### 3.1464 $$\int (d+e x)^5 (a^2+2 a b x+b^2 x^2)^2 \, dx$$

Optimal. Leaf size=119 $-\frac{4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac{3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac{4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5}$

[Out]

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

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Rubi [A]  time = 0.208124, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 43} $-\frac{4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac{3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac{4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 0.048776, size = 350, normalized size = 2.94 $\frac{1}{4} b^2 e^3 x^8 \left (3 a^2 e^2+10 a b d e+5 b^2 d^2\right )+\frac{2}{7} b e^2 x^7 \left (15 a^2 b d e^2+2 a^3 e^3+20 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{6} e x^6 \left (60 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4+40 a b^3 d^3 e+5 b^4 d^4\right )+\frac{1}{5} d x^5 \left (60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4+20 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{2} a d^2 x^4 \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )+\frac{2}{3} a^2 d^3 x^3 \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^3 d^4 x^2 (5 a e+4 b d)+a^4 d^5 x+\frac{1}{9} b^3 e^4 x^9 (4 a e+5 b d)+\frac{1}{10} b^4 e^5 x^{10}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*d^5*x + (a^3*d^4*(4*b*d + 5*a*e)*x^2)/2 + (2*a^2*d^3*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2)*x^3)/3 + (a*d^2*
(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3)*x^4)/2 + (d*(b^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d
^2*e^2 + 40*a^3*b*d*e^3 + 5*a^4*e^4)*x^5)/5 + (e*(5*b^4*d^4 + 40*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 20*a^3*b*d
*e^3 + a^4*e^4)*x^6)/6 + (2*b*e^2*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 2*a^3*e^3)*x^7)/7 + (b^2*e^3*
(5*b^2*d^2 + 10*a*b*d*e + 3*a^2*e^2)*x^8)/4 + (b^3*e^4*(5*b*d + 4*a*e)*x^9)/9 + (b^4*e^5*x^10)/10

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Maple [B]  time = 0.041, size = 361, normalized size = 3. \begin{align*}{\frac{{e}^{5}{b}^{4}{x}^{10}}{10}}+{\frac{ \left ( 4\,a{b}^{3}{e}^{5}+5\,d{e}^{4}{b}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{5}{b}^{2}{a}^{2}+20\,d{e}^{4}a{b}^{3}+10\,{d}^{2}{e}^{3}{b}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,{e}^{5}{a}^{3}b+30\,d{e}^{4}{b}^{2}{a}^{2}+40\,{d}^{2}{e}^{3}a{b}^{3}+10\,{d}^{3}{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{5}{a}^{4}+20\,d{e}^{4}{a}^{3}b+60\,{d}^{2}{e}^{3}{b}^{2}{a}^{2}+40\,{d}^{3}{e}^{2}a{b}^{3}+5\,{d}^{4}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{4}+40\,{d}^{2}{e}^{3}{a}^{3}b+60\,{d}^{3}{e}^{2}{b}^{2}{a}^{2}+20\,{d}^{4}ea{b}^{3}+{d}^{5}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{4}+40\,{d}^{3}{e}^{2}{a}^{3}b+30\,{d}^{4}e{b}^{2}{a}^{2}+4\,{d}^{5}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{4}+20\,{d}^{4}e{a}^{3}b+6\,{d}^{5}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{d}^{4}e{a}^{4}+4\,{d}^{5}{a}^{3}b \right ){x}^{2}}{2}}+{d}^{5}{a}^{4}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/10*e^5*b^4*x^10+1/9*(4*a*b^3*e^5+5*b^4*d*e^4)*x^9+1/8*(6*a^2*b^2*e^5+20*a*b^3*d*e^4+10*b^4*d^2*e^3)*x^8+1/7*
(4*a^3*b*e^5+30*a^2*b^2*d*e^4+40*a*b^3*d^2*e^3+10*b^4*d^3*e^2)*x^7+1/6*(a^4*e^5+20*a^3*b*d*e^4+60*a^2*b^2*d^2*
e^3+40*a*b^3*d^3*e^2+5*b^4*d^4*e)*x^6+1/5*(5*a^4*d*e^4+40*a^3*b*d^2*e^3+60*a^2*b^2*d^3*e^2+20*a*b^3*d^4*e+b^4*
d^5)*x^5+1/4*(10*a^4*d^2*e^3+40*a^3*b*d^3*e^2+30*a^2*b^2*d^4*e+4*a*b^3*d^5)*x^4+1/3*(10*a^4*d^3*e^2+20*a^3*b*d
^4*e+6*a^2*b^2*d^5)*x^3+1/2*(5*a^4*d^4*e+4*a^3*b*d^5)*x^2+d^5*a^4*x

