\(\int (A+B x) (d+e x)^4 (b x+c x^2) \, dx\) [1]

Optimal result

Integrand size = 22, antiderivative size = 118 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=-\frac {d (B d-A e) (c d-b e) (d+e x)^5}{5 e^4}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^6}{6 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^7}{7 e^4}+\frac {B c (d+e x)^8}{8 e^4} \] Output:

-1/5*d*(-A*e+B*d)*(-b*e+c*d)*(e*x+d)^5/e^4+1/6*(B*d*(-2*b*e+3*c*d)-A*e*(-b 
*e+2*c*d))*(e*x+d)^6/e^4-1/7*(-A*c*e-B*b*e+3*B*c*d)*(e*x+d)^7/e^4+1/8*B*c* 
(e*x+d)^8/e^4
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.50 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{2} A b d^4 x^2+\frac {1}{3} d^3 (b B d+A c d+4 A b e) x^3+\frac {1}{4} d^2 (2 A e (2 c d+3 b e)+B d (c d+4 b e)) x^4+\frac {2}{5} d e (A e (3 c d+2 b e)+B d (2 c d+3 b e)) x^5+\frac {1}{6} e^2 (A e (4 c d+b e)+2 B d (3 c d+2 b e)) x^6+\frac {1}{7} e^3 (4 B c d+b B e+A c e) x^7+\frac {1}{8} B c e^4 x^8 \] Input:

Integrate[(A + B*x)*(d + e*x)^4*(b*x + c*x^2),x]
 

Output:

(A*b*d^4*x^2)/2 + (d^3*(b*B*d + A*c*d + 4*A*b*e)*x^3)/3 + (d^2*(2*A*e*(2*c 
*d + 3*b*e) + B*d*(c*d + 4*b*e))*x^4)/4 + (2*d*e*(A*e*(3*c*d + 2*b*e) + B* 
d*(2*c*d + 3*b*e))*x^5)/5 + (e^2*(A*e*(4*c*d + b*e) + 2*B*d*(3*c*d + 2*b*e 
))*x^6)/6 + (e^3*(4*B*c*d + b*B*e + A*c*e)*x^7)/7 + (B*c*e^4*x^8)/8
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int (A+B x) \left (b x+c x^2\right ) (d+e x)^4 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {(d+e x)^6 (A c e+b B e-3 B c d)}{e^3}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^3}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{e^3}+\frac {B c (d+e x)^7}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac {(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac {d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac {B c (d+e x)^8}{8 e^4}\)

Input:

Int[(A + B*x)*(d + e*x)^4*(b*x + c*x^2),x]
 

Output:

-1/5*(d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^5)/e^4 + ((B*d*(3*c*d - 2*b*e) - 
 A*e*(2*c*d - b*e))*(d + e*x)^6)/(6*e^4) - ((3*B*c*d - b*B*e - A*c*e)*(d + 
 e*x)^7)/(7*e^4) + (B*c*(d + e*x)^8)/(8*e^4)
 

