### 3.2133 $$\int (d+e x)^2 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=272 $\frac{3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7}+\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}$

[Out]

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3)/(3*e^7) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4)/(4*e
^7) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^5)/(5*e^7) - ((2*c*d - b*
e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^6)/(6*e^7) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*
d - a*e))*(d + e*x)^7)/(7*e^7) - (3*c^2*(2*c*d - b*e)*(d + e*x)^8)/(8*e^7) + (c^3*(d + e*x)^9)/(9*e^7)

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Rubi [A]  time = 0.271228, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^7}+\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac{3 (d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{3 e^7}-\frac{3 c^2 (d+e x)^8 (2 c d-b e)}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(a + b*x + c*x^2)^3,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3)/(3*e^7) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4)/(4*e
^7) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^5)/(5*e^7) - ((2*c*d - b*
e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^6)/(6*e^7) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*
d - a*e))*(d + e*x)^7)/(7*e^7) - (3*c^2*(2*c*d - b*e)*(d + e*x)^8)/(8*e^7) + (c^3*(d + e*x)^9)/(9*e^7)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^4}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^5}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^7}{e^6}+\frac{c^3 (d+e x)^8}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{3 e^7}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^7}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^5}{5 e^7}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^6}{6 e^7}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac{3 c^2 (2 c d-b e) (d+e x)^8}{8 e^7}+\frac{c^3 (d+e x)^9}{9 e^7}\\ \end{align*}

Mathematica [A]  time = 0.0948533, size = 282, normalized size = 1.04 $\frac{1}{4} x^4 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+\frac{1}{2} a^2 d x^2 (2 a e+3 b d)+a^3 d^2 x+\frac{1}{7} c x^7 \left (3 c e (a e+2 b d)+3 b^2 e^2+c^2 d^2\right )+\frac{1}{6} x^6 \left (3 b c \left (2 a e^2+c d^2\right )+6 a c^2 d e+6 b^2 c d e+b^3 e^2\right )+\frac{1}{5} x^5 \left (3 b^2 \left (a e^2+c d^2\right )+12 a b c d e+3 a c \left (a e^2+c d^2\right )+2 b^3 d e\right )+\frac{1}{3} a x^3 \left (6 a b d e+a \left (a e^2+3 c d^2\right )+3 b^2 d^2\right )+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(a + b*x + c*x^2)^3,x]

[Out]

a^3*d^2*x + (a^2*d*(3*b*d + 2*a*e)*x^2)/2 + (a*(3*b^2*d^2 + 6*a*b*d*e + a*(3*c*d^2 + a*e^2))*x^3)/3 + ((b^3*d^
2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b*(2*c*d^2 + a*e^2))*x^4)/4 + ((2*b^3*d*e + 12*a*b*c*d*e + 3*b^2*(c*d^2 +
a*e^2) + 3*a*c*(c*d^2 + a*e^2))*x^5)/5 + ((6*b^2*c*d*e + 6*a*c^2*d*e + b^3*e^2 + 3*b*c*(c*d^2 + 2*a*e^2))*x^6)
/6 + (c*(c^2*d^2 + 3*b^2*e^2 + 3*c*e*(2*b*d + a*e))*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

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Maple [A]  time = 0.041, size = 359, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{2}b{c}^{2}+2\,de{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ({d}^{2}{c}^{3}+6\,deb{c}^{2}+{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{2}b{c}^{2}+2\,de \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,de \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{5}}{5}}+{\frac{ \left ({d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,de \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{a}^{2}b{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +6\,deb{a}^{2}+{e}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{3}+3\,{d}^{2}b{a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x+a)^3,x)

[Out]

1/9*c^3*e^2*x^9+1/8*(3*b*c^2*e^2+2*c^3*d*e)*x^8+1/7*(d^2*c^3+6*d*e*b*c^2+e^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^
7+1/6*(3*d^2*b*c^2+2*d*e*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+e^2*(4*a*b*c+b*(2*a*c+b^2)))*x^6+1/5*(d^2*(a*c^2+2*b^2*
c+c*(2*a*c+b^2))+2*d*e*(4*a*b*c+b*(2*a*c+b^2))+e^2*(a*(2*a*c+b^2)+2*b^2*a+a^2*c))*x^5+1/4*(d^2*(4*a*b*c+b*(2*a
*c+b^2))+2*d*e*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+3*a^2*b*e^2)*x^4+1/3*(d^2*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+6*d*e*b*a
^2+e^2*a^3)*x^3+1/2*(2*a^3*d*e+3*a^2*b*d^2)*x^2+a^3*d^2*x

