\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^{3/2} \sqrt {e+f x^2}} \, dx\) [477]

Optimal result

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

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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