\(\int \frac {A+B x^2+C x^4}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [16]

Optimal result

Integrand size = 49, antiderivative size = 523 \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {A \sqrt {a+b x^2} \sqrt {c+d x^2}}{a c x \sqrt {e+f x^2}}-\frac {A \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {(a C e+A b f-a B f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} f \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {C e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-A*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/x/(f*x^2+e)^(1/2)-A*(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^(3/2)/e/(d 
*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-(A*b*f-B*a*f+C*a*e)*(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^(3/2)/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)+C*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]

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

Integrate[(A + B*x^2 + C*x^4)/(x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e 
+ f*x^2]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4)/(x^2*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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle A \int \frac {1}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+C \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {B \sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{c \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\)

Input:

Int[(A + B*x^2 + C*x^4)/(x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^ 
2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {C \,x^{4}+x^{2} B +A}{x^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]

Input:

int((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1 
/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral((A + B*x**2 + C*x**4)/(x**2*sqrt(a + b*x**2)*sqrt(c + d*x**2)*sqr 
t(e + f*x**2)), x)
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1 
/2),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 
+ e)*x^2), x)
 

Giac [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1 
/2),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 
+ e)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^2\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/(x^2*(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 + C*x^4)/(x^2*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^ 
2)^(1/2)), x)
 

Reduce [F]

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

int((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((C*x^4+B*x^2+A)/x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)