Integrand size = 26, antiderivative size = 321 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\frac {d (b c-2 a d) x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 a c^2}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c x^3}-\frac {(b c-2 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 a c^2 x}-\frac {\sqrt {d} (b c-2 a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
1/3*d*(-2*a*d+b*c)*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^2-1/3*(d*x^2+c)*(e* (b*x^2+a)/(d*x^2+c))^(1/2)/c/x^3-1/3*(-2*a*d+b*c)*(d*x^2+c)*(e*(b*x^2+a)/( d*x^2+c))^(1/2)/a/c^2/x-1/3*(-2*a*d+b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c) ^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^ (1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1 /2)-1/3*b*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1 /2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^( 1/2)/a/c^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=-\frac {\sqrt {\frac {b}{a}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (b c x^2+a \left (c-2 d x^2\right )\right )-i b c (-b c+2 a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (-b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^2 x^3 \left (a+b x^2\right )} \]
-1/3*(Sqrt[b/a]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*( c + d*x^2)*(b*c*x^2 + a*(c - 2*d*x^2)) - I*b*c*(-(b*c) + 2*a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)] + I*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c^2*x^3*(a + b*x^2))
Time = 0.50 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2058, 377, 445, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {\sqrt {b x^2+a}}{x^4 \sqrt {d x^2+c}}dx}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\int \frac {-b d x^2+b c-2 a d}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\int \frac {b d \left (a c-(b c-2 a d) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {b d \int \frac {a c-(b c-2 a d) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {b d \left (a c \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-(b c-2 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-(b c-2 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-(b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-(b c-2 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-2 a d)}{a c x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}\right )}{\sqrt {a+b x^2}}\) |
(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-1/3*(Sqrt[a + b*x^2]* Sqrt[c + d*x^2])/(c*x^3) + (-(((b*c - 2*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])/(a*c*x)) - (b*d*(-((b*c - 2*a*d)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^ 2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d* x^2]))) + (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(a*c))/(3*c)))/Sqrt[a + b*x^2]
3.3.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 5.43 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (-2 a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{3 c^{2} x^{3} a}-\frac {d b \left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 \left (2 a d -b c \right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{3 a \,c^{2} \left (b \,x^{2}+a \right )}\) | \(378\) |
| default | \(-\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+\sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}-b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}+\sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}-\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}+2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{2} x^{3} a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(443\) |
-1/3*(d*x^2+c)*(-2*a*d*x^2+b*c*x^2+a*c)/c^2/x^3/a*(e*(b*x^2+a)/(d*x^2+c))^ (1/2)-1/3*d/a*b/c^2*(a*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2 )/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d*e+b*c*e)/c/b/e)^(1/2))-2*(2*a*d-b*c)*a*c*e/(-b/a)^(1/2)*(1+b*x^2/a)^ (1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)/(e*d *a+e*b*c+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^( 1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))))*(e*(b*x^2 +a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)/(b*x^2+a)
Time = 0.11 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\frac {{\left (b^{2} c d - 2 \, a b d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c d - {\left (a^{2} + 2 \, a b\right )} d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, a^{2} c^{2} x^{3}} \]
1/3*((b^2*c*d - 2*a*b*d^2)*sqrt(a*c*e/d^2)*x^3*sqrt(-b/a)*elliptic_e(arcsi n(x*sqrt(-b/a)), a*d/(b*c)) - (b^2*c*d - (a^2 + 2*a*b)*d^2)*sqrt(a*c*e/d^2 )*x^3*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((a*b*c*d - 2*a^2*d^2)*x^4 + a^2*c^2 + (a*b*c^2 - a^2*c*d)*x^2)*sqrt((b*e*x^2 + a*e)/ (d*x^2 + c)))/(a^2*c^2*x^3)
Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{4}} \,d x } \]
\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^4} \, dx=\int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^4} \,d x \]