### 3.2149 $$\int (d+e x) (a+b x+c x^2)^4 \, dx$$

Optimal. Leaf size=268 $\frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10}$

[Out]

a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d +
3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*
b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e +
6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10

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Rubi [A]  time = 0.290969, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {631} $\frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^4,x]

[Out]

a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d +
3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*
b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e +
6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10

Rule 631

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

Rubi steps

\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx &=\int \left (a^4 d+a^3 (4 b d+a e) x+2 a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^2+2 a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^4+\left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+2 c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^6+2 c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^7+c^3 (c d+4 b e) x^8+c^4 e x^9\right ) \, dx\\ &=a^4 d x+\frac{1}{2} a^3 (4 b d+a e) x^2+\frac{2}{3} a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^3+\frac{1}{2} a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^4+\frac{1}{5} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^5+\frac{1}{6} \left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^6+\frac{2}{7} c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^7+\frac{1}{4} c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^8+\frac{1}{9} c^3 (c d+4 b e) x^9+\frac{1}{10} c^4 e x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0694483, size = 268, normalized size = 1. $\frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^4*d*x + (a^3*(4*b*d + a*e)*x^2)/2 + (2*a^2*(3*b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + (a*(2*b^3*d + 6*a*b*c*d +
3*a*b^2*e + 2*a^2*c*e)*x^4)/2 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*e + 12*a^2*b*c*e)*x^5)/5 + ((4*
b^3*c*d + 12*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^6)/6 + (2*c*(3*b^2*c*d + 2*a*c^2*d + 2*b^3*e +
6*a*b*c*e)*x^7)/7 + (c^2*(2*b*c*d + 3*b^2*e + 2*a*c*e)*x^8)/4 + (c^3*(c*d + 4*b*e)*x^9)/9 + (c^4*e*x^10)/10

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Maple [A]  time = 0.038, size = 343, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}e{x}^{10}}{10}}+{\frac{ \left ( 4\,eb{c}^{3}+d{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,db{c}^{3}+e \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( d \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) +e \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( d \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) +e \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) +e \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) +4\,e{a}^{3}b \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{4}+4\,d{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*e*x^10+1/9*(4*b*c^3*e+c^4*d)*x^9+1/8*(4*d*b*c^3+e*(2*(2*a*c+b^2)*c^2+4*b^2*c^2))*x^8+1/7*(d*(2*(2*a*c
+b^2)*c^2+4*b^2*c^2)+e*(4*b*a*c^2+4*(2*a*c+b^2)*b*c))*x^7+1/6*(d*(4*b*a*c^2+4*(2*a*c+b^2)*b*c)+e*(2*a^2*c^2+8*
a*c*b^2+(2*a*c+b^2)^2))*x^6+1/5*(d*(2*a^2*c^2+8*a*c*b^2+(2*a*c+b^2)^2)+e*(4*a^2*b*c+4*a*b*(2*a*c+b^2)))*x^5+1/
4*(d*(4*a^2*b*c+4*a*b*(2*a*c+b^2))+e*(2*a^2*(2*a*c+b^2)+4*b^2*a^2))*x^4+1/3*(d*(2*a^2*(2*a*c+b^2)+4*b^2*a^2)+4
*e*a^3*b)*x^3+1/2*(a^4*e+4*a^3*b*d)*x^2+a^4*d*x

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Maxima [A]  time = 1.01288, size = 373, normalized size = 1.39 \begin{align*} \frac{1}{10} \, c^{4} e x^{10} + \frac{1}{9} \,{\left (c^{4} d + 4 \, b c^{3} e\right )} x^{9} + \frac{1}{4} \,{\left (2 \, b c^{3} d +{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{8} + \frac{2}{7} \,{\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (4 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{6} + a^{4} d x + \frac{1}{5} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{5} + \frac{1}{2} \,{\left (2 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{4} + \frac{2}{3} \,{\left (2 \, a^{3} b e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*e*x^10 + 1/9*(c^4*d + 4*b*c^3*e)*x^9 + 1/4*(2*b*c^3*d + (3*b^2*c^2 + 2*a*c^3)*e)*x^8 + 2/7*((3*b^2*c^
2 + 2*a*c^3)*d + 2*(b^3*c + 3*a*b*c^2)*e)*x^7 + 1/6*(4*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*
e)*x^6 + a^4*d*x + 1/5*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 4*(a*b^3 + 3*a^2*b*c)*e)*x^5 + 1/2*(2*(a*b^3 + 3*a^
2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*e)*x^4 + 2/3*(2*a^3*b*e + (3*a^2*b^2 + 2*a^3*c)*d)*x^3 + 1/2*(4*a^3*b*d + a^4
*e)*x^2

