Integrand size = 29, antiderivative size = 339 \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {e (c d-b e) x}{a d \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x^2}}+\frac {-d-2 e x^2}{a d^2 x \sqrt {d+e x^2}}-\frac {2 c^2 \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}-\frac {2 c^2 \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}} \]
e*(-b*e+c*d)*x/a/d/(c*d^2+e*(a*e-b*d))/(e*x^2+d)^(1/2)+(-2*e*x^2-d)/a/d^2/ x/(e*x^2+d)^(1/2)-2*c^2*arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e *x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(1+b/(-4*a*c+b^2)^(1/2))/a/(2* c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-2*c^2*arc tan(x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^ 2)^(1/2))^(1/2))*(1-b/(-4*a*c+b^2)^(1/2))/a/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2) ))^(3/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 17.54 (sec) , antiderivative size = 2158, normalized size of antiderivative = 6.37 \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
-((d + 2*e*x^2)/(a*d^2*x*Sqrt[d + e*x^2])) - ((c + (b*c)/Sqrt[b^2 - 4*a*c] )*x*(45*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c]) *e)*x^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))] + (30*e*x ^2*Sqrt[-(((-b + Sqrt[b^2 - 4*a*c])*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x ^2*(d + e*x^2))/(d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)^2))])/d - 45*ArcSin [Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a *c] - 2*c*x^2)))]] - (30*e*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4* a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/d - (45*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2*ArcSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4 *a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d*(-b + Sqrt[b^ 2 - 4*a*c] - 2*c*x^2)) - (30*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^4*Ar cSin[Sqrt[-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))]])/(d^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)) + 4*(-((( 2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c* x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^ 2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d + (-b + Sqr t[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2)))] + (4*e*x^ 2*(-(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*x^2)/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))))^(5/2)*Sqrt[((-b + Sqrt[b^2 - 4*a*c])*(d + e*x^2))/(d*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2))]*Hypergeometric2F1[2, 2, 7/2, -(((2*c*d +...
Time = 1.60 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1624, 245, 208, 2246, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1624 |
| \(\displaystyle \frac {e^2 \int \frac {1}{x^2 \left (e x^2+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}+\frac {\int \frac {-c e x^2+c d-b e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {e^2 \left (-\frac {2 e \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{d}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}+\frac {\int \frac {-c e x^2+c d-b e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {-c e x^2+c d-b e}{x^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2246 |
| \(\displaystyle \frac {\int \left (\frac {c d-b e}{a x^2 \sqrt {e x^2+d}}+\frac {e b^2-c d b-c (c d-b e) x^2-a c e}{a \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}\right )dx}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {c \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {c \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x^2} (c d-b e)}{a d x}}{a e^2-b d e+c d^2}+\frac {e^2 \left (-\frac {2 e x}{d^2 \sqrt {d+e x^2}}-\frac {1}{d x \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}\) |
(e^2*(-(1/(d*x*Sqrt[d + e*x^2])) - (2*e*x)/(d^2*Sqrt[d + e*x^2])))/(c*d^2 - b*d*e + a*e^2) + (-(((c*d - b*e)*Sqrt[d + e*x^2])/(a*d*x)) - (c*(c*d - b *e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])]) /(a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (c*(c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt [2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4 *a*c])*e]))/(c*d^2 - b*d*e + a*e^2)
3.4.98.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[e^2/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^m* (d + e*x^2)^q, x], x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^m*(d + e *x^2)^(q + 1)*(Simp[c*d - b*e - c*e*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1]
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x ] && PolyQ[Px, x] && IntegerQ[p]
Time = 1.14 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.51
| method | result | size |
| default | \(\frac {-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}}{a}-\frac {\sqrt {2}\, d \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (2 a \,c^{2}-b^{2} c \right ) d^{2}+e \left (-3 a b c +b^{3}\right ) d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\sqrt {2}\, d \sqrt {e \,x^{2}+d}\, \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a \,c^{2}+b^{2} c \right ) d^{2}+e \left (3 a b c -b^{3}\right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-2 \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, e x \left (b e -c d \right )\right )}{2 a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(513\) |
| pseudoelliptic | \(-\frac {\frac {\sqrt {2}\, d^{2} \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (2 a \,c^{2}-b^{2} c \right ) d^{2}+e \left (-3 a b c +b^{3}\right ) d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, x \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\left (-\frac {\sqrt {2}\, d^{2} x \sqrt {e \,x^{2}+d}\, \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a \,c^{2}+b^{2} c \right ) d^{2}+e \left (3 a b c -b^{3}\right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\left (d^{3} c -e \left (-c \,x^{2}+b \right ) d^{2}+e^{2} \left (-b \,x^{2}+a \right ) d +2 x^{2} e^{3} a \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a x \left (a \,e^{2}-b d e +c \,d^{2}\right ) d^{2}}\) | \(515\) |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}}{d^{2} a x}-\frac {\frac {e^{2} a \sqrt {\left (x +\frac {\sqrt {-e d}}{e}\right )^{2} e -2 \sqrt {-e d}\, \left (x +\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {\sqrt {-e d}}{e}\right )}+\frac {e^{2} a \sqrt {\left (x -\frac {\sqrt {-e d}}{e}\right )^{2} e +2 \sqrt {-e d}\, \left (x -\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x -\frac {\sqrt {-e d}}{e}\right )}-\frac {d \sqrt {2}\, \left (-\left (\left (a c e -b^{2} e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-3 b c d e +2 c^{2} d^{2}\right ) a +b^{2} d \left (b e -c d \right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right ) \left (\left (a c e -b^{2} e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (3 b c d e -2 c^{2} d^{2}\right ) a -b^{3} d e +b^{2} c \,d^{2}\right )\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}}{a d}\) | \(573\) |
1/a*(-1/d/x/(e*x^2+d)^(1/2)-2*e/d^2*x/(e*x^2+d)^(1/2))-1/2/a/((-2*a*e+b*d+ (-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)/(e*x^2+d)^(1/2)/((2*a*e-b*d+(-4*d^2 *(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(2^(1/2)*d*(((a*c-b^2)*e+b*c*d)*(-4*d^2*(a *c-1/4*b^2))^(1/2)+(2*a*c^2-b^2*c)*d^2+e*(-3*a*b*c+b^3)*d)*((-2*a*e+b*d+(- 4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(e*x^2+d)^(1/2)*arctanh(a/x*(e*x^2+d) ^(1/2)*2^(1/2)/((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2))-((2*a*e -b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(2^(1/2)*d*(e*x^2+d)^(1/2)*((( a*c-b^2)*e+b*c*d)*(-4*d^2*(a*c-1/4*b^2))^(1/2)+(-2*a*c^2+b^2*c)*d^2+e*(3*a *b*c-b^3)*d)*arctan(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a*c- 1/4*b^2))^(1/2))*a)^(1/2))-2*(-4*d^2*(a*c-1/4*b^2))^(1/2)*((-2*a*e+b*d+(-4 *d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*e*x*(b*e-c*d)))/(-4*d^2*(a*c-1/4*b^2)) ^(1/2)/(a*e^2-b*d*e+c*d^2)/d
Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]