\(\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e-f x^2}} \, dx\) [385]

Optimal result

Integrand size = 38, antiderivative size = 316 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e-f x^2}} \, dx=-\frac {a \sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e-f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e+a f)}\right )}{b c \sqrt {b e+a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e-f x^2}}+\frac {a \sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e-f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b e}{b e+a f},\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e+a f)}\right )}{b c \sqrt {b e+a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e-f x^2}} \] Output:

-a*e^(1/2)*(d*x^2+c)^(1/2)*(a*(-f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((a*f 
+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),((-a*d+b*c)*e/c/(a*f+b*e))^(1/2))/b/ 
c/(a*f+b*e)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(-f*x^2+e)^(1/2)+a*e^(1/ 
2)*(d*x^2+c)^(1/2)*(a*(-f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticPi((a*f+b*e)^( 
1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),b*e/(a*f+b*e),((-a*d+b*c)*e/c/(a*f+b*e))^(1 
/2))/b/c/(a*f+b*e)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(-f*x^2+e)^(1/2)
 

Mathematica [F]

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

Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e - f*x^2]),x]
 

Output:

Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e - f*x^2]), x]
 

Rubi [F]

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.

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

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e-f x^2}}dx\)

Input:

Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e - f*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [F]

\[\int \frac {x^{2}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {-f \,x^{2}+e}}d x\]

Input:

int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x)
 

Output:

int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x)
 

Fricas [F(-1)]

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

integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(-f*x**2+e)**(1/2),x)
 

Output:

Integral(x**2/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*sqrt(e - f*x**2)), x)
 

Maxima [F]

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

integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x, algorith 
m="maxima")
 

Output:

integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-f*x^2 + e)), x)
 

Giac [F]

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

integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x, algorith 
m="giac")
 

Output:

integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-f*x^2 + e)), x)
 

Mupad [F(-1)]

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

int(x^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e - f*x^2)^(1/2)),x)
 

Output:

int(x^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e - f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e-f x^2}} \, dx=\int \frac {\sqrt {-f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{-b d f \,x^{6}-a d f \,x^{4}-b c f \,x^{4}+b d e \,x^{4}-a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \] Input:

int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(-f*x^2+e)^(1/2),x)
 

Output:

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