\(\int \frac {1}{x^2 (a+b x^2+c x^4)^{3/2}} \, dx\) [1020]

Optimal result

Integrand size = 20, antiderivative size = 385 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x \sqrt {a+b x^2+c x^4}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{a^{3/2} \left (b^2-4 a c\right ) x \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (2 b-3 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a+b x^2+c x^4}} \] Output:

(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/x/(c*x^4+b*x^2+a)^(1/2)-2*(-3*a*c+b^2)* 
(c*x^4+b*x^2+a)^(1/2)/a^(3/2)/(-4*a*c+b^2)/x/(a^(1/2)+c^(1/2)*x^2)-2*c^(1/ 
4)*(-3*a*c+b^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^ 
2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^ 
(1/2))^(1/2))/a^(7/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*(2*b-3*a^(1/2 
)*c^(1/2))*c^(1/4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2) 
*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2 
)/c^(1/2))^(1/2))/a^(7/4)/(b-2*a^(1/2)*c^(1/2))/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.89 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-4 a^2 c+2 b^2 x^2 \left (b+c x^2\right )+a \left (b^2-7 b c x^2-6 c^2 x^4\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[1/(x^2*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

-1/2*(2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(-4*a^2*c + 2*b^2*x^2*(b + c*x^2) 
+ a*(b^2 - 7*b*c*x^2 - 6*c^2*x^4)) - I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a* 
c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt 
[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[ 
I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a* 
c])/(b - Sqrt[b^2 - 4*a*c])] + I*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 
 3*a*c*Sqrt[b^2 - 4*a*c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sq 
rt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 
 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x] 
, (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(a^2*(b^2 - 4*a*c)*Sqr 
t[c/(b + Sqrt[b^2 - 4*a*c])]*x*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1441, 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.

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

\(\Big \downarrow \) 1441

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1604

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

Int[1/(x^2*(a + b*x^2 + c*x^4)^(3/2)),x]
 

Output:

(b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*x*Sqrt[a + b*x^2 + c*x^4]) + ((-2 
*(b^2 - 3*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x) + (c*((-2*(b^2 - 3*a*c)*(-(( 
x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + 
Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE 
[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqr 
t[a + b*x^2 + c*x^4])))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 3*a*c))/ 
Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[ 
c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c 
]))/4])/(2*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/a)/(a*(b^2 - 4*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 1441

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(-(d*x)^(m + 1))*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 
1)/(2*a*d*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) 
  Int[(d*x)^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[b^2*(m + 2*p + 3) - 2*a*c*(m 
 + 4*p + 5) + b*c*(m + 4*p + 7)*x^2, x], x], x] /; FreeQ[{a, b, c, d, m}, x 
] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -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])
 

Maple [A] (verified)

Time = 6.16 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{a^{2} x}-\frac {2 c \left (\frac {\left (2 a c -b^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right ) a^{2}}+\frac {b \left (3 a c -b^{2}\right ) x}{2 a^{2} \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {b}{a^{2}}+\frac {b \left (3 a c -b^{2}\right )}{a^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {c}{a^{2}}+\frac {c \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) a^{2}}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(536\)
elliptic	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{a^{2} x}-\frac {2 c \left (\frac {\left (2 a c -b^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right ) a^{2}}+\frac {b \left (3 a c -b^{2}\right ) x}{2 a^{2} \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {b}{a^{2}}+\frac {b \left (3 a c -b^{2}\right )}{a^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {c}{a^{2}}+\frac {c \left (2 a c -b^{2}\right )}{\left (4 a c -b^{2}\right ) a^{2}}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(536\)
risch	\(\text {Expression too large to display}\)	\(1450\)

Input:

int(1/x^2/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} x^{5} + {\left (b^{4} - 3 \, a b^{2} c\right )} x^{3} + {\left (a b^{3} - 3 \, a^{2} b c\right )} x - {\left ({\left (a b^{2} c - 3 \, a^{2} c^{2}\right )} x^{5} + {\left (a b^{3} - 3 \, a^{2} b c\right )} x^{3} + {\left (a^{2} b^{2} - 3 \, a^{3} c\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (6 \, a b c^{2} - {\left (a b^{2} + 2 \, b^{3}\right )} c\right )} x^{5} - {\left (a b^{3} + 2 \, b^{4} - 6 \, a b^{2} c\right )} x^{3} - {\left (a^{2} b^{2} + 2 \, a b^{3} - 6 \, a^{2} b c\right )} x - {\left ({\left (6 \, a^{2} c^{2} + {\left (a^{2} b - 2 \, a b^{2}\right )} c\right )} x^{5} + {\left (a^{2} b^{2} - 2 \, a b^{3} + 6 \, a^{2} b c\right )} x^{3} + {\left (a^{3} b - 2 \, a^{2} b^{2} + 6 \, a^{3} c\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left (2 \, {\left (a b^{2} c - 3 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + {\left (2 \, a b^{3} - 7 \, a^{2} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{5} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{3} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}} \] Input:

integrate(1/x^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

1/2*(2*sqrt(1/2)*((b^3*c - 3*a*b*c^2)*x^5 + (b^4 - 3*a*b^2*c)*x^3 + (a*b^3 
 - 3*a^2*b*c)*x - ((a*b^2*c - 3*a^2*c^2)*x^5 + (a*b^3 - 3*a^2*b*c)*x^3 + ( 
a^2*b^2 - 3*a^3*c)*x)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 
 4*a*c)/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a 
*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) 
+ sqrt(1/2)*((6*a*b*c^2 - (a*b^2 + 2*b^3)*c)*x^5 - (a*b^3 + 2*b^4 - 6*a*b^ 
2*c)*x^3 - (a^2*b^2 + 2*a*b^3 - 6*a^2*b*c)*x - ((6*a^2*c^2 + (a^2*b - 2*a* 
b^2)*c)*x^5 + (a^2*b^2 - 2*a*b^3 + 6*a^2*b*c)*x^3 + (a^3*b - 2*a^2*b^2 + 6 
*a^3*c)*x)*sqrt((b^2 - 4*a*c)/a^2))*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2 
) - b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - 
b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) - 2*(2*(a*b 
^2*c - 3*a^2*c^2)*x^4 + a^2*b^2 - 4*a^3*c + (2*a*b^3 - 7*a^2*b*c)*x^2)*sqr 
t(c*x^4 + b*x^2 + a))/((a^3*b^2*c - 4*a^4*c^2)*x^5 + (a^3*b^3 - 4*a^4*b*c) 
*x^3 + (a^4*b^2 - 4*a^5*c)*x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/x**2/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral(1/(x**2*(a + b*x**2 + c*x**4)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

integrate(1/x^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^2), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

integrate(1/x^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int(1/(x^2*(a + b*x^2 + c*x^4)^(3/2)),x)
 

Output:

int(1/(x^2*(a + b*x^2 + c*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{10}+2 b c \,x^{8}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{4}+a^{2} x^{2}}d x \] Input:

int(1/x^2/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a**2*x**2 + 2*a*b*x**4 + 2*a*c*x**6 + b**2* 
x**6 + 2*b*c*x**8 + c**2*x**10),x)