\(\int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [360]

Optimal result

Integrand size = 37, antiderivative size = 550 \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b d \sqrt {e+f x^2}}-\frac {c \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d f \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e (b e+a f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b f^2 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e (b d e+b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d f^2 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

Mathematica [F]

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

Integrate[x^4/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[x^4/(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.

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

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}dx\)

Input:

Int[x^4/(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]

\[\int \frac {x^{4}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]

Input:

int(x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int(x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate(x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
 

Output:

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

Maxima [F]

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

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

Output:

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
 

Giac [F]

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

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

Output:

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

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

int(x^4/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int(x^4/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

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

int(x^4/(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**4)/(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)