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Maxima [B]  time = 1.14052, size = 486, normalized size = 4.08 \begin{align*} \frac{1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac{1}{9} \,{\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac{1}{4} \,{\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac{2}{7} \,{\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/10*b^4*e^5*x^10 + a^4*d^5*x + 1/9*(5*b^4*d*e^4 + 4*a*b^3*e^5)*x^9 + 1/4*(5*b^4*d^2*e^3 + 10*a*b^3*d*e^4 + 3*
a^2*b^2*e^5)*x^8 + 2/7*(5*b^4*d^3*e^2 + 20*a*b^3*d^2*e^3 + 15*a^2*b^2*d*e^4 + 2*a^3*b*e^5)*x^7 + 1/6*(5*b^4*d^
4*e + 40*a*b^3*d^3*e^2 + 60*a^2*b^2*d^2*e^3 + 20*a^3*b*d*e^4 + a^4*e^5)*x^6 + 1/5*(b^4*d^5 + 20*a*b^3*d^4*e +
60*a^2*b^2*d^3*e^2 + 40*a^3*b*d^2*e^3 + 5*a^4*d*e^4)*x^5 + 1/2*(2*a*b^3*d^5 + 15*a^2*b^2*d^4*e + 20*a^3*b*d^3*
e^2 + 5*a^4*d^2*e^3)*x^4 + 2/3*(3*a^2*b^2*d^5 + 10*a^3*b*d^4*e + 5*a^4*d^3*e^2)*x^3 + 1/2*(4*a^3*b*d^5 + 5*a^4
*d^4*e)*x^2

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Fricas [B]  time = 1.48742, size = 853, normalized size = 7.17 \begin{align*} \frac{1}{10} x^{10} e^{5} b^{4} + \frac{5}{9} x^{9} e^{4} d b^{4} + \frac{4}{9} x^{9} e^{5} b^{3} a + \frac{5}{4} x^{8} e^{3} d^{2} b^{4} + \frac{5}{2} x^{8} e^{4} d b^{3} a + \frac{3}{4} x^{8} e^{5} b^{2} a^{2} + \frac{10}{7} x^{7} e^{2} d^{3} b^{4} + \frac{40}{7} x^{7} e^{3} d^{2} b^{3} a + \frac{30}{7} x^{7} e^{4} d b^{2} a^{2} + \frac{4}{7} x^{7} e^{5} b a^{3} + \frac{5}{6} x^{6} e d^{4} b^{4} + \frac{20}{3} x^{6} e^{2} d^{3} b^{3} a + 10 x^{6} e^{3} d^{2} b^{2} a^{2} + \frac{10}{3} x^{6} e^{4} d b a^{3} + \frac{1}{6} x^{6} e^{5} a^{4} + \frac{1}{5} x^{5} d^{5} b^{4} + 4 x^{5} e d^{4} b^{3} a + 12 x^{5} e^{2} d^{3} b^{2} a^{2} + 8 x^{5} e^{3} d^{2} b a^{3} + x^{5} e^{4} d a^{4} + x^{4} d^{5} b^{3} a + \frac{15}{2} x^{4} e d^{4} b^{2} a^{2} + 10 x^{4} e^{2} d^{3} b a^{3} + \frac{5}{2} x^{4} e^{3} d^{2} a^{4} + 2 x^{3} d^{5} b^{2} a^{2} + \frac{20}{3} x^{3} e d^{4} b a^{3} + \frac{10}{3} x^{3} e^{2} d^{3} a^{4} + 2 x^{2} d^{5} b a^{3} + \frac{5}{2} x^{2} e d^{4} a^{4} + x d^{5} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/10*x^10*e^5*b^4 + 5/9*x^9*e^4*d*b^4 + 4/9*x^9*e^5*b^3*a + 5/4*x^8*e^3*d^2*b^4 + 5/2*x^8*e^4*d*b^3*a + 3/4*x^
8*e^5*b^2*a^2 + 10/7*x^7*e^2*d^3*b^4 + 40/7*x^7*e^3*d^2*b^3*a + 30/7*x^7*e^4*d*b^2*a^2 + 4/7*x^7*e^5*b*a^3 + 5
/6*x^6*e*d^4*b^4 + 20/3*x^6*e^2*d^3*b^3*a + 10*x^6*e^3*d^2*b^2*a^2 + 10/3*x^6*e^4*d*b*a^3 + 1/6*x^6*e^5*a^4 +
1/5*x^5*d^5*b^4 + 4*x^5*e*d^4*b^3*a + 12*x^5*e^2*d^3*b^2*a^2 + 8*x^5*e^3*d^2*b*a^3 + x^5*e^4*d*a^4 + x^4*d^5*b
^3*a + 15/2*x^4*e*d^4*b^2*a^2 + 10*x^4*e^2*d^3*b*a^3 + 5/2*x^4*e^3*d^2*a^4 + 2*x^3*d^5*b^2*a^2 + 20/3*x^3*e*d^
4*b*a^3 + 10/3*x^3*e^2*d^3*a^4 + 2*x^2*d^5*b*a^3 + 5/2*x^2*e*d^4*a^4 + x*d^5*a^4

