\(\int \frac {1}{(d+e x^2)^{7/2} (a-c x^4)} \, dx\) [348]

Optimal result

Integrand size = 22, antiderivative size = 293 \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=-\frac {e^2 x}{5 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{5/2}}-\frac {2 e^2 \left (7 c d^2-2 a e^2\right ) x}{15 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}-\frac {e^2 \left (73 c^2 d^4-21 a c d^2 e^2+8 a^2 e^4\right ) x}{15 d^3 \left (c d^2-a e^2\right )^3 \sqrt {d+e x^2}}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}}+\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2}} \] Output:

-1/5*e^2*x/d/(-a*e^2+c*d^2)/(e*x^2+d)^(5/2)-2/15*e^2*(-2*a*e^2+7*c*d^2)*x/ 
d^2/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2)-1/15*e^2*(8*a^2*e^4-21*a*c*d^2*e^2+73 
*c^2*d^4)*x/d^3/(-a*e^2+c*d^2)^3/(e*x^2+d)^(1/2)+1/2*c^(3/2)*arctan((c^(1/ 
2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/(c^(1/2)*d-a^(1/2 
)*e)^(7/2)+1/2*c^(3/2)*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^ 
2+d)^(1/2))/a^(3/4)/(c^(1/2)*d+a^(1/2)*e)^(7/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.13 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\frac {e^{3/2} \left (-\frac {2 \sqrt {e} x \left (-3 a c d^2 e^2 \left (15 d^2+20 d e x^2+7 e^2 x^4\right )+a^2 e^4 \left (15 d^2+20 d e x^2+8 e^2 x^4\right )+c^2 d^4 \left (90 d^2+160 d e x^2+73 e^2 x^4\right )\right )}{d^3 \left (d+e x^2\right )^{5/2}}+15 c^2 \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {3 c d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+a d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-10 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-14 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+3 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{30 \left (c d^2-a e^2\right )^3} \] Input:

Integrate[1/((d + e*x^2)^(7/2)*(a - c*x^4)),x]
 

Output:

(e^(3/2)*((-2*Sqrt[e]*x*(-3*a*c*d^2*e^2*(15*d^2 + 20*d*e*x^2 + 7*e^2*x^4) 
+ a^2*e^4*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4) + c^2*d^4*(90*d^2 + 160*d*e*x^ 
2 + 73*e^2*x^4)))/(d^3*(d + e*x^2)^(5/2)) + 15*c^2*RootSum[c*d^4 - 4*c*d^3 
*#1 + 6*c*d^2*#1^2 - 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1^4 & , (3*c*d^4*Log[ 
d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + a*d^2*e^2*Log[d + 2*e*x^ 
2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 10*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[ 
e]*x*Sqrt[d + e*x^2] - #1]*#1 - 14*a*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*S 
qrt[d + e*x^2] - #1]*#1 + 3*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1]*#1^2 + a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - # 
1]*#1^2)/(c*d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ]))/(30 
*(c*d^2 - a*e^2)^3)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1487, 209, 209, 208, 2257, 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.

\(\displaystyle \int \frac {1}{\left (a-c x^4\right ) \left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1487

\(\displaystyle \frac {c \int \frac {d-e x^2}{\left (e x^2+d\right )^{5/2} \left (a-c x^4\right )}dx}{c d^2-a e^2}-\frac {e^2 \int \frac {1}{\left (e x^2+d\right )^{7/2}}dx}{c d^2-a e^2}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {c \int \frac {d-e x^2}{\left (e x^2+d\right )^{5/2} \left (a-c x^4\right )}dx}{c d^2-a e^2}-\frac {e^2 \left (\frac {4 \int \frac {1}{\left (e x^2+d\right )^{5/2}}dx}{5 d}+\frac {x}{5 d \left (d+e x^2\right )^{5/2}}\right )}{c d^2-a e^2}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {c \int \frac {d-e x^2}{\left (e x^2+d\right )^{5/2} \left (a-c x^4\right )}dx}{c d^2-a e^2}-\frac {e^2 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {x}{5 d \left (d+e x^2\right )^{5/2}}\right )}{c d^2-a e^2}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {c \int \frac {d-e x^2}{\left (e x^2+d\right )^{5/2} \left (a-c x^4\right )}dx}{c d^2-a e^2}-\frac {e^2 \left (\frac {4 \left (\frac {2 x}{3 d^2 \sqrt {d+e x^2}}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {x}{5 d \left (d+e x^2\right )^{5/2}}\right )}{c d^2-a e^2}\)

