2\,A\,a\,b\,d^2\,e+\frac {A\,b^2\,d^3}{3}\right )+x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,b\,d\,e^2}{5}+\frac {2\,A\,a\,b\,e^3}{5}+\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}\right )+A\,a^2\,d^3\,x+\frac {a\,d^2\,x^2\,\left (3\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^2\,x^6\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{6}+\frac {B\,b^2\,e^3\,x^7}{7} \]
x^4*((A*a^2*e^3)/4 + (B*b^2*d^3)/4 + (3*A*b^2*d^2*e)/4 + (3*B*a^2*d*e^2)/4 + (3*A*a*b*d*e^2)/2 + (3*B*a*b*d^2*e)/2) + x^3*((A*b^2*d^3)/3 + (2*B*a*b* d^3)/3 + A*a^2*d*e^2 + B*a^2*d^2*e + 2*A*a*b*d^2*e) + x^5*((B*a^2*e^3)/5 + (2*A*a*b*e^3)/5 + (3*A*b^2*d*e^2)/5 + (3*B*b^2*d^2*e)/5 + (6*B*a*b*d*e^2) /5) + A*a^2*d^3*x + (a*d^2*x^2*(3*A*a*e + 2*A*b*d + B*a*d))/2 + (b*e^2*x^6 *(A*b*e + 2*B*a*e + 3*B*b*d))/6 + (B*b^2*e^3*x^7)/7