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Maxima [A]  time = 0.978291, size = 371, normalized size = 1.36 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{8} \,{\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, b c^{2} d^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e +{\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{6} + a^{3} d^{2} x + \frac{1}{5} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d e + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, a^{2} b e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} + 6 \,{\left (a b^{2} + a^{2} c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b d e + a^{3} e^{2} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

1/9*c^3*e^2*x^9 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/7*(c^3*d^2 + 6*b*c^2*d*e + 3*(b^2*c + a*c^2)*e^2)*x^7
+ 1/6*(3*b*c^2*d^2 + 6*(b^2*c + a*c^2)*d*e + (b^3 + 6*a*b*c)*e^2)*x^6 + a^3*d^2*x + 1/5*(3*(b^2*c + a*c^2)*d^2
+ 2*(b^3 + 6*a*b*c)*d*e + 3*(a*b^2 + a^2*c)*e^2)*x^5 + 1/4*(3*a^2*b*e^2 + (b^3 + 6*a*b*c)*d^2 + 6*(a*b^2 + a^
2*c)*d*e)*x^4 + 1/3*(6*a^2*b*d*e + a^3*e^2 + 3*(a*b^2 + a^2*c)*d^2)*x^3 + 1/2*(3*a^2*b*d^2 + 2*a^3*d*e)*x^2

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Fricas [A]  time = 1.80794, size = 743, normalized size = 2.73 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{3}{8} x^{8} e^{2} c^{2} b + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{6}{7} x^{7} e d c^{2} b + \frac{3}{7} x^{7} e^{2} c b^{2} + \frac{3}{7} x^{7} e^{2} c^{2} a + \frac{1}{2} x^{6} d^{2} c^{2} b + x^{6} e d c b^{2} + \frac{1}{6} x^{6} e^{2} b^{3} + x^{6} e d c^{2} a + x^{6} e^{2} c b a + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} e d b^{3} + \frac{3}{5} x^{5} d^{2} c^{2} a + \frac{12}{5} x^{5} e d c b a + \frac{3}{5} x^{5} e^{2} b^{2} a + \frac{3}{5} x^{5} e^{2} c a^{2} + \frac{1}{4} x^{4} d^{2} b^{3} + \frac{3}{2} x^{4} d^{2} c b a + \frac{3}{2} x^{4} e d b^{2} a + \frac{3}{2} x^{4} e d c a^{2} + \frac{3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + x^{3} d^{2} c a^{2} + 2 x^{3} e d b a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/9*x^9*e^2*c^3 + 1/4*x^8*e*d*c^3 + 3/8*x^8*e^2*c^2*b + 1/7*x^7*d^2*c^3 + 6/7*x^7*e*d*c^2*b + 3/7*x^7*e^2*c*b^
2 + 3/7*x^7*e^2*c^2*a + 1/2*x^6*d^2*c^2*b + x^6*e*d*c*b^2 + 1/6*x^6*e^2*b^3 + x^6*e*d*c^2*a + x^6*e^2*c*b*a +
3/5*x^5*d^2*c*b^2 + 2/5*x^5*e*d*b^3 + 3/5*x^5*d^2*c^2*a + 12/5*x^5*e*d*c*b*a + 3/5*x^5*e^2*b^2*a + 3/5*x^5*e^2
*c*a^2 + 1/4*x^4*d^2*b^3 + 3/2*x^4*d^2*c*b*a + 3/2*x^4*e*d*b^2*a + 3/2*x^4*e*d*c*a^2 + 3/4*x^4*e^2*b*a^2 + x^3
*d^2*b^2*a + x^3*d^2*c*a^2 + 2*x^3*e*d*b*a^2 + 1/3*x^3*e^2*a^3 + 3/2*x^2*d^2*b*a^2 + x^2*e*d*a^3 + x*d^2*a^3