\(\Big \downarrow \) 2257

\(\displaystyle \frac {c \int \left (\frac {\sqrt {c} d-\sqrt {a} e}{2 \sqrt {a} \sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right ) \left (e x^2+d\right )^{5/2}}-\frac {-\sqrt {c} d-\sqrt {a} e}{2 \sqrt {a} \sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \left (e x^2+d\right )^{5/2}}\right )dx}{c d^2-a e^2}-\frac {e^2 \left (\frac {4 \left (\frac {2 x}{3 d^2 \sqrt {d+e x^2}}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {x}{5 d \left (d+e x^2\right )^{5/2}}\right )}{c d^2-a e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}-\frac {e x \left (5 \sqrt {c} d-2 \sqrt {a} e\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{6 \sqrt {a} \sqrt {c} d^2 \sqrt {d+e x^2} \left (\sqrt {c} d-\sqrt {a} e\right )^2}+\frac {e x \left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 \sqrt {a} e+5 \sqrt {c} d\right )}{6 \sqrt {a} \sqrt {c} d^2 \sqrt {d+e x^2} \left (\sqrt {a} e+\sqrt {c} d\right )^2}-\frac {e x \left (\sqrt {a} e+\sqrt {c} d\right )}{6 \sqrt {a} \sqrt {c} d \left (d+e x^2\right )^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )}+\frac {e x \left (\sqrt {c} d-\sqrt {a} e\right )}{6 \sqrt {a} \sqrt {c} d \left (d+e x^2\right )^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )}\right )}{c d^2-a e^2}-\frac {e^2 \left (\frac {4 \left (\frac {2 x}{3 d^2 \sqrt {d+e x^2}}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}+\frac {x}{5 d \left (d+e x^2\right )^{5/2}}\right )}{c d^2-a e^2}\)

Input:

Int[1/((d + e*x^2)^(7/2)*(a - c*x^4)),x]
 

Output:

-((e^2*(x/(5*d*(d + e*x^2)^(5/2)) + (4*(x/(3*d*(d + e*x^2)^(3/2)) + (2*x)/ 
(3*d^2*Sqrt[d + e*x^2])))/(5*d)))/(c*d^2 - a*e^2)) + (c*((e*(Sqrt[c]*d - S 
qrt[a]*e)*x)/(6*Sqrt[a]*Sqrt[c]*d*(Sqrt[c]*d + Sqrt[a]*e)*(d + e*x^2)^(3/2 
)) - (e*(Sqrt[c]*d + Sqrt[a]*e)*x)/(6*Sqrt[a]*Sqrt[c]*d*(Sqrt[c]*d - Sqrt[ 
a]*e)*(d + e*x^2)^(3/2)) - (e*(5*Sqrt[c]*d - 2*Sqrt[a]*e)*(Sqrt[c]*d + Sqr 
t[a]*e)*x)/(6*Sqrt[a]*Sqrt[c]*d^2*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[d + e*x^2 
]) + (e*(Sqrt[c]*d - Sqrt[a]*e)*(5*Sqrt[c]*d + 2*Sqrt[a]*e)*x)/(6*Sqrt[a]* 
Sqrt[c]*d^2*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[d + e*x^2]) + (Sqrt[c]*(Sqrt[c] 
*d + Sqrt[a]*e)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e 
*x^2])])/(2*a^(3/4)*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + (Sqrt[c]*(Sqrt[c]*d - 
 Sqrt[a]*e)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^ 
2])])/(2*a^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))))/(c*d^2 - a*e^2)
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 1487

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Simp[e^2 
/(c*d^2 + a*e^2)   Int[(d + e*x^2)^q, x], x] + Simp[c/(c*d^2 + a*e^2)   Int 
[(d + e*x^2)^(q + 1)*((d - e*x^2)/(a + c*x^4)), x], x] /; FreeQ[{a, c, d, e 
}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q] && LtQ[q, -1]
 

rule 2009

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

rule 2257

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a 
, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.39

method	result	size
pseudoelliptic	\(\frac {\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, c^{2} d^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (e^{3} a^{2}+3 d^{2} e a c +3 \sqrt {d^{2} a c}\, a \,e^{2}+\sqrt {d^{2} a c}\, c \,d^{2}\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (c^{2} d^{4} \left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (e^{3} a^{2}+3 d^{2} e a c -3 \sqrt {d^{2} a c}\, a \,e^{2}-\sqrt {d^{2} a c}\, c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+2 \left (6 c^{2} d^{6}+\frac {32 c^{2} d^{5} e \,x^{2}}{3}-3 c \left (-\frac {73 c \,x^{4}}{45}+a \right ) e^{2} d^{4}-4 a c \,d^{3} e^{3} x^{2}+a \,e^{4} \left (-\frac {7 c \,x^{4}}{5}+a \right ) d^{2}+\frac {4 a^{2} d \,e^{5} x^{2}}{3}+\frac {8 a^{2} e^{6} x^{4}}{15}\right ) x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\, e^{2}\right )}{2 \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (e \,x^{2}+d \right )^{\frac {5}{2}} \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (a \,e^{2}-c \,d^{2}\right )^{3} d^{3}}\)	\(408\)
default	\(\text {Expression too large to display}\)	\(5526\)

