Integrand size = 29, antiderivative size = 512 \[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{5 a x^5}+\frac {4 e \sqrt {d+e x^2}}{15 a d x^3}+\frac {(b d-a e) \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^2 x}-\frac {2 e (b d-a e) \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (b^2 d-a c d-a b e\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {c \left (b^2 d-a c d-a b e+\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (b^2 d-a c d-a b e-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/5*(e*x^2+d)^(1/2)/a/x^5+4/15*e*(e*x^2+d)^(1/2)/a/d/x^3+1/3*(-a*e+b*d)*( e*x^2+d)^(1/2)/a^2/d/x^3-8/15*e^2*(e*x^2+d)^(1/2)/a/d^2/x-2/3*e*(-a*e+b*d) *(e*x^2+d)^(1/2)/a^2/d^2/x-(-a*b*e-a*c*d+b^2*d)*(e*x^2+d)^(1/2)/a^3/d/x-c* 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))*(b^2*d-a*c*d-a*b*e+(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d) /(-4*a*c+b^2)^(1/2))/a^3/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(b-(-4*a*c +b^2)^(1/2))^(1/2)-c*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))*(b^2*d-a*c*d-a*b*e+(-2*a^2*c*e+a* b^2*e+3*a*b*c*d-b^3*d)/(-4*a*c+b^2)^(1/2))/a^3/(b+(-4*a*c+b^2)^(1/2))^(1/2 )/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(10511\) vs. \(2(512)=1024\).
Time = 16.52 (sec) , antiderivative size = 10511, normalized size of antiderivative = 20.53 \[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
Time = 2.51 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1618, 245, 245, 242, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1618 |
| \(\displaystyle \frac {d \int \frac {1}{x^6 \sqrt {e x^2+d}}dx}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {d \left (-\frac {4 e \int \frac {1}{x^4 \sqrt {e x^2+d}}dx}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {d \left (-\frac {4 e \left (-\frac {2 e \int \frac {1}{x^2 \sqrt {e x^2+d}}dx}{3 d}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {d \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{a}-\frac {\int \frac {c d x^2+b d-a e}{x^4 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a}\) |
| \(\Big \downarrow \) 2246 |
| \(\displaystyle \frac {d \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{a}-\frac {\int \left (\frac {b d-a e}{a x^4 \sqrt {e x^2+d}}+\frac {-d b^2+a e b+a c d}{a^2 x^2 \sqrt {e x^2+d}}+\frac {d b^3-a e b^2-2 a c d b+c \left (d b^2-a e b-a c d\right ) x^2+a^2 c e}{a^2 \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}\right )dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (-\frac {4 e \left (\frac {2 e \sqrt {d+e x^2}}{3 d^2 x}-\frac {\sqrt {d+e x^2}}{3 d x^3}\right )}{5 d}-\frac {\sqrt {d+e x^2}}{5 d x^5}\right )}{a}-\frac {\frac {c \left (\frac {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {c \left (-\frac {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d+e x^2} \left (-a b e-a c d+b^2 d\right )}{a^2 d x}+\frac {2 e \sqrt {d+e x^2} (b d-a e)}{3 a d^2 x}-\frac {\sqrt {d+e x^2} (b d-a e)}{3 a d x^3}}{a}\) |
(d*(-1/5*Sqrt[d + e*x^2]/(d*x^5) - (4*e*(-1/3*Sqrt[d + e*x^2]/(d*x^3) + (2 *e*Sqrt[d + e*x^2])/(3*d^2*x)))/(5*d)))/a - (-1/3*((b*d - a*e)*Sqrt[d + e* x^2])/(a*d*x^3) + (2*e*(b*d - a*e)*Sqrt[d + e*x^2])/(3*a*d^2*x) + ((b^2*d - a*c*d - a*b*e)*Sqrt[d + e*x^2])/(a^2*d*x) + (c*(b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*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^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (c*(b^2*d - a*c*d - a*b*e - (b^3*d - 3*a*b*c*d - a*b^2*e + 2 *a^2*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^2*Sqrt[b + Sqrt[b ^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a
3.4.66.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
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[d/a Int[(f*x)^m*(d + e*x^2)^(q - 1), x], x] - Simp[1/(a*f^2) Int[(f*x)^(m + 2)*(d + e*x^2)^(q - 1)*(Simp[b*d - a*e + c*d*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && GtQ[q, 0] && LtQ[m, 0]
