\(\int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx\) [1004]

Optimal result

Integrand size = 21, antiderivative size = 377 \[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {a+b x^2-c x^4}}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {a+b x^2-c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {i \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|-\frac {b+\sqrt {b^2+4 a c}}{-b+\sqrt {b^2+4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),-\frac {b+\sqrt {b^2+4 a c}}{-b+\sqrt {b^2+4 a c}}\right )\right )}{2 \sqrt {2} c \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} \sqrt {a+b x^2-c x^4}} \] Input:

Integrate[x^2/Sqrt[a + b*x^2 - c*x^4],x]
 

Output:

((-1/2*I)*(-b + Sqrt[b^2 + 4*a*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])]*(EllipticE[I*ArcSinh[Sqr 
t[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]))] - 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]))]))/( 
Sqrt[2]*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1460, 389, 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.

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

\(\Big \downarrow \) 1460

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

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

Int[x^2/Sqrt[a + b*x^2 - c*x^4],x]
 

Output:

(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 - 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 + Sq 
rt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*c^(3/2)) + ((b - Sqrt[ 
b^2 + 4*a*c])*Sqrt[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))))/Sqrt[a + b*x^2 - c*x^4]
 

Defintions of rubi rules used

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 389

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[1/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b   Int 
[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N 
eQ[b*c - a*d, 0] &&  !SimplerSqrtQ[-b/a, -d/c]
 

rule 1460

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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[x^2/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sq 
rt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a 
*c, 0] && NegQ[c/a]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.58

method	result	size
default	\(-\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(217\)
elliptic	\(-\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\)	\(217\)

Input:

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

Output:

-1/2*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 = 0.07 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {-c} c x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {-c} c x \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b \sqrt {-c} x\right )} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} + b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} + 4 \, a c}{c^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} c}{2 \, c^{2} x} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {-c x^{4} + b x^{2} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {-c x^{4} + b x^{2} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx=\int \frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, x^{2}}{-c \,x^{4}+b \,x^{2}+a}d x \] Input:

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

Output:

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