Input:

int(1/(e*x^2+d)^(7/2)/(-c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

1/2/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)/(e*x^2+d)^(5/2)/(d^2*a*c)^(1/2)*(((a* 
e+(d^2*a*c)^(1/2))*a)^(1/2)*c^2*d^4*(e*x^2+d)^(5/2)*(e^3*a^2+3*d^2*e*a*c+3 
*(d^2*a*c)^(1/2)*a*e^2+(d^2*a*c)^(1/2)*c*d^2)*arctan((e*x^2+d)^(1/2)/x*a/( 
(-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(c^2*d^4 
*(e*x^2+d)^(5/2)*(e^3*a^2+3*d^2*e*a*c-3*(d^2*a*c)^(1/2)*a*e^2-(d^2*a*c)^(1 
/2)*c*d^2)*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))+2* 
(6*c^2*d^6+32/3*c^2*d^5*e*x^2-3*c*(-73/45*c*x^4+a)*e^2*d^4-4*a*c*d^3*e^3*x 
^2+a*e^4*(-7/5*c*x^4+a)*d^2+4/3*a^2*d*e^5*x^2+8/15*a^2*e^6*x^4)*x*((a*e+(d 
^2*a*c)^(1/2))*a)^(1/2)*(d^2*a*c)^(1/2)*e^2))/((a*e+(d^2*a*c)^(1/2))*a)^(1 
/2)/(a*e^2-c*d^2)^3/d^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^(7/2)/(-c*x^4+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=- \int \frac {1}{- a d^{3} \sqrt {d + e x^{2}} - 3 a d^{2} e x^{2} \sqrt {d + e x^{2}} - 3 a d e^{2} x^{4} \sqrt {d + e x^{2}} - a e^{3} x^{6} \sqrt {d + e x^{2}} + c d^{3} x^{4} \sqrt {d + e x^{2}} + 3 c d^{2} e x^{6} \sqrt {d + e x^{2}} + 3 c d e^{2} x^{8} \sqrt {d + e x^{2}} + c e^{3} x^{10} \sqrt {d + e x^{2}}}\, dx \] Input:

integrate(1/(e*x**2+d)**(7/2)/(-c*x**4+a),x)
 

Output:

-Integral(1/(-a*d**3*sqrt(d + e*x**2) - 3*a*d**2*e*x**2*sqrt(d + e*x**2) - 
 3*a*d*e**2*x**4*sqrt(d + e*x**2) - a*e**3*x**6*sqrt(d + e*x**2) + c*d**3* 
x**4*sqrt(d + e*x**2) + 3*c*d**2*e*x**6*sqrt(d + e*x**2) + 3*c*d*e**2*x**8 
*sqrt(d + e*x**2) + c*e**3*x**10*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\int { -\frac {1}{{\left (c x^{4} - a\right )} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(7/2)/(-c*x^4+a),x, algorithm="maxima")
 

Output:

-integrate(1/((c*x^4 - a)*(e*x^2 + d)^(7/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^2+d)^(7/2)/(-c*x^4+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\int \frac {1}{\left (a-c\,x^4\right )\,{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int(1/((a - c*x^4)*(d + e*x^2)^(7/2)),x)
 

Output:

int(1/((a - c*x^4)*(d + e*x^2)^(7/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \left (a-c x^4\right )} \, dx=\int \frac {1}{\sqrt {e \,x^{2}+d}\, a \,d^{3}+3 \sqrt {e \,x^{2}+d}\, a \,d^{2} e \,x^{2}+3 \sqrt {e \,x^{2}+d}\, a d \,e^{2} x^{4}+\sqrt {e \,x^{2}+d}\, a \,e^{3} x^{6}-\sqrt {e \,x^{2}+d}\, c \,d^{3} x^{4}-3 \sqrt {e \,x^{2}+d}\, c \,d^{2} e \,x^{6}-3 \sqrt {e \,x^{2}+d}\, c d \,e^{2} x^{8}-\sqrt {e \,x^{2}+d}\, c \,e^{3} x^{10}}d x \] Input:

int(1/(e*x^2+d)^(7/2)/(-c*x^4+a),x)
 

Output:

int(1/(sqrt(d + e*x**2)*a*d**3 + 3*sqrt(d + e*x**2)*a*d**2*e*x**2 + 3*sqrt 
(d + e*x**2)*a*d*e**2*x**4 + sqrt(d + e*x**2)*a*e**3*x**6 - sqrt(d + e*x** 
2)*c*d**3*x**4 - 3*sqrt(d + e*x**2)*c*d**2*e*x**6 - 3*sqrt(d + e*x**2)*c*d 
*e**2*x**8 - sqrt(d + e*x**2)*c*e**3*x**10),x)