Defintions of rubi rules used

rule 1195

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.61

method	result	size
norman	\(\frac {B \,e^{4} c \,x^{8}}{8}+\left (\frac {1}{7} A c \,e^{4}+\frac {1}{7} B \,e^{4} b +\frac {4}{7} B c d \,e^{3}\right ) x^{7}+\left (\frac {1}{6} A b \,e^{4}+\frac {2}{3} A c d \,e^{3}+\frac {2}{3} B b d \,e^{3}+B c \,d^{2} e^{2}\right ) x^{6}+\left (\frac {4}{5} A b d \,e^{3}+\frac {6}{5} A c \,d^{2} e^{2}+\frac {6}{5} B b \,d^{2} e^{2}+\frac {4}{5} B c \,d^{3} e \right ) x^{5}+\left (\frac {3}{2} A b \,d^{2} e^{2}+A c \,d^{3} e +B b \,d^{3} e +\frac {1}{4} B c \,d^{4}\right ) x^{4}+\left (\frac {4}{3} A b \,d^{3} e +\frac {1}{3} A c \,d^{4}+\frac {1}{3} B b \,d^{4}\right ) x^{3}+\frac {A \,d^{4} b \,x^{2}}{2}\)	\(190\)
default	\(\frac {B \,e^{4} c \,x^{8}}{8}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) c +B \,e^{4} b \right ) x^{7}}{7}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c +\left (A \,e^{4}+4 B d \,e^{3}\right ) b \right ) x^{6}}{6}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b \right ) x^{5}}{5}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b \right ) x^{4}}{4}+\frac {\left (A c \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) b \right ) x^{3}}{3}+\frac {A \,d^{4} b \,x^{2}}{2}\)	\(200\)
gosper	\(\frac {x^{2} \left (105 B \,e^{4} c \,x^{6}+120 x^{5} A c \,e^{4}+120 x^{5} B \,e^{4} b +480 x^{5} B c d \,e^{3}+140 x^{4} A b \,e^{4}+560 x^{4} A c d \,e^{3}+560 x^{4} B b d \,e^{3}+840 x^{4} B c \,d^{2} e^{2}+672 x^{3} A b d \,e^{3}+1008 x^{3} A c \,d^{2} e^{2}+1008 x^{3} B b \,d^{2} e^{2}+672 x^{3} B c \,d^{3} e +1260 x^{2} A b \,d^{2} e^{2}+840 x^{2} A c \,d^{3} e +840 x^{2} B b \,d^{3} e +210 B c \,d^{4} x^{2}+1120 x A b \,d^{3} e +280 A c \,d^{4} x +280 x B b \,d^{4}+420 A \,d^{4} b \right )}{840}\)	\(218\)
risch	\(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {1}{7} x^{7} B \,e^{4} b +\frac {4}{7} x^{7} B c d \,e^{3}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {2}{3} x^{6} B b d \,e^{3}+x^{6} B c \,d^{2} e^{2}+\frac {4}{5} x^{5} A b d \,e^{3}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {6}{5} x^{5} B b \,d^{2} e^{2}+\frac {4}{5} x^{5} B c \,d^{3} e +\frac {3}{2} x^{4} A b \,d^{2} e^{2}+x^{4} A c \,d^{3} e +x^{4} B b \,d^{3} e +\frac {1}{4} B c \,d^{4} x^{4}+\frac {4}{3} x^{3} A b \,d^{3} e +\frac {1}{3} A c \,d^{4} x^{3}+\frac {1}{3} x^{3} B b \,d^{4}+\frac {1}{2} A \,d^{4} b \,x^{2}\)	\(219\)
parallelrisch	\(\frac {1}{8} B \,e^{4} c \,x^{8}+\frac {1}{7} x^{7} A c \,e^{4}+\frac {1}{7} x^{7} B \,e^{4} b +\frac {4}{7} x^{7} B c d \,e^{3}+\frac {1}{6} x^{6} A b \,e^{4}+\frac {2}{3} x^{6} A c d \,e^{3}+\frac {2}{3} x^{6} B b d \,e^{3}+x^{6} B c \,d^{2} e^{2}+\frac {4}{5} x^{5} A b d \,e^{3}+\frac {6}{5} x^{5} A c \,d^{2} e^{2}+\frac {6}{5} x^{5} B b \,d^{2} e^{2}+\frac {4}{5} x^{5} B c \,d^{3} e +\frac {3}{2} x^{4} A b \,d^{2} e^{2}+x^{4} A c \,d^{3} e +x^{4} B b \,d^{3} e +\frac {1}{4} B c \,d^{4} x^{4}+\frac {4}{3} x^{3} A b \,d^{3} e +\frac {1}{3} A c \,d^{4} x^{3}+\frac {1}{3} x^{3} B b \,d^{4}+\frac {1}{2} A \,d^{4} b \,x^{2}\)	\(219\)
orering	\(\frac {x \left (105 B \,e^{4} c \,x^{6}+120 x^{5} A c \,e^{4}+120 x^{5} B \,e^{4} b +480 x^{5} B c d \,e^{3}+140 x^{4} A b \,e^{4}+560 x^{4} A c d \,e^{3}+560 x^{4} B b d \,e^{3}+840 x^{4} B c \,d^{2} e^{2}+672 x^{3} A b d \,e^{3}+1008 x^{3} A c \,d^{2} e^{2}+1008 x^{3} B b \,d^{2} e^{2}+672 x^{3} B c \,d^{3} e +1260 x^{2} A b \,d^{2} e^{2}+840 x^{2} A c \,d^{3} e +840 x^{2} B b \,d^{3} e +210 B c \,d^{4} x^{2}+1120 x A b \,d^{3} e +280 A c \,d^{4} x +280 x B b \,d^{4}+420 A \,d^{4} b \right ) \left (c \,x^{2}+b x \right )}{840 c x +840 b}\)	\(232\)