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Fricas [A]  time = 1.75107, size = 713, normalized size = 2.66 \begin{align*} \frac{1}{10} x^{10} e c^{4} + \frac{1}{9} x^{9} d c^{4} + \frac{4}{9} x^{9} e c^{3} b + \frac{1}{2} x^{8} d c^{3} b + \frac{3}{4} x^{8} e c^{2} b^{2} + \frac{1}{2} x^{8} e c^{3} a + \frac{6}{7} x^{7} d c^{2} b^{2} + \frac{4}{7} x^{7} e c b^{3} + \frac{4}{7} x^{7} d c^{3} a + \frac{12}{7} x^{7} e c^{2} b a + \frac{2}{3} x^{6} d c b^{3} + \frac{1}{6} x^{6} e b^{4} + 2 x^{6} d c^{2} b a + 2 x^{6} e c b^{2} a + x^{6} e c^{2} a^{2} + \frac{1}{5} x^{5} d b^{4} + \frac{12}{5} x^{5} d c b^{2} a + \frac{4}{5} x^{5} e b^{3} a + \frac{6}{5} x^{5} d c^{2} a^{2} + \frac{12}{5} x^{5} e c b a^{2} + x^{4} d b^{3} a + 3 x^{4} d c b a^{2} + \frac{3}{2} x^{4} e b^{2} a^{2} + x^{4} e c a^{3} + 2 x^{3} d b^{2} a^{2} + \frac{4}{3} x^{3} d c a^{3} + \frac{4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e*c^4 + 1/9*x^9*d*c^4 + 4/9*x^9*e*c^3*b + 1/2*x^8*d*c^3*b + 3/4*x^8*e*c^2*b^2 + 1/2*x^8*e*c^3*a + 6/
7*x^7*d*c^2*b^2 + 4/7*x^7*e*c*b^3 + 4/7*x^7*d*c^3*a + 12/7*x^7*e*c^2*b*a + 2/3*x^6*d*c*b^3 + 1/6*x^6*e*b^4 + 2
*x^6*d*c^2*b*a + 2*x^6*e*c*b^2*a + x^6*e*c^2*a^2 + 1/5*x^5*d*b^4 + 12/5*x^5*d*c*b^2*a + 4/5*x^5*e*b^3*a + 6/5*
x^5*d*c^2*a^2 + 12/5*x^5*e*c*b*a^2 + x^4*d*b^3*a + 3*x^4*d*c*b*a^2 + 3/2*x^4*e*b^2*a^2 + x^4*e*c*a^3 + 2*x^3*d
*b^2*a^2 + 4/3*x^3*d*c*a^3 + 4/3*x^3*e*b*a^3 + 2*x^2*d*b*a^3 + 1/2*x^2*e*a^4 + x*d*a^4

