### 3.491 $$\int (d+e x)^4 (a+c x^2)^4 \, dx$$

Optimal. Leaf size=270 $\frac{1}{9} c^2 x^9 \left (6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right )+\frac{4}{7} a c x^7 \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+\frac{1}{5} a^2 x^5 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+3 a^2 c^2 d e^3 x^8+\frac{2}{3} a^3 d^2 x^3 \left (3 a e^2+2 c d^2\right )+\frac{8}{3} a^3 c d e^3 x^6+a^4 d^4 x+a^4 d e^3 x^4+\frac{2}{11} c^3 e^2 x^{11} \left (2 a e^2+3 c d^2\right )+\frac{8}{5} a c^3 d e^3 x^{10}+\frac{2 d^3 e \left (a+c x^2\right )^5}{5 c}+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}$

[Out]

a^4*d^4*x + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^4*d*e^3*x^4 + (a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4
)*x^5)/5 + (8*a^3*c*d*e^3*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4)*x^7)/7 + 3*a^2*c^2*d*e^3*x^8 + (
c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x^9)/9 + (8*a*c^3*d*e^3*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^
11)/11 + (c^4*d*e^3*x^12)/3 + (c^4*e^4*x^13)/13 + (2*d^3*e*(a + c*x^2)^5)/(5*c)

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Rubi [A]  time = 0.252391, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {696, 1810} $\frac{1}{9} c^2 x^9 \left (6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right )+\frac{4}{7} a c x^7 \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+\frac{1}{5} a^2 x^5 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+3 a^2 c^2 d e^3 x^8+\frac{2}{3} a^3 d^2 x^3 \left (3 a e^2+2 c d^2\right )+\frac{8}{3} a^3 c d e^3 x^6+a^4 d^4 x+a^4 d e^3 x^4+\frac{2}{11} c^3 e^2 x^{11} \left (2 a e^2+3 c d^2\right )+\frac{8}{5} a c^3 d e^3 x^{10}+\frac{2 d^3 e \left (a+c x^2\right )^5}{5 c}+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(a + c*x^2)^4,x]

[Out]

a^4*d^4*x + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^4*d*e^3*x^4 + (a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4
)*x^5)/5 + (8*a^3*c*d*e^3*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4)*x^7)/7 + 3*a^2*c^2*d*e^3*x^8 + (
c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x^9)/9 + (8*a*c^3*d*e^3*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^
11)/11 + (c^4*d*e^3*x^12)/3 + (c^4*e^4*x^13)/13 + (2*d^3*e*(a + c*x^2)^5)/(5*c)

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*m*d^(m - 1)*(a + c*x^2)^(p + 1))
/(2*c*(p + 1)), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx &=\frac{2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-4 d^3 e x+(d+e x)^4\right ) \, dx\\ &=\frac{2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^4+2 a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^2+4 a^4 d e^3 x^3+a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^4+16 a^3 c d e^3 x^5+4 a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^6+24 a^2 c^2 d e^3 x^7+c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^8+16 a c^3 d e^3 x^9+2 c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{10}+4 c^4 d e^3 x^{11}+c^4 e^4 x^{12}\right ) \, dx\\ &=a^4 d^4 x+\frac{2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^4 d e^3 x^4+\frac{1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac{8}{3} a^3 c d e^3 x^6+\frac{4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+3 a^2 c^2 d e^3 x^8+\frac{1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac{8}{5} a c^3 d e^3 x^{10}+\frac{2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}+\frac{2 d^3 e \left (a+c x^2\right )^5}{5 c}\\ \end{align*}

Mathematica [A]  time = 0.0445337, size = 300, normalized size = 1.11 $\frac{1}{9} c^2 x^9 \left (6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right )+\frac{4}{7} a c x^7 \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+\frac{1}{5} a^2 x^5 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+\frac{4}{3} a^2 c d e x^6 \left (2 a e^2+3 c d^2\right )+a^3 d e x^4 \left (a e^2+4 c d^2\right )+\frac{2}{3} a^3 d^2 x^3 \left (3 a e^2+2 c d^2\right )+2 a^4 d^3 e x^2+a^4 d^4 x+\frac{2}{11} c^3 e^2 x^{11} \left (2 a e^2+3 c d^2\right )+\frac{2}{5} c^3 d e x^{10} \left (4 a e^2+c d^2\right )+a c^2 d e x^8 \left (3 a e^2+2 c d^2\right )+\frac{1}{3} c^4 d e^3 x^{12}+\frac{1}{13} c^4 e^4 x^{13}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^4*d^4*x + 2*a^4*d^3*e*x^2 + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^3*d*e*(4*c*d^2 + a*e^2)*x^4 + (a^2*(6*
c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (4*a^2*c*d*e*(3*c*d^2 + 2*a*e^2)*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c
*d^2*e^2 + a^2*e^4)*x^7)/7 + a*c^2*d*e*(2*c*d^2 + 3*a*e^2)*x^8 + (c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x
^9)/9 + (2*c^3*d*e*(c*d^2 + 4*a*e^2)*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (c^4*d*e^3*x^12)/3 +
(c^4*e^4*x^13)/13