Input:

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x,method=_RETURNVERBOSE)
 

Output:

1/8*B*e^4*c*x^8+(1/7*A*c*e^4+1/7*B*e^4*b+4/7*B*c*d*e^3)*x^7+(1/6*A*b*e^4+2 
/3*A*c*d*e^3+2/3*B*b*d*e^3+B*c*d^2*e^2)*x^6+(4/5*A*b*d*e^3+6/5*A*c*d^2*e^2 
+6/5*B*b*d^2*e^2+4/5*B*c*d^3*e)*x^5+(3/2*A*b*d^2*e^2+A*c*d^3*e+B*b*d^3*e+1 
/4*B*c*d^4)*x^4+(4/3*A*b*d^3*e+1/3*A*c*d^4+1/3*B*b*d^4)*x^3+1/2*A*d^4*b*x^ 
2
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.51 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{2} \, A b d^{4} x^{2} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \, {\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \, {\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, A b d^{3} e + {\left (B b + A c\right )} d^{4}\right )} x^{3} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x, algorithm="fricas")
 

Output:

1/8*B*c*e^4*x^8 + 1/2*A*b*d^4*x^2 + 1/7*(4*B*c*d*e^3 + (B*b + A*c)*e^4)*x^ 
7 + 1/6*(6*B*c*d^2*e^2 + A*b*e^4 + 4*(B*b + A*c)*d*e^3)*x^6 + 2/5*(2*B*c*d 
^3*e + 2*A*b*d*e^3 + 3*(B*b + A*c)*d^2*e^2)*x^5 + 1/4*(B*c*d^4 + 6*A*b*d^2 
*e^2 + 4*(B*b + A*c)*d^3*e)*x^4 + 1/3*(4*A*b*d^3*e + (B*b + A*c)*d^4)*x^3
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (112) = 224\).

Time = 0.03 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.95 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {A b d^{4} x^{2}}{2} + \frac {B c e^{4} x^{8}}{8} + x^{7} \left (\frac {A c e^{4}}{7} + \frac {B b e^{4}}{7} + \frac {4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac {A b e^{4}}{6} + \frac {2 A c d e^{3}}{3} + \frac {2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \cdot \left (\frac {4 A b d e^{3}}{5} + \frac {6 A c d^{2} e^{2}}{5} + \frac {6 B b d^{2} e^{2}}{5} + \frac {4 B c d^{3} e}{5}\right ) + x^{4} \cdot \left (\frac {3 A b d^{2} e^{2}}{2} + A c d^{3} e + B b d^{3} e + \frac {B c d^{4}}{4}\right ) + x^{3} \cdot \left (\frac {4 A b d^{3} e}{3} + \frac {A c d^{4}}{3} + \frac {B b d^{4}}{3}\right ) \] Input:

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x),x)
 

Output:

A*b*d**4*x**2/2 + B*c*e**4*x**8/8 + x**7*(A*c*e**4/7 + B*b*e**4/7 + 4*B*c* 
d*e**3/7) + x**6*(A*b*e**4/6 + 2*A*c*d*e**3/3 + 2*B*b*d*e**3/3 + B*c*d**2* 
e**2) + x**5*(4*A*b*d*e**3/5 + 6*A*c*d**2*e**2/5 + 6*B*b*d**2*e**2/5 + 4*B 
*c*d**3*e/5) + x**4*(3*A*b*d**2*e**2/2 + A*c*d**3*e + B*b*d**3*e + B*c*d** 
4/4) + x**3*(4*A*b*d**3*e/3 + A*c*d**4/3 + B*b*d**4/3)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.51 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {1}{2} \, A b d^{4} x^{2} + \frac {1}{7} \, {\left (4 \, B c d e^{3} + {\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \, {\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac {2}{5} \, {\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \, {\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (4 \, A b d^{3} e + {\left (B b + A c\right )} d^{4}\right )} x^{3} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x, algorithm="maxima")
 

Output:

1/8*B*c*e^4*x^8 + 1/2*A*b*d^4*x^2 + 1/7*(4*B*c*d*e^3 + (B*b + A*c)*e^4)*x^ 
7 + 1/6*(6*B*c*d^2*e^2 + A*b*e^4 + 4*(B*b + A*c)*d*e^3)*x^6 + 2/5*(2*B*c*d 
^3*e + 2*A*b*d*e^3 + 3*(B*b + A*c)*d^2*e^2)*x^5 + 1/4*(B*c*d^4 + 6*A*b*d^2 
*e^2 + 4*(B*b + A*c)*d^3*e)*x^4 + 1/3*(4*A*b*d^3*e + (B*b + A*c)*d^4)*x^3
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.85 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{8} \, B c e^{4} x^{8} + \frac {4}{7} \, B c d e^{3} x^{7} + \frac {1}{7} \, B b e^{4} x^{7} + \frac {1}{7} \, A c e^{4} x^{7} + B c d^{2} e^{2} x^{6} + \frac {2}{3} \, B b d e^{3} x^{6} + \frac {2}{3} \, A c d e^{3} x^{6} + \frac {1}{6} \, A b e^{4} x^{6} + \frac {4}{5} \, B c d^{3} e x^{5} + \frac {6}{5} \, B b d^{2} e^{2} x^{5} + \frac {6}{5} \, A c d^{2} e^{2} x^{5} + \frac {4}{5} \, A b d e^{3} x^{5} + \frac {1}{4} \, B c d^{4} x^{4} + B b d^{3} e x^{4} + A c d^{3} e x^{4} + \frac {3}{2} \, A b d^{2} e^{2} x^{4} + \frac {1}{3} \, B b d^{4} x^{3} + \frac {1}{3} \, A c d^{4} x^{3} + \frac {4}{3} \, A b d^{3} e x^{3} + \frac {1}{2} \, A b d^{4} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/8*B*c*e^4*x^8 + 4/7*B*c*d*e^3*x^7 + 1/7*B*b*e^4*x^7 + 1/7*A*c*e^4*x^7 + 
B*c*d^2*e^2*x^6 + 2/3*B*b*d*e^3*x^6 + 2/3*A*c*d*e^3*x^6 + 1/6*A*b*e^4*x^6 
+ 4/5*B*c*d^3*e*x^5 + 6/5*B*b*d^2*e^2*x^5 + 6/5*A*c*d^2*e^2*x^5 + 4/5*A*b* 
d*e^3*x^5 + 1/4*B*c*d^4*x^4 + B*b*d^3*e*x^4 + A*c*d^3*e*x^4 + 3/2*A*b*d^2* 
e^2*x^4 + 1/3*B*b*d^4*x^3 + 1/3*A*c*d^4*x^3 + 4/3*A*b*d^3*e*x^3 + 1/2*A*b* 
d^4*x^2
 

Mupad [B] (verification not implemented)

