\(\int \frac {1}{x^4 \sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [706]

Optimal result

Integrand size = 23, antiderivative size = 301 \[ \int \frac {1}{x^4 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {\sqrt {a c+(b c+a d) x^2+b d x^4}}{3 a c x^3}+\frac {2 (b c+a d) \sqrt {a c+(b c+a d) x^2+b d x^4}}{3 a c^2 x \left (a+b x^2\right )}+\frac {2 \sqrt {b} (b c+a d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\sqrt {b} d \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} c \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/3*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/a/c/x^3+2/3*(a*d+b*c)*(a*c+(a*d+b*c 
)*x^2+b*d*x^4)^(1/2)/a/c^2/x/(b*x^2+a)+2/3*b^(1/2)*(a*d+b*c)*(b*x^2+a)*(a* 
(d*x^2+c)/c/(b*x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2) 
,(1-a*d/b/c)^(1/2))/a^(5/2)/c/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/3*b^(1/2 
)*d*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*InverseJacobiAM(arctan(b^(1/ 
2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(3/2)/c/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/ 
2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.72 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^4 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 b c x^2+2 a d x^2\right )+2 i b c (b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (2 b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 x^3 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-(a*c) + 2*b*c*x^2 + 2*a*d*x^2) + (2*I 
)*b*c*(b*c + 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*(2*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^2*x^3*Sqrt[(a + b*x^2)*(c + d*x^2)])
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.95, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2048, 1443, 25, 1604, 25, 27, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 1443

\(\displaystyle \frac {\int -\frac {b d x^2+2 (b c+a d)}{x^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {b d x^2+2 (b c+a d)}{x^2 \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}\)

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {b d \int \frac {2 (b c+a d) x^2+a c}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{a c}-\frac {2 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}\)

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {2 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\frac {b d \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}\right )}{a c}-\frac {2 (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{a c x}}{3 a c}-\frac {\sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 a c x^3}\)

Input:

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

Output:

-1/3*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]/(a*c*x^3) - ((-2*(b*c + a*d)*Sq 
rt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(a*c*x) + (b*d*((-2*(b*c + a*d)*(-((x 
*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d] 
*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a* 
c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]* 
EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d 
)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c 
+ a*d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d]) + (a^(1/4)*c^(1/4)*(2*b*c + Sqr 
t[a]*Sqrt[b]*Sqrt[c]*Sqrt[d] + 2*a*d)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x 
^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt 
[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 
- (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sq 
rt[a*c + (b*c + a*d)*x^2 + b*d*x^4])))/(a*c))/(3*a*c)
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1443

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim 
p[1/(a*d^2*(m + 1))   Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* 
x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 
- 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [A] (verified)

Time = 3.99 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right ) \left (-2 x^{2} d a -2 b c \,x^{2}+a c \right )}{3 a^{2} c^{2} x^{3} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}-\frac {b d \left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\right )}{3 a^{2} c^{2}}\)	\(293\)
default	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 a c \,x^{3}}+\frac {2 \left (a d +b c \right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 a^{2} c^{2} x}-\frac {b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 a c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {2 \left (a d +b c \right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 a^{2} c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(307\)
elliptic	\(-\frac {\sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 a c \,x^{3}}+\frac {2 \left (a d +b c \right ) \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 a^{2} c^{2} x}-\frac {b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 a c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {2 \left (a d +b c \right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 a^{2} c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(307\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^4 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {2 \, {\left (b^{2} c + a b d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c + {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} {\left (a^{2} c - 2 \, {\left (a b c + a^{2} d\right )} x^{2}\right )}}{3 \, a^{3} c^{2} x^{3}} \] Input:

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

Output:

-1/3*(2*(b^2*c + a*b*d)*sqrt(a*c)*x^3*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt( 
-b/a)), a*d/(b*c)) - (2*b^2*c + (a^2 + 2*a*b)*d)*sqrt(a*c)*x^3*sqrt(-b/a)* 
elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) + sqrt(b*d*x^4 + (b*c + a*d)*x 
^2 + a*c)*(a^2*c - 2*(a*b*c + a^2*d)*x^2))/(a^3*c^2*x^3)
 

Sympy [F]

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

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

Output:

Integral(1/(x**4*sqrt((a + b*x**2)*(c + d*x**2))), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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)