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Maple [A]  time = 0.043, size = 313, normalized size = 1.2 \begin{align*}{\frac{{c}^{4}{e}^{4}{x}^{13}}{13}}+{\frac{{c}^{4}d{e}^{3}{x}^{12}}{3}}+{\frac{ \left ( 4\,{e}^{4}a{c}^{3}+6\,{d}^{2}{e}^{2}{c}^{4} \right ){x}^{11}}{11}}+{\frac{ \left ( 16\,d{e}^{3}a{c}^{3}+4\,{d}^{3}e{c}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{4}{a}^{2}{c}^{2}+24\,{d}^{2}{e}^{2}a{c}^{3}+{c}^{4}{d}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 24\,d{e}^{3}{a}^{2}{c}^{2}+16\,{d}^{3}ea{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,{e}^{4}{a}^{3}c+36\,{d}^{2}{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{4}a{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 16\,d{e}^{3}{a}^{3}c+24\,{d}^{3}e{a}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}{a}^{4}+24\,{d}^{2}{e}^{2}{a}^{3}c+6\,{d}^{4}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{4}+16\,{d}^{3}e{a}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{4}+4\,{d}^{4}{a}^{3}c \right ){x}^{3}}{3}}+2\,{d}^{3}e{a}^{4}{x}^{2}+{a}^{4}{d}^{4}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*e^4*x^13+1/3*c^4*d*e^3*x^12+1/11*(4*a*c^3*e^4+6*c^4*d^2*e^2)*x^11+1/10*(16*a*c^3*d*e^3+4*c^4*d^3*e)*x
^10+1/9*(6*a^2*c^2*e^4+24*a*c^3*d^2*e^2+c^4*d^4)*x^9+1/8*(24*a^2*c^2*d*e^3+16*a*c^3*d^3*e)*x^8+1/7*(4*a^3*c*e^
4+36*a^2*c^2*d^2*e^2+4*a*c^3*d^4)*x^7+1/6*(16*a^3*c*d*e^3+24*a^2*c^2*d^3*e)*x^6+1/5*(a^4*e^4+24*a^3*c*d^2*e^2+
6*a^2*c^2*d^4)*x^5+1/4*(4*a^4*d*e^3+16*a^3*c*d^3*e)*x^4+1/3*(6*a^4*d^2*e^2+4*a^3*c*d^4)*x^3+2*d^3*e*a^4*x^2+a^
4*d^4*x

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Maxima [A]  time = 1.17414, size = 413, normalized size = 1.53 \begin{align*} \frac{1}{13} \, c^{4} e^{4} x^{13} + \frac{1}{3} \, c^{4} d e^{3} x^{12} + \frac{2}{11} \,{\left (3 \, c^{4} d^{2} e^{2} + 2 \, a c^{3} e^{4}\right )} x^{11} + \frac{2}{5} \,{\left (c^{4} d^{3} e + 4 \, a c^{3} d e^{3}\right )} x^{10} + 2 \, a^{4} d^{3} e x^{2} + \frac{1}{9} \,{\left (c^{4} d^{4} + 24 \, a c^{3} d^{2} e^{2} + 6 \, a^{2} c^{2} e^{4}\right )} x^{9} + a^{4} d^{4} x +{\left (2 \, a c^{3} d^{3} e + 3 \, a^{2} c^{2} d e^{3}\right )} x^{8} + \frac{4}{7} \,{\left (a c^{3} d^{4} + 9 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{7} + \frac{4}{3} \,{\left (3 \, a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, a^{2} c^{2} d^{4} + 24 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} x^{5} +{\left (4 \, a^{3} c d^{3} e + a^{4} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, a^{3} c d^{4} + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*e^4*x^13 + 1/3*c^4*d*e^3*x^12 + 2/11*(3*c^4*d^2*e^2 + 2*a*c^3*e^4)*x^11 + 2/5*(c^4*d^3*e + 4*a*c^3*d*
e^3)*x^10 + 2*a^4*d^3*e*x^2 + 1/9*(c^4*d^4 + 24*a*c^3*d^2*e^2 + 6*a^2*c^2*e^4)*x^9 + a^4*d^4*x + (2*a*c^3*d^3*
e + 3*a^2*c^2*d*e^3)*x^8 + 4/7*(a*c^3*d^4 + 9*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^7 + 4/3*(3*a^2*c^2*d^3*e + 2*a^3*
c*d*e^3)*x^6 + 1/5*(6*a^2*c^2*d^4 + 24*a^3*c*d^2*e^2 + a^4*e^4)*x^5 + (4*a^3*c*d^3*e + a^4*d*e^3)*x^4 + 2/3*(2
*a^3*c*d^4 + 3*a^4*d^2*e^2)*x^3

