\(\int \frac {1}{(a+b x^2)^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [479]

Optimal result

Integrand size = 34, antiderivative size = 320 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b \sqrt {c} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{a \sqrt {b c-a d} (b e-a f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}-\frac {\sqrt {c} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{\sqrt {b c-a d} e (b e-a f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \] Output:

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

Mathematica [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[1/((a + b*x^2)^(3/2)*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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}}dx\)

Input:

Int[1/((a + b*x^2)^(3/2)*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]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]

Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm=" 
fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^2*d*f*x^8 + (b 
^2*d*e + (b^2*c + 2*a*b*d)*f)*x^6 + ((b^2*c + 2*a*b*d)*e + (2*a*b*c + a^2* 
d)*f)*x^4 + a^2*c*e + (a^2*c*f + (2*a*b*c + a^2*d)*e)*x^2), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:

integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
 

Output:

Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm=" 
maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm=" 
giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d f \,x^{8}+2 a b d f \,x^{6}+b^{2} c f \,x^{6}+b^{2} d e \,x^{6}+a^{2} d f \,x^{4}+2 a b c f \,x^{4}+2 a b d e \,x^{4}+b^{2} c e \,x^{4}+a^{2} c f \,x^{2}+a^{2} d e \,x^{2}+2 a b c e \,x^{2}+a^{2} c e}d x \] Input:

int(1/(b*x^2+a)^(3/2)/(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))/(a**2*c*e + a**2* 
c*f*x**2 + a**2*d*e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 
 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2* 
d*e*x**6 + b**2*d*f*x**8),x)