\(\int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [17]

Optimal result

Integrand size = 54, antiderivative size = 653 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {A \sqrt {c+d x^2} \sqrt {e+f x^2}}{c e x \sqrt {a+b x^2}}+\frac {D x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d f \sqrt {a+b x^2}}-\frac {\sqrt {b e-a f} (a c D e+2 A b d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 a b c d \sqrt {e} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {\left (a^2 D e f+2 b^2 f (B e-A f)+a b e (D e-2 C f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt {e} f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {e} (a d D f+b (d D e-2 C d f+c D f)) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b e}{b e-a f},\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c d f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

-A*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/c/e/x/(b*x^2+a)^(1/2)+1/2*D*x*(d*x^2+c) 
^(1/2)*(f*x^2+e)^(1/2)/d/f/(b*x^2+a)^(1/2)-1/2*(-a*f+b*e)^(1/2)*(2*A*b*d*f 
+D*a*c*e)*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticE((-a*f+ 
b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),((-a*d+b*c)*e/c/(-a*f+b*e))^(1/2))/a/ 
b/c/d/e^(1/2)/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/2*(a^2*D 
*e*f+2*b^2*f*(-A*f+B*e)+a*b*e*(-2*C*f+D*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e 
/(b*x^2+a))^(1/2)*EllipticF((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),((- 
a*d+b*c)*e/c/(-a*f+b*e))^(1/2))/b^2/c/e^(1/2)/f/(-a*f+b*e)^(1/2)/(a*(d*x^2 
+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/2*a*e^(1/2)*(a*d*D*f+b*(-2*C*d*f+ 
D*c*f+D*d*e))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticPi(( 
-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),b*e/(-a*f+b*e),((-a*d+b*c)*e/c/( 
-a*f+b*e))^(1/2))/b^2/c/d/f/(-a*f+b*e)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 
2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(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 + D*x^6)/(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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle A \int \frac {1}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+C \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+D \int \frac {x^4}{\sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {B \sqrt {e} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{c \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(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]

\[\int \frac {D x^{6}+C \,x^{4}+x^{2} B +A}{x^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]

Input:

int((D*x^6+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((D*x^6+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]

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

integrate((D*x^6+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((D*x^6 + 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]

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

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

Maxima [F]

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

integrate((D*x^6+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((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqr 
t(f*x^2 + e)*x^2), x)
 

Giac [F]

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

integrate((D*x^6+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((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqr 
t(f*x^2 + e)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^2\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(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^6*D)/(x^2*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*( 
e + f*x^2)^(1/2)), x)
 

Reduce [F]

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

int((D*x^6+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((D*x^6+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)