Time = 10.75 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.54 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=x^4\,\left (\frac {B\,c\,d^4}{4}+A\,c\,d^3\,e+B\,b\,d^3\,e+\frac {3\,A\,b\,d^2\,e^2}{2}\right )+x^6\,\left (\frac {A\,b\,e^4}{6}+\frac {2\,A\,c\,d\,e^3}{3}+\frac {2\,B\,b\,d\,e^3}{3}+B\,c\,d^2\,e^2\right )+x^3\,\left (\frac {A\,c\,d^4}{3}+\frac {B\,b\,d^4}{3}+\frac {4\,A\,b\,d^3\,e}{3}\right )+x^7\,\left (\frac {A\,c\,e^4}{7}+\frac {B\,b\,e^4}{7}+\frac {4\,B\,c\,d\,e^3}{7}\right )+\frac {2\,d\,e\,x^5\,\left (2\,A\,b\,e^2+2\,B\,c\,d^2+3\,A\,c\,d\,e+3\,B\,b\,d\,e\right )}{5}+\frac {A\,b\,d^4\,x^2}{2}+\frac {B\,c\,e^4\,x^8}{8} \] Input:

int((b*x + c*x^2)*(A + B*x)*(d + e*x)^4,x)
 

Output:

x^4*((B*c*d^4)/4 + A*c*d^3*e + B*b*d^3*e + (3*A*b*d^2*e^2)/2) + x^6*((A*b* 
e^4)/6 + (2*A*c*d*e^3)/3 + (2*B*b*d*e^3)/3 + B*c*d^2*e^2) + x^3*((A*c*d^4) 
/3 + (B*b*d^4)/3 + (4*A*b*d^3*e)/3) + x^7*((A*c*e^4)/7 + (B*b*e^4)/7 + (4* 
B*c*d*e^3)/7) + (2*d*e*x^5*(2*A*b*e^2 + 2*B*c*d^2 + 3*A*c*d*e + 3*B*b*d*e) 
)/5 + (A*b*d^4*x^2)/2 + (B*c*e^4*x^8)/8
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.88 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {x^{2} \left (105 b c \,e^{4} x^{6}+120 a c \,e^{4} x^{5}+120 b^{2} e^{4} x^{5}+480 b c d \,e^{3} x^{5}+140 a b \,e^{4} x^{4}+560 a c d \,e^{3} x^{4}+560 b^{2} d \,e^{3} x^{4}+840 b c \,d^{2} e^{2} x^{4}+672 a b d \,e^{3} x^{3}+1008 a c \,d^{2} e^{2} x^{3}+1008 b^{2} d^{2} e^{2} x^{3}+672 b c \,d^{3} e \,x^{3}+1260 a b \,d^{2} e^{2} x^{2}+840 a c \,d^{3} e \,x^{2}+840 b^{2} d^{3} e \,x^{2}+210 b c \,d^{4} x^{2}+1120 a b \,d^{3} e x +280 a c \,d^{4} x +280 b^{2} d^{4} x +420 a b \,d^{4}\right )}{840} \] Input:

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x),x)
 

Output:

(x**2*(420*a*b*d**4 + 1120*a*b*d**3*e*x + 1260*a*b*d**2*e**2*x**2 + 672*a* 
b*d*e**3*x**3 + 140*a*b*e**4*x**4 + 280*a*c*d**4*x + 840*a*c*d**3*e*x**2 + 
 1008*a*c*d**2*e**2*x**3 + 560*a*c*d*e**3*x**4 + 120*a*c*e**4*x**5 + 280*b 
**2*d**4*x + 840*b**2*d**3*e*x**2 + 1008*b**2*d**2*e**2*x**3 + 560*b**2*d* 
e**3*x**4 + 120*b**2*e**4*x**5 + 210*b*c*d**4*x**2 + 672*b*c*d**3*e*x**3 + 
 840*b*c*d**2*e**2*x**4 + 480*b*c*d*e**3*x**5 + 105*b*c*e**4*x**6))/840