\(\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx\) [383]

Optimal result

Integrand size = 38, antiderivative size = 310 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=-\frac {a \sqrt {c-d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c+a d) e}\right )}{b \sqrt {c} \sqrt {b c+a d} \sqrt {\frac {a \left (c-d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {c-d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c+a d},\arcsin \left (\frac {\sqrt {b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c+a d) e}\right )}{b \sqrt {c} \sqrt {b c+a d} \sqrt {\frac {a \left (c-d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

-a*(-d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((a*d+b*c)^(1 
/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(a*d+b*c)/e)^(1/2))/b/c^(1/2)/ 
(a*d+b*c)^(1/2)/(a*(-d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+a*(-d*x^2 
+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticPi((a*d+b*c)^(1/2)*x/c^( 
1/2)/(b*x^2+a)^(1/2),b*c/(a*d+b*c),(c*(-a*f+b*e)/(a*d+b*c)/e)^(1/2))/b/c^( 
1/2)/(a*d+b*c)^(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. \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {x^2}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}\,\sqrt {f\,x^2+e}} \,d x \] 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)