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Sympy [A]  time = 0.116787, size = 332, normalized size = 1.22 \begin{align*} a^{3} d^{2} x + \frac{c^{3} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 b c^{2} e^{2}}{8} + \frac{c^{3} d e}{4}\right ) + x^{7} \left (\frac{3 a c^{2} e^{2}}{7} + \frac{3 b^{2} c e^{2}}{7} + \frac{6 b c^{2} d e}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{6} \left (a b c e^{2} + a c^{2} d e + \frac{b^{3} e^{2}}{6} + b^{2} c d e + \frac{b c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} c e^{2}}{5} + \frac{3 a b^{2} e^{2}}{5} + \frac{12 a b c d e}{5} + \frac{3 a c^{2} d^{2}}{5} + \frac{2 b^{3} d e}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{3 a^{2} b e^{2}}{4} + \frac{3 a^{2} c d e}{2} + \frac{3 a b^{2} d e}{2} + \frac{3 a b c d^{2}}{2} + \frac{b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + 2 a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac{3 a^{2} b d^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x+a)**3,x)

[Out]

a**3*d**2*x + c**3*e**2*x**9/9 + x**8*(3*b*c**2*e**2/8 + c**3*d*e/4) + x**7*(3*a*c**2*e**2/7 + 3*b**2*c*e**2/7
+ 6*b*c**2*d*e/7 + c**3*d**2/7) + x**6*(a*b*c*e**2 + a*c**2*d*e + b**3*e**2/6 + b**2*c*d*e + b*c**2*d**2/2) +
x**5*(3*a**2*c*e**2/5 + 3*a*b**2*e**2/5 + 12*a*b*c*d*e/5 + 3*a*c**2*d**2/5 + 2*b**3*d*e/5 + 3*b**2*c*d**2/5)
+ x**4*(3*a**2*b*e**2/4 + 3*a**2*c*d*e/2 + 3*a*b**2*d*e/2 + 3*a*b*c*d**2/2 + b**3*d**2/4) + x**3*(a**3*e**2/3
+ 2*a**2*b*d*e + a**2*c*d**2 + a*b**2*d**2) + x**2*(a**3*d*e + 3*a**2*b*d**2/2)

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Giac [A]  time = 1.1392, size = 446, normalized size = 1.64 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{8} \, b c^{2} x^{8} e^{2} + \frac{6}{7} \, b c^{2} d x^{7} e + \frac{1}{2} \, b c^{2} d^{2} x^{6} + \frac{3}{7} \, b^{2} c x^{7} e^{2} + \frac{3}{7} \, a c^{2} x^{7} e^{2} + b^{2} c d x^{6} e + a c^{2} d x^{6} e + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{3}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{6} \, b^{3} x^{6} e^{2} + a b c x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{12}{5} \, a b c d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{3}{2} \, a b c d^{2} x^{4} + \frac{3}{5} \, a b^{2} x^{5} e^{2} + \frac{3}{5} \, a^{2} c x^{5} e^{2} + \frac{3}{2} \, a b^{2} d x^{4} e + \frac{3}{2} \, a^{2} c d x^{4} e + a b^{2} d^{2} x^{3} + a^{2} c d^{2} x^{3} + \frac{3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac{3}{2} \, a^{2} b d^{2} x^{2} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/9*c^3*x^9*e^2 + 1/4*c^3*d*x^8*e + 1/7*c^3*d^2*x^7 + 3/8*b*c^2*x^8*e^2 + 6/7*b*c^2*d*x^7*e + 1/2*b*c^2*d^2*x^
6 + 3/7*b^2*c*x^7*e^2 + 3/7*a*c^2*x^7*e^2 + b^2*c*d*x^6*e + a*c^2*d*x^6*e + 3/5*b^2*c*d^2*x^5 + 3/5*a*c^2*d^2*
x^5 + 1/6*b^3*x^6*e^2 + a*b*c*x^6*e^2 + 2/5*b^3*d*x^5*e + 12/5*a*b*c*d*x^5*e + 1/4*b^3*d^2*x^4 + 3/2*a*b*c*d^2
*x^4 + 3/5*a*b^2*x^5*e^2 + 3/5*a^2*c*x^5*e^2 + 3/2*a*b^2*d*x^4*e + 3/2*a^2*c*d*x^4*e + a*b^2*d^2*x^3 + a^2*c*d
^2*x^3 + 3/4*a^2*b*x^4*e^2 + 2*a^2*b*d*x^3*e + 3/2*a^2*b*d^2*x^2 + 1/3*a^3*x^3*e^2 + a^3*d*x^2*e + a^3*d^2*x