\(\int \frac {1}{(a+b x^2-c x^4)^{3/2}} \, dx\) [317]

Optimal result

Integrand size = 17, antiderivative size = 400 \[ \int \frac {1}{\left (a+b x^2-c x^4\right )^{3/2}} \, dx=\frac {x \left (b^2+2 a c-b c x^2\right )}{a \left (b^2+4 a c\right ) \sqrt {a+b x^2-c x^4}}-\frac {b \sqrt {c} x \sqrt {a+b x^2-c x^4}}{a \left (b^2+4 a c\right ) \left (\sqrt {-a}+\sqrt {c} x^2\right )}+\frac {b \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a+b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right )|\frac {1}{4} \left (2+\frac {b}{\sqrt {-a} \sqrt {c}}\right )\right )}{\sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {a+b x^2-c x^4}}-\frac {\left (b-2 \sqrt {-a} \sqrt {c}\right ) \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a+b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right ),\frac {1}{4} \left (2+\frac {b}{\sqrt {-a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {a+b x^2-c x^4}} \] 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+b/(-a)^(1/2)/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+b/(-a)^(1/2)/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 = 10.54 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+b x^2-c x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x \left (b^2+2 a c-b c x^2\right )-i \sqrt {2} b \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+i \sqrt {2} \left (-b^2-4 a c+b \sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}-2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {-b+\sqrt {b^2+4 a c}+2 c x^2}{-b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{4 a \left (b^2+4 a c\right ) \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {a+b x^2-c x^4}} \] 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*Sqrt[2 
]*b*(-b + Sqrt[b^2 + 4*a*c])*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2)/(b + S 
qrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*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*Sqrt[2]*(-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 + Sq 
rt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*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])])/(4*a*(b^2 + 4*a*c)*Sq 
rt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1405, 25, 27, 1514, 399, 321, 327}

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

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 + Sqrt[ 
b^2 + 4*a*c])]*(-1/2*(b*(b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c] 
]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + 
Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*c^(3/2)) + (Sqrt[b + 
 Sqrt[b^2 + 4*a*c]]*(b^2 + 4*a*c - b*Sqrt[b^2 + 4*a*c])*EllipticF[ArcSin[( 
Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/( 
b - Sqrt[b^2 + 4*a*c])])/(2*Sqrt[2]*c^(3/2))))/(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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 = 0.56 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.18

method	result	size
default	\(\frac {2 c \left (-\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}-\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c +b^{2}\right ) \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(474\)
elliptic	\(\frac {2 c \left (-\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}-\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c +b^{2}\right ) \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(474\)

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)+1/4*(1/a-(2*a*c+b^2)/a/(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*b/(4*a*c+b^2)*c*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 = 0.08 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+b x^2-c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left ({\left (a b c x^{4} - a b^{2} x^{2} - a^{2} b\right )} \sqrt {a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - {\left (b^{2} c x^{4} - b^{3} x^{2} - a b^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left ({\left (2 \, a^{2} - a b\right )} c x^{4} - 2 \, a^{3} + a^{2} b - {\left (2 \, a^{2} b - a b^{2}\right )} x^{2}\right )} \sqrt {a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + {\left ({\left (2 \, a b + b^{2}\right )} c x^{4} - 2 \, a^{2} b - a b^{2} - {\left (2 \, a b^{2} + b^{3}\right )} x^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + 2 \, {\left (a b c x^{3} - {\left (a b^{2} + 2 \, a^{2} c\right )} x\right )} \sqrt {-c x^{4} + b x^{2} + a}}{2 \, {\left (a^{3} b^{2} + 4 \, a^{4} c - {\left (a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} + 4 \, a^{3} b c\right )} x^{2}\right )}} \] 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)*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*(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/(a + b*x^2 - c*x^4)^(3/2),x)
 

Output:

int(1/(a + b*x^2 - 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)