\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^2 \sqrt {e+f x^2}} \, dx\) [320]

Optimal result

Integrand size = 37, antiderivative size = 503 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^2 \sqrt {e+f x^2}} \, dx=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x \sqrt {e+f x^2}}-\frac {c \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {d \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {b d e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

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

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

Output:

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

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

\(\Big \downarrow \) 450

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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