Integrand size = 24, antiderivative size = 124 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{6 e^4}-\frac {3 c (2 c d-b e) (d+e x)^7}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4} \]
-1/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^5/e^4+1/6*(6*c^2*d^2+b^2*e^2 -2*c*e*(-a*e+3*b*d))*(e*x+d)^6/e^4-3/7*c*(-b*e+2*c*d)*(e*x+d)^7/e^4+1/4*c^ 2*(e*x+d)^8/e^4
Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=a b d^4 x+\frac {1}{2} d^3 \left (b^2 d+2 a c d+4 a b e\right ) x^2+\frac {1}{3} d^2 \left (3 b c d^2+4 b^2 d e+8 a c d e+6 a b e^2\right ) x^3+\frac {1}{2} d \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^4+\frac {1}{5} e \left (8 c^2 d^3+b e^2 (4 b d+a e)+2 c d e (9 b d+4 a e)\right ) x^5+\frac {1}{6} e^2 \left (12 c^2 d^2+b^2 e^2+2 c e (6 b d+a e)\right ) x^6+\frac {1}{7} c e^3 (8 c d+3 b e) x^7+\frac {1}{4} c^2 e^4 x^8 \]
a*b*d^4*x + (d^3*(b^2*d + 2*a*c*d + 4*a*b*e)*x^2)/2 + (d^2*(3*b*c*d^2 + 4* b^2*d*e + 8*a*c*d*e + 6*a*b*e^2)*x^3)/3 + (d*(c^2*d^3 + 6*c*d*e*(b*d + a*e ) + b*e^2*(3*b*d + 2*a*e))*x^4)/2 + (e*(8*c^2*d^3 + b*e^2*(4*b*d + a*e) + 2*c*d*e*(9*b*d + 4*a*e))*x^5)/5 + (e^2*(12*c^2*d^2 + b^2*e^2 + 2*c*e*(6*b* d + a*e))*x^6)/6 + (c*e^3*(8*c*d + 3*b*e)*x^7)/7 + (c^2*e^4*x^8)/4
Time = 0.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
\(\displaystyle \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^3}+\frac {(d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {3 c (d+e x)^6 (2 c d-b e)}{e^3}+\frac {2 c^2 (d+e x)^7}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4}\) |
-1/5*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/e^4 + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^6)/(6*e^4) - (3*c*(2*c*d - b*e )*(d + e*x)^7)/(7*e^4) + (c^2*(d + e*x)^8)/(4*e^4)
3.15.94.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(241\) vs. \(2(116)=232\).
Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95
method | result | size |
norman | \(\frac {c^{2} e^{4} x^{8}}{4}+\left (\frac {3}{7} c \,e^{4} b +\frac {8}{7} c^{2} d \,e^{3}\right ) x^{7}+\left (\frac {1}{3} c \,e^{4} a +\frac {1}{6} b^{2} e^{4}+2 b c d \,e^{3}+2 c^{2} d^{2} e^{2}\right ) x^{6}+\left (\frac {1}{5} a b \,e^{4}+\frac {8}{5} a c d \,e^{3}+\frac {4}{5} b^{2} d \,e^{3}+\frac {18}{5} b c \,d^{2} e^{2}+\frac {8}{5} c^{2} d^{3} e \right ) x^{5}+\left (a b d \,e^{3}+3 a c \,d^{2} e^{2}+\frac {3}{2} b^{2} d^{2} e^{2}+3 b c \,d^{3} e +\frac {1}{2} c^{2} d^{4}\right ) x^{4}+\left (2 a b \,d^{2} e^{2}+\frac {8}{3} a c \,d^{3} e +\frac {4}{3} b^{2} d^{3} e +b \,d^{4} c \right ) x^{3}+\left (2 a b \,d^{3} e +a c \,d^{4}+\frac {1}{2} b^{2} d^{4}\right ) x^{2}+b \,d^{4} a x\) | \(242\) |
gosper | \(\frac {1}{2} x^{2} b^{2} d^{4}+\frac {1}{2} x^{4} c^{2} d^{4}+\frac {1}{6} x^{6} b^{2} e^{4}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {18}{5} x^{5} b c \,d^{2} e^{2}+2 x^{6} b c d \,e^{3}+\frac {8}{5} x^{5} a c d \,e^{3}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {8}{5} x^{5} c^{2} d^{3} e +\frac {3}{2} x^{4} b^{2} d^{2} e^{2}+\frac {4}{3} x^{3} b^{2} d^{3} e +x^{3} b \,d^{4} c +x^{2} a c \,d^{4}+b \,d^{4} a x +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} c \,e^{4} a +2 x^{6} c^{2} d^{2} e^{2}+\frac {1}{5} x^{5} a b \,e^{4}+\frac {8}{3} x^{3} a c \,d^{3} e +2 x^{2} a b \,d^{3} e +2 x^{3} a b \,d^{2} e^{2}+3 x^{4} a c \,d^{2} e^{2}+3 x^{4} b c \,d^{3} e +x^{4} a b d \,e^{3}\) | \(281\) |