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Fricas [A]  time = 1.67179, size = 699, normalized size = 2.59 \begin{align*} \frac{1}{13} x^{13} e^{4} c^{4} + \frac{1}{3} x^{12} e^{3} d c^{4} + \frac{6}{11} x^{11} e^{2} d^{2} c^{4} + \frac{4}{11} x^{11} e^{4} c^{3} a + \frac{2}{5} x^{10} e d^{3} c^{4} + \frac{8}{5} x^{10} e^{3} d c^{3} a + \frac{1}{9} x^{9} d^{4} c^{4} + \frac{8}{3} x^{9} e^{2} d^{2} c^{3} a + \frac{2}{3} x^{9} e^{4} c^{2} a^{2} + 2 x^{8} e d^{3} c^{3} a + 3 x^{8} e^{3} d c^{2} a^{2} + \frac{4}{7} x^{7} d^{4} c^{3} a + \frac{36}{7} x^{7} e^{2} d^{2} c^{2} a^{2} + \frac{4}{7} x^{7} e^{4} c a^{3} + 4 x^{6} e d^{3} c^{2} a^{2} + \frac{8}{3} x^{6} e^{3} d c a^{3} + \frac{6}{5} x^{5} d^{4} c^{2} a^{2} + \frac{24}{5} x^{5} e^{2} d^{2} c a^{3} + \frac{1}{5} x^{5} e^{4} a^{4} + 4 x^{4} e d^{3} c a^{3} + x^{4} e^{3} d a^{4} + \frac{4}{3} x^{3} d^{4} c a^{3} + 2 x^{3} e^{2} d^{2} a^{4} + 2 x^{2} e d^{3} a^{4} + x d^{4} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*e^4*c^4 + 1/3*x^12*e^3*d*c^4 + 6/11*x^11*e^2*d^2*c^4 + 4/11*x^11*e^4*c^3*a + 2/5*x^10*e*d^3*c^4 + 8/
5*x^10*e^3*d*c^3*a + 1/9*x^9*d^4*c^4 + 8/3*x^9*e^2*d^2*c^3*a + 2/3*x^9*e^4*c^2*a^2 + 2*x^8*e*d^3*c^3*a + 3*x^8
*e^3*d*c^2*a^2 + 4/7*x^7*d^4*c^3*a + 36/7*x^7*e^2*d^2*c^2*a^2 + 4/7*x^7*e^4*c*a^3 + 4*x^6*e*d^3*c^2*a^2 + 8/3*
x^6*e^3*d*c*a^3 + 6/5*x^5*d^4*c^2*a^2 + 24/5*x^5*e^2*d^2*c*a^3 + 1/5*x^5*e^4*a^4 + 4*x^4*e*d^3*c*a^3 + x^4*e^3
*d*a^4 + 4/3*x^3*d^4*c*a^3 + 2*x^3*e^2*d^2*a^4 + 2*x^2*e*d^3*a^4 + x*d^4*a^4

