\(\int \frac {1}{(-a+b x^2+c x^4)^{3/2}} \, dx\) [319]

Optimal result

Integrand size = 18, antiderivative size = 407 \[ \int \frac {1}{\left (-a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (b^2+2 a c+b c x^2\right )}{a \left (b^2+4 a c\right ) \sqrt {-a+b x^2+c x^4}}+\frac {b \sqrt {c} x \sqrt {-a+b x^2+c x^4}}{a \left (b^2+4 a c\right ) \left (\sqrt {-a}+\sqrt {c} x^2\right )}+\frac {b \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a-b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right )|\frac {1}{4} \left (2+\frac {a b}{(-a)^{3/2} \sqrt {c}}\right )\right )}{\sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {-a+b x^2+c x^4}}-\frac {\left (b+2 \sqrt {-a} \sqrt {c}\right ) \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a-b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right ),\frac {1}{4} \left (2+\frac {a b}{(-a)^{3/2} \sqrt {c}}\right )\right )}{2 \sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {-a+b x^2+c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.66 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (-a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-4 \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} x \left (b^2+2 a c+b c x^2\right )+i b \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2+4 a c}+4 c x^2}{b-\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-i \left (-b^2-4 a c+b \sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2+4 a c}+4 c x^2}{b-\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 a \left (b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {-a+b x^2+c x^4}} \] Input:

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

Output:

(-4*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*x*(b^2 + 2*a*c + b*c*x^2) + I*b*(-b + 
Sqrt[b^2 + 4*a*c])*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 + Sq 
rt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])] - I*(-b^2 - 4*a*c + b*Sqrt[b^2 + 
 4*a*c])*Sqrt[(b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*S 
qrt[(2*b - 2*Sqrt[b^2 + 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Ellipti 
cF[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])])/(4*a*(b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 + 
 4*a*c])]*Sqrt[-a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.63, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1405, 25, 27, 1514, 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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

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

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 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 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 1514

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.14

method	result	size
default	\(-\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {1}{a}+\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}+\frac {b c \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\left (4 a c +b^{2}\right ) \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(464\)
elliptic	\(-\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {1}{a}+\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}+\frac {b c \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\left (4 a c +b^{2}\right ) \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right )}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(464\)

Input:

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

Output:

-2*c*(1/2*b/a/(4*a*c+b^2)*x^3+1/2*(2*a*c+b^2)/a/(4*a*c+b^2)/c*x)/((x^4+1/c 
*b*x^2-1/c*a)*c)^(1/2)+1/2*(-1/a+(2*a*c+b^2)/a/(4*a*c+b^2))/(-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*(-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)) 
+b/(4*a*c+b^2)*c/(-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*(-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 
*(-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 = 474, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (-a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (a b c x^{4} + a b^{2} x^{2} - a^{2} b\right )} \sqrt {-a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + {\left (b^{2} c x^{4} + b^{3} x^{2} - a b^{2}\right )} \sqrt {-a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left ({\left (2 \, a^{2} + a b\right )} c x^{4} - 2 \, a^{3} - a^{2} b + {\left (2 \, a^{2} b + a b^{2}\right )} x^{2}\right )} \sqrt {-a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - {\left ({\left (2 \, a b - b^{2}\right )} c x^{4} - 2 \, a^{2} b + a b^{2} + {\left (2 \, a b^{2} - b^{3}\right )} x^{2}\right )} \sqrt {-a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (a b c x^{3} + {\left (a b^{2} + 2 \, a^{2} c\right )} x\right )} \sqrt {c x^{4} + b x^{2} - a}}{2 \, {\left (a^{3} b^{2} + 4 \, a^{4} c - {\left (a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{4} - {\left (a^{2} b^{3} + 4 \, a^{3} b c\right )} x^{2}\right )}} \] Input:

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

Output:

1/2*(sqrt(1/2)*((a*b*c*x^4 + a*b^2*x^2 - a^2*b)*sqrt(-a)*sqrt((b^2 + 4*a*c 
)/a^2) + (b^2*c*x^4 + b^3*x^2 - a*b^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)*(((2*a^2 + a*b)*c*x^4 - 2*a^3 - a^2*b + (2*a^2*b + a*b^2)*x^2)*sqrt( 
-a)*sqrt((b^2 + 4*a*c)/a^2) - ((2*a*b - b^2)*c*x^4 - 2*a^2*b + a*b^2 + (2* 
a*b^2 - b^3)*x^2)*sqrt(-a))*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) + b)/a)*ellipt 
ic_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*(a*b*c*x^3 + (a*b^2 + 2 
*a^2*c)*x)*sqrt(c*x^4 + b*x^2 - a))/(a^3*b^2 + 4*a^4*c - (a^2*b^2*c + 4*a^ 
3*c^2)*x^4 - (a^2*b^3 + 4*a^3*b*c)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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