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Sympy [A]  time = 0.113585, size = 313, normalized size = 1.17 \begin{align*} a^{4} d x + \frac{c^{4} e x^{10}}{10} + x^{9} \left (\frac{4 b c^{3} e}{9} + \frac{c^{4} d}{9}\right ) + x^{8} \left (\frac{a c^{3} e}{2} + \frac{3 b^{2} c^{2} e}{4} + \frac{b c^{3} d}{2}\right ) + x^{7} \left (\frac{12 a b c^{2} e}{7} + \frac{4 a c^{3} d}{7} + \frac{4 b^{3} c e}{7} + \frac{6 b^{2} c^{2} d}{7}\right ) + x^{6} \left (a^{2} c^{2} e + 2 a b^{2} c e + 2 a b c^{2} d + \frac{b^{4} e}{6} + \frac{2 b^{3} c d}{3}\right ) + x^{5} \left (\frac{12 a^{2} b c e}{5} + \frac{6 a^{2} c^{2} d}{5} + \frac{4 a b^{3} e}{5} + \frac{12 a b^{2} c d}{5} + \frac{b^{4} d}{5}\right ) + x^{4} \left (a^{3} c e + \frac{3 a^{2} b^{2} e}{2} + 3 a^{2} b c d + a b^{3} d\right ) + x^{3} \left (\frac{4 a^{3} b e}{3} + \frac{4 a^{3} c d}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac{a^{4} e}{2} + 2 a^{3} b d\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d*x + c**4*e*x**10/10 + x**9*(4*b*c**3*e/9 + c**4*d/9) + x**8*(a*c**3*e/2 + 3*b**2*c**2*e/4 + b*c**3*d/2)
+ x**7*(12*a*b*c**2*e/7 + 4*a*c**3*d/7 + 4*b**3*c*e/7 + 6*b**2*c**2*d/7) + x**6*(a**2*c**2*e + 2*a*b**2*c*e +
2*a*b*c**2*d + b**4*e/6 + 2*b**3*c*d/3) + x**5*(12*a**2*b*c*e/5 + 6*a**2*c**2*d/5 + 4*a*b**3*e/5 + 12*a*b**2*
c*d/5 + b**4*d/5) + x**4*(a**3*c*e + 3*a**2*b**2*e/2 + 3*a**2*b*c*d + a*b**3*d) + x**3*(4*a**3*b*e/3 + 4*a**3*
c*d/3 + 2*a**2*b**2*d) + x**2*(a**4*e/2 + 2*a**3*b*d)

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Giac [A]  time = 1.09179, size = 435, normalized size = 1.62 \begin{align*} \frac{1}{10} \, c^{4} x^{10} e + \frac{1}{9} \, c^{4} d x^{9} + \frac{4}{9} \, b c^{3} x^{9} e + \frac{1}{2} \, b c^{3} d x^{8} + \frac{3}{4} \, b^{2} c^{2} x^{8} e + \frac{1}{2} \, a c^{3} x^{8} e + \frac{6}{7} \, b^{2} c^{2} d x^{7} + \frac{4}{7} \, a c^{3} d x^{7} + \frac{4}{7} \, b^{3} c x^{7} e + \frac{12}{7} \, a b c^{2} x^{7} e + \frac{2}{3} \, b^{3} c d x^{6} + 2 \, a b c^{2} d x^{6} + \frac{1}{6} \, b^{4} x^{6} e + 2 \, a b^{2} c x^{6} e + a^{2} c^{2} x^{6} e + \frac{1}{5} \, b^{4} d x^{5} + \frac{12}{5} \, a b^{2} c d x^{5} + \frac{6}{5} \, a^{2} c^{2} d x^{5} + \frac{4}{5} \, a b^{3} x^{5} e + \frac{12}{5} \, a^{2} b c x^{5} e + a b^{3} d x^{4} + 3 \, a^{2} b c d x^{4} + \frac{3}{2} \, a^{2} b^{2} x^{4} e + a^{3} c x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac{4}{3} \, a^{3} c d x^{3} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^4*x^10*e + 1/9*c^4*d*x^9 + 4/9*b*c^3*x^9*e + 1/2*b*c^3*d*x^8 + 3/4*b^2*c^2*x^8*e + 1/2*a*c^3*x^8*e + 6/
7*b^2*c^2*d*x^7 + 4/7*a*c^3*d*x^7 + 4/7*b^3*c*x^7*e + 12/7*a*b*c^2*x^7*e + 2/3*b^3*c*d*x^6 + 2*a*b*c^2*d*x^6 +
1/6*b^4*x^6*e + 2*a*b^2*c*x^6*e + a^2*c^2*x^6*e + 1/5*b^4*d*x^5 + 12/5*a*b^2*c*d*x^5 + 6/5*a^2*c^2*d*x^5 + 4/
5*a*b^3*x^5*e + 12/5*a^2*b*c*x^5*e + a*b^3*d*x^4 + 3*a^2*b*c*d*x^4 + 3/2*a^2*b^2*x^4*e + a^3*c*x^4*e + 2*a^2*b
^2*d*x^3 + 4/3*a^3*c*d*x^3 + 4/3*a^3*b*x^3*e + 2*a^3*b*d*x^2 + 1/2*a^4*x^2*e + a^4*d*x