\(\int \frac {1}{(-a+b x^2-c x^4)^{3/2}} \, dx\) [313]

Optimal result

Integrand size = 19, antiderivative size = 366 \[ \int \frac {1}{\left (-a+b x^2-c x^4\right )^{3/2}} \, dx=-\frac {x \left (b^2-2 a c-b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {-a+b x^2-c x^4}}+\frac {b \sqrt {c} x \sqrt {-a+b x^2-c x^4}}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a-b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2+\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {-a+b x^2-c x^4}}-\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a-b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2+\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt {-a+b x^2-c x^4}} \] 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)/(a^(1/2)+c^(1/2)*x^2)+b*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)*Elliptic 
E(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2+b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4 
)/(-4*a*c+b^2)/(-c*x^4+b*x^2-a)^(1/2)-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^(3/4)/(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 = 10.52 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\left (-a+b x^2-c x^4\right )^{3/2}} \, dx=-\frac {4 \sqrt {-\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (b^2-2 a c-b c x^2\right )-i \sqrt {2} b \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}-2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {-b+\sqrt {b^2-4 a c}+2 c x^2}{-b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \sqrt {2} \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}-2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {-b+\sqrt {b^2-4 a c}+2 c x^2}{-b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right ) \sqrt {-\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {-a+b x^2-c x^4}} \] Input:

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

Output:

-1/4*(4*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*x*(b^2 - 2*a*c - b*c*x^2) - I*S 
qrt[2]*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[(-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 
 + 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])]*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*(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 = 0.76 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1405, 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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

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

Defintions of rubi rules used

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 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 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 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]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {2 c \left (-\frac {b \,x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c -b^{2}\right ) x}{2 a \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}-\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {1}{a}+\frac {2 a c -b^{2}}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {4+\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}-a}}+\frac {b c \sqrt {4+\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\left (4 a c -b^{2}\right ) \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}-a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(479\)
elliptic	\(\frac {2 c \left (-\frac {b \,x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c -b^{2}\right ) x}{2 a \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}-\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {1}{a}+\frac {2 a c -b^{2}}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {4+\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}-a}}+\frac {b c \sqrt {4+\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\left (4 a c -b^{2}\right ) \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right )}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}-a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(479\)

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))-Ellip 
ticE(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 = 0.08 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (-a+b x^2-c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (b^{2} c x^{4} - b^{3} x^{2} + a b^{2} + {\left (a b c x^{4} - a b^{2} x^{2} + a^{2} b\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \sqrt {-a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (2 \, a b - b^{2}\right )} c x^{4} + 2 \, a^{2} b - a b^{2} - {\left (2 \, a b^{2} - b^{3}\right )} x^{2} - {\left ({\left (2 \, a^{2} + a b\right )} c x^{4} + 2 \, a^{3} + a^{2} b - {\left (2 \, a^{2} b + a b^{2}\right )} x^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}}\right )} \sqrt {-a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 2 \, a c}{2 \, a c}) + 2 \, {\left (a b c x^{3} - {\left (a b^{2} - 2 \, a^{2} c\right )} x\right )} \sqrt {-c x^{4} + b x^{2} - a}}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}} \] Input:

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

Output:

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