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Sympy [B]  time = 0.120092, size = 401, normalized size = 3.37 \begin{align*} a^{4} d^{5} x + \frac{b^{4} e^{5} x^{10}}{10} + x^{9} \left (\frac{4 a b^{3} e^{5}}{9} + \frac{5 b^{4} d e^{4}}{9}\right ) + x^{8} \left (\frac{3 a^{2} b^{2} e^{5}}{4} + \frac{5 a b^{3} d e^{4}}{2} + \frac{5 b^{4} d^{2} e^{3}}{4}\right ) + x^{7} \left (\frac{4 a^{3} b e^{5}}{7} + \frac{30 a^{2} b^{2} d e^{4}}{7} + \frac{40 a b^{3} d^{2} e^{3}}{7} + \frac{10 b^{4} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac{a^{4} e^{5}}{6} + \frac{10 a^{3} b d e^{4}}{3} + 10 a^{2} b^{2} d^{2} e^{3} + \frac{20 a b^{3} d^{3} e^{2}}{3} + \frac{5 b^{4} d^{4} e}{6}\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} b d^{2} e^{3} + 12 a^{2} b^{2} d^{3} e^{2} + 4 a b^{3} d^{4} e + \frac{b^{4} d^{5}}{5}\right ) + x^{4} \left (\frac{5 a^{4} d^{2} e^{3}}{2} + 10 a^{3} b d^{3} e^{2} + \frac{15 a^{2} b^{2} d^{4} e}{2} + a b^{3} d^{5}\right ) + x^{3} \left (\frac{10 a^{4} d^{3} e^{2}}{3} + \frac{20 a^{3} b d^{4} e}{3} + 2 a^{2} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{4} d^{4} e}{2} + 2 a^{3} b d^{5}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d**5*x + b**4*e**5*x**10/10 + x**9*(4*a*b**3*e**5/9 + 5*b**4*d*e**4/9) + x**8*(3*a**2*b**2*e**5/4 + 5*a*b
**3*d*e**4/2 + 5*b**4*d**2*e**3/4) + x**7*(4*a**3*b*e**5/7 + 30*a**2*b**2*d*e**4/7 + 40*a*b**3*d**2*e**3/7 + 1
0*b**4*d**3*e**2/7) + x**6*(a**4*e**5/6 + 10*a**3*b*d*e**4/3 + 10*a**2*b**2*d**2*e**3 + 20*a*b**3*d**3*e**2/3
+ 5*b**4*d**4*e/6) + x**5*(a**4*d*e**4 + 8*a**3*b*d**2*e**3 + 12*a**2*b**2*d**3*e**2 + 4*a*b**3*d**4*e + b**4*
d**5/5) + x**4*(5*a**4*d**2*e**3/2 + 10*a**3*b*d**3*e**2 + 15*a**2*b**2*d**4*e/2 + a*b**3*d**5) + x**3*(10*a**
4*d**3*e**2/3 + 20*a**3*b*d**4*e/3 + 2*a**2*b**2*d**5) + x**2*(5*a**4*d**4*e/2 + 2*a**3*b*d**5)