risch | \(\frac {1}{2} x^{2} b^{2} d^{4}+\frac {1}{2} x^{4} c^{2} d^{4}+\frac {1}{6} x^{6} b^{2} e^{4}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {18}{5} x^{5} b c \,d^{2} e^{2}+2 x^{6} b c d \,e^{3}+\frac {8}{5} x^{5} a c d \,e^{3}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {8}{5} x^{5} c^{2} d^{3} e +\frac {3}{2} x^{4} b^{2} d^{2} e^{2}+\frac {4}{3} x^{3} b^{2} d^{3} e +x^{3} b \,d^{4} c +x^{2} a c \,d^{4}+b \,d^{4} a x +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} c \,e^{4} a +2 x^{6} c^{2} d^{2} e^{2}+\frac {1}{5} x^{5} a b \,e^{4}+\frac {8}{3} x^{3} a c \,d^{3} e +2 x^{2} a b \,d^{3} e +2 x^{3} a b \,d^{2} e^{2}+3 x^{4} a c \,d^{2} e^{2}+3 x^{4} b c \,d^{3} e +x^{4} a b d \,e^{3}\) | \(281\) |
parallelrisch | \(\frac {1}{2} x^{2} b^{2} d^{4}+\frac {1}{2} x^{4} c^{2} d^{4}+\frac {1}{6} x^{6} b^{2} e^{4}+\frac {1}{4} c^{2} e^{4} x^{8}+\frac {18}{5} x^{5} b c \,d^{2} e^{2}+2 x^{6} b c d \,e^{3}+\frac {8}{5} x^{5} a c d \,e^{3}+\frac {4}{5} x^{5} b^{2} d \,e^{3}+\frac {8}{5} x^{5} c^{2} d^{3} e +\frac {3}{2} x^{4} b^{2} d^{2} e^{2}+\frac {4}{3} x^{3} b^{2} d^{3} e +x^{3} b \,d^{4} c +x^{2} a c \,d^{4}+b \,d^{4} a x +\frac {3}{7} x^{7} c \,e^{4} b +\frac {8}{7} x^{7} c^{2} d \,e^{3}+\frac {1}{3} x^{6} c \,e^{4} a +2 x^{6} c^{2} d^{2} e^{2}+\frac {1}{5} x^{5} a b \,e^{4}+\frac {8}{3} x^{3} a c \,d^{3} e +2 x^{2} a b \,d^{3} e +2 x^{3} a b \,d^{2} e^{2}+3 x^{4} a c \,d^{2} e^{2}+3 x^{4} b c \,d^{3} e +x^{4} a b d \,e^{3}\) | \(281\) |
default | \(\frac {c^{2} e^{4} x^{8}}{4}+\frac {\left (\left (b \,e^{4}+8 c d \,e^{3}\right ) c +2 c \,e^{4} b \right ) x^{7}}{7}+\frac {\left (\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) c +\left (b \,e^{4}+8 c d \,e^{3}\right ) b +2 c \,e^{4} a \right ) x^{6}}{6}+\frac {\left (\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c +\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) b +\left (b \,e^{4}+8 c d \,e^{3}\right ) a \right ) x^{5}}{5}+\frac {\left (\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b +\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (b \,d^{4} c +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) b +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a \right ) x^{3}}{3}+\frac {\left (b^{2} d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a \right ) x^{2}}{2}+b \,d^{4} a x\) | \(290\) |
1/4*c^2*e^4*x^8+(3/7*c*e^4*b+8/7*c^2*d*e^3)*x^7+(1/3*c*e^4*a+1/6*b^2*e^4+2 *b*c*d*e^3+2*c^2*d^2*e^2)*x^6+(1/5*a*b*e^4+8/5*a*c*d*e^3+4/5*b^2*d*e^3+18/ 5*b*c*d^2*e^2+8/5*c^2*d^3*e)*x^5+(a*b*d*e^3+3*a*c*d^2*e^2+3/2*b^2*d^2*e^2+ 3*b*c*d^3*e+1/2*c^2*d^4)*x^4+(2*a*b*d^2*e^2+8/3*a*c*d^3*e+4/3*b^2*d^3*e+b* d^4*c)*x^3+(2*a*b*d^3*e+a*c*d^4+1/2*b^2*d^4)*x^2+b*d^4*a*x
Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {1}{7} \, {\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac {1}{6} \, {\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b d^{3} e + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{2} \]
1/4*c^2*e^4*x^8 + 1/7*(8*c^2*d*e^3 + 3*b*c*e^4)*x^7 + a*b*d^4*x + 1/6*(12* c^2*d^2*e^2 + 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^6 + 1/5*(8*c^2*d^3*e + 1 8*b*c*d^2*e^2 + a*b*e^4 + 4*(b^2 + 2*a*c)*d*e^3)*x^5 + 1/2*(c^2*d^4 + 6*b* c*d^3*e + 2*a*b*d*e^3 + 3*(b^2 + 2*a*c)*d^2*e^2)*x^4 + 1/3*(3*b*c*d^4 + 6* a*b*d^2*e^2 + 4*(b^2 + 2*a*c)*d^3*e)*x^3 + 1/2*(4*a*b*d^3*e + (b^2 + 2*a*c )*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (114) = 228\).