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Sympy [A]  time = 0.120948, size = 340, normalized size = 1.26 \begin{align*} a^{4} d^{4} x + 2 a^{4} d^{3} e x^{2} + \frac{c^{4} d e^{3} x^{12}}{3} + \frac{c^{4} e^{4} x^{13}}{13} + x^{11} \left (\frac{4 a c^{3} e^{4}}{11} + \frac{6 c^{4} d^{2} e^{2}}{11}\right ) + x^{10} \left (\frac{8 a c^{3} d e^{3}}{5} + \frac{2 c^{4} d^{3} e}{5}\right ) + x^{9} \left (\frac{2 a^{2} c^{2} e^{4}}{3} + \frac{8 a c^{3} d^{2} e^{2}}{3} + \frac{c^{4} d^{4}}{9}\right ) + x^{8} \left (3 a^{2} c^{2} d e^{3} + 2 a c^{3} d^{3} e\right ) + x^{7} \left (\frac{4 a^{3} c e^{4}}{7} + \frac{36 a^{2} c^{2} d^{2} e^{2}}{7} + \frac{4 a c^{3} d^{4}}{7}\right ) + x^{6} \left (\frac{8 a^{3} c d e^{3}}{3} + 4 a^{2} c^{2} d^{3} e\right ) + x^{5} \left (\frac{a^{4} e^{4}}{5} + \frac{24 a^{3} c d^{2} e^{2}}{5} + \frac{6 a^{2} c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 4 a^{3} c d^{3} e\right ) + x^{3} \left (2 a^{4} d^{2} e^{2} + \frac{4 a^{3} c d^{4}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**4*x + 2*a**4*d**3*e*x**2 + c**4*d*e**3*x**12/3 + c**4*e**4*x**13/13 + x**11*(4*a*c**3*e**4/11 + 6*c**4
*d**2*e**2/11) + x**10*(8*a*c**3*d*e**3/5 + 2*c**4*d**3*e/5) + x**9*(2*a**2*c**2*e**4/3 + 8*a*c**3*d**2*e**2/3
+ c**4*d**4/9) + x**8*(3*a**2*c**2*d*e**3 + 2*a*c**3*d**3*e) + x**7*(4*a**3*c*e**4/7 + 36*a**2*c**2*d**2*e**2
/7 + 4*a*c**3*d**4/7) + x**6*(8*a**3*c*d*e**3/3 + 4*a**2*c**2*d**3*e) + x**5*(a**4*e**4/5 + 24*a**3*c*d**2*e**
2/5 + 6*a**2*c**2*d**4/5) + x**4*(a**4*d*e**3 + 4*a**3*c*d**3*e) + x**3*(2*a**4*d**2*e**2 + 4*a**3*c*d**4/3)

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Giac [A]  time = 1.33728, size = 421, normalized size = 1.56 \begin{align*} \frac{1}{13} \, c^{4} x^{13} e^{4} + \frac{1}{3} \, c^{4} d x^{12} e^{3} + \frac{6}{11} \, c^{4} d^{2} x^{11} e^{2} + \frac{2}{5} \, c^{4} d^{3} x^{10} e + \frac{1}{9} \, c^{4} d^{4} x^{9} + \frac{4}{11} \, a c^{3} x^{11} e^{4} + \frac{8}{5} \, a c^{3} d x^{10} e^{3} + \frac{8}{3} \, a c^{3} d^{2} x^{9} e^{2} + 2 \, a c^{3} d^{3} x^{8} e + \frac{4}{7} \, a c^{3} d^{4} x^{7} + \frac{2}{3} \, a^{2} c^{2} x^{9} e^{4} + 3 \, a^{2} c^{2} d x^{8} e^{3} + \frac{36}{7} \, a^{2} c^{2} d^{2} x^{7} e^{2} + 4 \, a^{2} c^{2} d^{3} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac{4}{7} \, a^{3} c x^{7} e^{4} + \frac{8}{3} \, a^{3} c d x^{6} e^{3} + \frac{24}{5} \, a^{3} c d^{2} x^{5} e^{2} + 4 \, a^{3} c d^{3} x^{4} e + \frac{4}{3} \, a^{3} c d^{4} x^{3} + \frac{1}{5} \, a^{4} x^{5} e^{4} + a^{4} d x^{4} e^{3} + 2 \, a^{4} d^{2} x^{3} e^{2} + 2 \, a^{4} d^{3} x^{2} e + a^{4} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*c^4*x^13*e^4 + 1/3*c^4*d*x^12*e^3 + 6/11*c^4*d^2*x^11*e^2 + 2/5*c^4*d^3*x^10*e + 1/9*c^4*d^4*x^9 + 4/11*a
*c^3*x^11*e^4 + 8/5*a*c^3*d*x^10*e^3 + 8/3*a*c^3*d^2*x^9*e^2 + 2*a*c^3*d^3*x^8*e + 4/7*a*c^3*d^4*x^7 + 2/3*a^2
*c^2*x^9*e^4 + 3*a^2*c^2*d*x^8*e^3 + 36/7*a^2*c^2*d^2*x^7*e^2 + 4*a^2*c^2*d^3*x^6*e + 6/5*a^2*c^2*d^4*x^5 + 4/
7*a^3*c*x^7*e^4 + 8/3*a^3*c*d*x^6*e^3 + 24/5*a^3*c*d^2*x^5*e^2 + 4*a^3*c*d^3*x^4*e + 4/3*a^3*c*d^4*x^3 + 1/5*a
^4*x^5*e^4 + a^4*d*x^4*e^3 + 2*a^4*d^2*x^3*e^2 + 2*a^4*d^3*x^2*e + a^4*d^4*x