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Giac [B]  time = 1.10441, size = 514, normalized size = 4.32 \begin{align*} \frac{1}{10} \, b^{4} x^{10} e^{5} + \frac{5}{9} \, b^{4} d x^{9} e^{4} + \frac{5}{4} \, b^{4} d^{2} x^{8} e^{3} + \frac{10}{7} \, b^{4} d^{3} x^{7} e^{2} + \frac{5}{6} \, b^{4} d^{4} x^{6} e + \frac{1}{5} \, b^{4} d^{5} x^{5} + \frac{4}{9} \, a b^{3} x^{9} e^{5} + \frac{5}{2} \, a b^{3} d x^{8} e^{4} + \frac{40}{7} \, a b^{3} d^{2} x^{7} e^{3} + \frac{20}{3} \, a b^{3} d^{3} x^{6} e^{2} + 4 \, a b^{3} d^{4} x^{5} e + a b^{3} d^{5} x^{4} + \frac{3}{4} \, a^{2} b^{2} x^{8} e^{5} + \frac{30}{7} \, a^{2} b^{2} d x^{7} e^{4} + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} + \frac{15}{2} \, a^{2} b^{2} d^{4} x^{4} e + 2 \, a^{2} b^{2} d^{5} x^{3} + \frac{4}{7} \, a^{3} b x^{7} e^{5} + \frac{10}{3} \, a^{3} b d x^{6} e^{4} + 8 \, a^{3} b d^{2} x^{5} e^{3} + 10 \, a^{3} b d^{3} x^{4} e^{2} + \frac{20}{3} \, a^{3} b d^{4} x^{3} e + 2 \, a^{3} b d^{5} x^{2} + \frac{1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac{5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{4} d^{4} x^{2} e + a^{4} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/10*b^4*x^10*e^5 + 5/9*b^4*d*x^9*e^4 + 5/4*b^4*d^2*x^8*e^3 + 10/7*b^4*d^3*x^7*e^2 + 5/6*b^4*d^4*x^6*e + 1/5*b
^4*d^5*x^5 + 4/9*a*b^3*x^9*e^5 + 5/2*a*b^3*d*x^8*e^4 + 40/7*a*b^3*d^2*x^7*e^3 + 20/3*a*b^3*d^3*x^6*e^2 + 4*a*b
^3*d^4*x^5*e + a*b^3*d^5*x^4 + 3/4*a^2*b^2*x^8*e^5 + 30/7*a^2*b^2*d*x^7*e^4 + 10*a^2*b^2*d^2*x^6*e^3 + 12*a^2*
b^2*d^3*x^5*e^2 + 15/2*a^2*b^2*d^4*x^4*e + 2*a^2*b^2*d^5*x^3 + 4/7*a^3*b*x^7*e^5 + 10/3*a^3*b*d*x^6*e^4 + 8*a^
3*b*d^2*x^5*e^3 + 10*a^3*b*d^3*x^4*e^2 + 20/3*a^3*b*d^4*x^3*e + 2*a^3*b*d^5*x^2 + 1/6*a^4*x^6*e^5 + a^4*d*x^5*
e^4 + 5/2*a^4*d^2*x^4*e^3 + 10/3*a^4*d^3*x^3*e^2 + 5/2*a^4*d^4*x^2*e + a^4*d^5*x