Time = 0.03 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.25 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=a b d^{4} x + \frac {c^{2} e^{4} x^{8}}{4} + x^{7} \cdot \left (\frac {3 b c e^{4}}{7} + \frac {8 c^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {a c e^{4}}{3} + \frac {b^{2} e^{4}}{6} + 2 b c d e^{3} + 2 c^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {a b e^{4}}{5} + \frac {8 a c d e^{3}}{5} + \frac {4 b^{2} d e^{3}}{5} + \frac {18 b c d^{2} e^{2}}{5} + \frac {8 c^{2} d^{3} e}{5}\right ) + x^{4} \left (a b d e^{3} + 3 a c d^{2} e^{2} + \frac {3 b^{2} d^{2} e^{2}}{2} + 3 b c d^{3} e + \frac {c^{2} d^{4}}{2}\right ) + x^{3} \cdot \left (2 a b d^{2} e^{2} + \frac {8 a c d^{3} e}{3} + \frac {4 b^{2} d^{3} e}{3} + b c d^{4}\right ) + x^{2} \cdot \left (2 a b d^{3} e + a c d^{4} + \frac {b^{2} d^{4}}{2}\right ) \]
a*b*d**4*x + c**2*e**4*x**8/4 + x**7*(3*b*c*e**4/7 + 8*c**2*d*e**3/7) + x* *6*(a*c*e**4/3 + b**2*e**4/6 + 2*b*c*d*e**3 + 2*c**2*d**2*e**2) + x**5*(a* b*e**4/5 + 8*a*c*d*e**3/5 + 4*b**2*d*e**3/5 + 18*b*c*d**2*e**2/5 + 8*c**2* d**3*e/5) + x**4*(a*b*d*e**3 + 3*a*c*d**2*e**2 + 3*b**2*d**2*e**2/2 + 3*b* c*d**3*e + c**2*d**4/2) + x**3*(2*a*b*d**2*e**2 + 8*a*c*d**3*e/3 + 4*b**2* d**3*e/3 + b*c*d**4) + x**2*(2*a*b*d**3*e + a*c*d**4 + b**2*d**4/2)
Time = 0.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {1}{7} \, {\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac {1}{6} \, {\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b d^{3} e + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{2} \]
1/4*c^2*e^4*x^8 + 1/7*(8*c^2*d*e^3 + 3*b*c*e^4)*x^7 + a*b*d^4*x + 1/6*(12* c^2*d^2*e^2 + 12*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^6 + 1/5*(8*c^2*d^3*e + 1 8*b*c*d^2*e^2 + a*b*e^4 + 4*(b^2 + 2*a*c)*d*e^3)*x^5 + 1/2*(c^2*d^4 + 6*b* c*d^3*e + 2*a*b*d*e^3 + 3*(b^2 + 2*a*c)*d^2*e^2)*x^4 + 1/3*(3*b*c*d^4 + 6* a*b*d^2*e^2 + 4*(b^2 + 2*a*c)*d^3*e)*x^3 + 1/2*(4*a*b*d^3*e + (b^2 + 2*a*c )*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (116) = 232\).
