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

Optimal result

Integrand size = 34, antiderivative size = 646 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {(b d e+b c f+a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{8 d f \sqrt {a+b x^2}}+\frac {1}{4} x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}-\frac {\sqrt {b c-a d} e (b d e+b c f+a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b \sqrt {c} d f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {b c-a d} (3 b d e+b c f-a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (a^2 d^2 f^2+b^2 (d e-c f)^2-2 a b d f (d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{8 b^2 \sqrt {c} d \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/8*(a*d*f+b*c*f+b*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/f/(b*x^2+a)^(1 
/2)+1/4*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)-1/8*(-a*d+b*c)^( 
1/2)*e*(a*d*f+b*c*f+b*d*e)*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2) 
*EllipticE((-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)/d/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/ 
2)+1/8*a*(-a*d+b*c)^(1/2)*(-a*d*f+b*c*f+3*b*d*e)*(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^2/c^(1/2)/d/(a*(d*x^2+c)/c/(b*x^2+a 
))^(1/2)/(f*x^2+e)^(1/2)-1/8*a*(a^2*d^2*f^2+b^2*(-c*f+d*e)^2-2*a*b*d*f*(c* 
f+d*e))*(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^2/c^(1/2)/d/(-a*d+b*c)^(1/2)/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 
2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[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.

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

\(\Big \downarrow \) 434

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x}{4}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a d f}{4}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c f}{4}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b d e}{4}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c f}{2}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a d e}{2}+\frac {\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c e}{2}+\frac {3 \left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c e}{4} \] Input:

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

Output:

(sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x + 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)*a* 
d*f + 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)*b*c*f + 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)*b*d*e + 2*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)*a* 
c*f + 2*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)*a*d*e + 2*int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*s 
qrt(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)*b*c*e + 3*int((sqrt(e + 
f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*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)*a*c 
*e)/4