\(\int \frac {\sqrt {e+f x^2}}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [341]

Optimal result

Integrand size = 37, antiderivative size = 296 \[ \int \frac {\sqrt {e+f x^2}}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a-b x^2} \sqrt {e+f x^2}}{a x \sqrt {c+d x^2}}-\frac {\sqrt {b c+a d} \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt {c} \sqrt {a-b x^2}}\right )|\frac {a (d e-c f)}{(b c+a d) e}\right )}{a \sqrt {c} \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {c} (b e+a f) \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt {c} \sqrt {a-b x^2}}\right ),\frac {a (d e-c f)}{(b c+a d) e}\right )}{a \sqrt {b c+a d} e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 450

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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