Time = 0.27 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.26 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {8}{7} \, c^{2} d e^{3} x^{7} + \frac {3}{7} \, b c e^{4} x^{7} + 2 \, c^{2} d^{2} e^{2} x^{6} + 2 \, b c d e^{3} x^{6} + \frac {1}{6} \, b^{2} e^{4} x^{6} + \frac {1}{3} \, a c e^{4} x^{6} + \frac {8}{5} \, c^{2} d^{3} e x^{5} + \frac {18}{5} \, b c d^{2} e^{2} x^{5} + \frac {4}{5} \, b^{2} d e^{3} x^{5} + \frac {8}{5} \, a c d e^{3} x^{5} + \frac {1}{5} \, a b e^{4} x^{5} + \frac {1}{2} \, c^{2} d^{4} x^{4} + 3 \, b c d^{3} e x^{4} + \frac {3}{2} \, b^{2} d^{2} e^{2} x^{4} + 3 \, a c d^{2} e^{2} x^{4} + a b d e^{3} x^{4} + b c d^{4} x^{3} + \frac {4}{3} \, b^{2} d^{3} e x^{3} + \frac {8}{3} \, a c d^{3} e x^{3} + 2 \, a b d^{2} e^{2} x^{3} + \frac {1}{2} \, b^{2} d^{4} x^{2} + a c d^{4} x^{2} + 2 \, a b d^{3} e x^{2} + a b d^{4} x \]
1/4*c^2*e^4*x^8 + 8/7*c^2*d*e^3*x^7 + 3/7*b*c*e^4*x^7 + 2*c^2*d^2*e^2*x^6 + 2*b*c*d*e^3*x^6 + 1/6*b^2*e^4*x^6 + 1/3*a*c*e^4*x^6 + 8/5*c^2*d^3*e*x^5 + 18/5*b*c*d^2*e^2*x^5 + 4/5*b^2*d*e^3*x^5 + 8/5*a*c*d*e^3*x^5 + 1/5*a*b*e ^4*x^5 + 1/2*c^2*d^4*x^4 + 3*b*c*d^3*e*x^4 + 3/2*b^2*d^2*e^2*x^4 + 3*a*c*d ^2*e^2*x^4 + a*b*d*e^3*x^4 + b*c*d^4*x^3 + 4/3*b^2*d^3*e*x^3 + 8/3*a*c*d^3 *e*x^3 + 2*a*b*d^2*e^2*x^3 + 1/2*b^2*d^4*x^2 + a*c*d^4*x^2 + 2*a*b*d^3*e*x ^2 + a*b*d^4*x
Time = 0.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.92 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right ) \, dx=x^5\,\left (\frac {4\,b^2\,d\,e^3}{5}+\frac {18\,b\,c\,d^2\,e^2}{5}+\frac {a\,b\,e^4}{5}+\frac {8\,c^2\,d^3\,e}{5}+\frac {8\,a\,c\,d\,e^3}{5}\right )+x^6\,\left (\frac {b^2\,e^4}{6}+2\,b\,c\,d\,e^3+2\,c^2\,d^2\,e^2+\frac {a\,c\,e^4}{3}\right )+x^4\,\left (\frac {3\,b^2\,d^2\,e^2}{2}+3\,b\,c\,d^3\,e+a\,b\,d\,e^3+\frac {c^2\,d^4}{2}+3\,a\,c\,d^2\,e^2\right )+x^2\,\left (\frac {b^2\,d^4}{2}+2\,a\,e\,b\,d^3+a\,c\,d^4\right )+x^3\,\left (\frac {4\,b^2\,d^3\,e}{3}+c\,b\,d^4+2\,a\,b\,d^2\,e^2+\frac {8\,a\,c\,d^3\,e}{3}\right )+\frac {c^2\,e^4\,x^8}{4}+\frac {c\,e^3\,x^7\,\left (3\,b\,e+8\,c\,d\right )}{7}+a\,b\,d^4\,x \]
x^5*((4*b^2*d*e^3)/5 + (8*c^2*d^3*e)/5 + (a*b*e^4)/5 + (8*a*c*d*e^3)/5 + ( 18*b*c*d^2*e^2)/5) + x^6*((b^2*e^4)/6 + 2*c^2*d^2*e^2 + (a*c*e^4)/3 + 2*b* c*d*e^3) + x^4*((c^2*d^4)/2 + (3*b^2*d^2*e^2)/2 + a*b*d*e^3 + 3*b*c*d^3*e + 3*a*c*d^2*e^2) + x^2*((b^2*d^4)/2 + a*c*d^4 + 2*a*b*d^3*e) + x^3*((4*b^2 *d^3*e)/3 + b*c*d^4 + (8*a*c*d^3*e)/3 + 2*a*b*d^2*e^2) + (c^2*e^4*x^8)/4 + (c*e^3*x^7*(3*b*e + 8*c*d))/7 + a*b*d^4*x