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 = 1.21 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (-2 a^{2} e^{2} x^{4}-5 a b d e \,x^{4}-15 a c \,d^{2} x^{4}+15 b^{2} d^{2} x^{4}+a^{2} d e \,x^{2}-5 a b \,d^{2} x^{2}+3 a^{2} d^{2}\right )}{15 d^{2} a^{3} x^{5}}+\frac {\sqrt {2}\, \left (-\left (\left (a^{2} c e +\left (-b^{2} e -2 b c d \right ) a +b^{3} d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-3 b c d e -2 c^{2} d^{2}\right ) a^{2}+b^{2} d \left (b e +4 c d \right ) a -b^{4} d^{2}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right ) \left (\left (a^{2} c e +\left (-b^{2} e -2 b c d \right ) a +b^{3} d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (3 b c d e +2 c^{2} d^{2}\right ) a^{2}+\left (-b^{3} d e -4 b^{2} c \,d^{2}\right ) a +b^{4} d^{2}\right )\right )}{2 a^{3} \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(500\) |
| pseudoelliptic | \(-\frac {\frac {5 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (\left (-2 a b c +b^{3}\right ) d +a e \left (a c -b^{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a^{2} c^{2}+4 a \,b^{2} c -b^{4}\right ) d^{2}+e \left (-3 c b \,a^{2}+a \,b^{3}\right ) d \right ) \sqrt {2}\, d^{2} x^{5} \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (-\frac {5 \sqrt {2}\, d^{2} \left (\left (\left (-2 a b c +b^{3}\right ) d +a e \left (a c -b^{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) d^{2}+e \left (3 c b \,a^{2}-a \,b^{3}\right ) d \right ) x^{5} \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (5 \left (-a c +b^{2}\right ) x^{4}-\frac {5 a b \,x^{2}}{3}+a^{2}\right ) d^{2}+\frac {a e \,x^{2} \left (-5 b \,x^{2}+a \right ) d}{3}-\frac {2 a^{2} e^{2} x^{4}}{3}\right ) \sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{5 \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x^{5} a^{3} d^{2}}\) | \(535\) |
| default | \(\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 d \,x^{5}}+\frac {2 e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 d^{2} x^{3}}}{a}+\frac {b \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 a^{2} d \,x^{3}}+\frac {\left (-a c +b^{2}\right ) \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}\right )}{a^{3}}-\frac {\left (\left (a^{2} c e +\left (-b^{2} e -2 b c d \right ) a +b^{3} 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^{2}+b^{2} d \left (b e +4 c d \right ) a -b^{4} 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 {2}\, \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 )-\left (\sqrt {2}\, \left (\left (a^{2} c e +\left (-b^{2} e -2 b c d \right ) a +b^{3} 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^{2}+\left (-b^{3} d e -4 b^{2} c \,d^{2}\right ) a +b^{4} d^{2}\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 )}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e}\, \left (a c -b^{2}\right )\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{2 a^{3} \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 )}}\) | \(623\) |
-1/15*(e*x^2+d)^(1/2)*(-2*a^2*e^2*x^4-5*a*b*d*e*x^4-15*a*c*d^2*x^4+15*b^2* d^2*x^4+a^2*d*e*x^2-5*a*b*d^2*x^2+3*a^2*d^2)/d^2/a^3/x^5+1/2/a^3/((2*a*e-b *d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a* c-1/4*b^2))^(1/2))*a)^(1/2)*(-((a^2*c*e+(-b^2*e-2*b*c*d)*a+b^3*d)*(-4*d^2* (a*c-1/4*b^2))^(1/2)+(-3*b*c*d*e-2*c^2*d^2)*a^2+b^2*d*(b*e+4*c*d)*a-b^4*d^ 2)*((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(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)*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))*((a^2*c*e+(-b^ 2*e-2*b*c*d)*a+b^3*d)*(-4*d^2*(a*c-1/4*b^2))^(1/2)+(3*b*c*d*e+2*c^2*d^2)*a ^2+(-b^3*d*e-4*b^2*c*d^2)*a+b^4*d^2))/(-4*d^2*(a*c-1/4*b^2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 5773 vs. \(2 (450) = 900\).
Time = 15.78 (sec) , antiderivative size = 5773, normalized size of antiderivative = 11.28 \[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x^{6} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^6 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {e\,x^2+d}}{x^6\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]