\(\int \frac {1}{x^4 \sqrt {\frac {e (a+b x^2)}{c+d x^2}}} \, dx\) [85]

Optimal result

Integrand size = 26, antiderivative size = 370 \[ \int \frac {1}{x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {-a-b x^2}{3 a x^3 \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}+\frac {(2 b c-a d) \left (a+b x^2\right )}{3 a^2 x \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}+\frac {\sqrt {d} (2 b c-a d) \left (a+b x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {d} \left (a+b x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}} \] Output:

1/3*(-b*x^2-a)/a/x^3/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)+1/3*(-a*d+2*b* 
c)*(b*x^2+a)/a^2/x/(d*x^2+c)/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)+1/3*d^ 
(1/2)*(-a*d+2*b*c)*(b*x^2+a)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2) 
,(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)/ 
(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)-1/3*b*c^(1/2)*d^(1/2)*(b*x^2+a)*Inv 
erseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/(c*(b*x^2+a) 
/a/(d*x^2+c))^(1/2)/(d*x^2+c)/(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.49 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b c x^2+a \left (c+d x^2\right )\right )-i b c (-2 b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i b c (-b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \] Input:

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

Output:

(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-2*b*c*x^2 + a*(c + d*x^2))) - I*b*c 
*(-2*b*c + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*Ar 
cSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*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 
)])/(3*a^2*Sqrt[b/a]*c*x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2058, 377, 25, 445, 25, 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.

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {a+b x^2} \left (-\frac {\frac {b d \left ((2 b c-a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{a c x}}{3 a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a x^3}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {a+b x^2} \left (-\frac {\frac {b d \left ((2 b c-a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{a c x}}{3 a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a x^3}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {a+b x^2} \left (-\frac {\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(2 b c-a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{a c x}}{3 a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a x^3}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\)

Input:

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

Output:

(Sqrt[a + b*x^2]*(-1/3*(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*x^3) - (-(((2* 
b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (b*d*((2*b*c - a*d) 
*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellip 
ticE[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]*El 
lipticF[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*a)))/(Sqrt[(e*(a 
+ b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 313

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]
 

rule 320

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]
 

rule 377

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]
 

rule 388

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]
 

rule 406

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]
 

rule 445

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]
 

rule 2058

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]
 

Maple [A] (verified)

Time = 7.40 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (a d \,x^{2}-2 b c \,x^{2}+a c \right )}{3 a^{2} x^{3} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}-\frac {d b \left (\frac {2 \left (a d -2 b c \right ) a c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}+\frac {a c \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{3 a^{2} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(377\)
default	\(-\frac {\left (b \,x^{2}+a \right ) \left (\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+2 b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+\sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+2 \sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-\sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} x^{3} c \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\)	\(444\)

Input:

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

Output:

-1/3*(b*x^2+a)*(a*d*x^2-2*b*c*x^2+a*c)/a^2/x^3/c/(e*(b*x^2+a)/(d*x^2+c))^( 
1/2)-1/3/a^2*d*b/c*(2*(a*d-2*b*c)*a*c*e/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*( 
1+1/c*x^2*d)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)/(a*d*e+b*c* 
e+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-E 
llipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2)))+a*c/(-b/a)^(1/2)* 
(1+1/a*x^2*b)^(1/2)*(1+1/c*x^2*d)^(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)))/(e*(b* 
x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^(1/2)/(d*x^2+c)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {{\left (2 \, b^{2} c d - a b d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c d + {\left (a^{2} - a b\right )} d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b c d - a^{2} d^{2}\right )} x^{4} - a^{2} c^{2} + 2 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, a^{3} c e x^{3}} \] Input:

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

Output:

-1/3*((2*b^2*c*d - a*b*d^2)*sqrt(a*c*e/d^2)*x^3*sqrt(-b/a)*elliptic_e(arcs 
in(x*sqrt(-b/a)), a*d/(b*c)) - (2*b^2*c*d + (a^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)) - ((2*a*b*c* 
d - a^2*d^2)*x^4 - a^2*c^2 + 2*(a*b*c^2 - a^2*c*d)*x^2)*sqrt((b*e*x^2 + a* 
e)/(d*x^2 + c)))/(a^3*c*e*x^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*d + int((sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b*d**2*x + 
 int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*x**4 + a*d*x**6 + b*c*x**6 + 
 b*d*x**8),x)*a*c**2*x))/(a*c*e*x)