\(\int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [134]

Optimal result

Integrand size = 48, antiderivative size = 1220 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx =\text {Too large to display} \] Output:

-1/9*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/d/x^9-1/63*(9*B*a*d-8*A*(a* 
e+b*d))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^2/d^2/x^7+1/315*(9*a*d*(6* 
B*a*e+6*B*b*d-7*C*a*d)-A*(48*b^2*d^2+47*a*b*d*e-a*(-48*a*e^2+49*c*d^2)))*( 
e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^3/d^3/x^5+1/315*(2*A*(32*b^3*d^3+30 
*a*b^2*d^2*e-a^2*e*(-32*a*e^2+31*c*d^2)-6*a*b*d*(-5*a*e^2+11*c*d^2))-3*a*d 
*(24*b^2*B*d^2-a*b*d*(-23*B*e+28*C*d)+a*(7*a*d*(-4*C*e+5*D*d)-B*(-24*a*e^2 
+25*c*d^2))))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^4/d^4/x^3-1/630*(-4* 
a*c+b^2)^(1/2)*(A*(128*b^4*d^4+104*a*b^3*d^3*e-2*a^2*b*d*e*(-52*a*e^2+111* 
c*d^2)-3*a*b^2*d^2*(-33*a*e^2+136*c*d^2)+a^2*(128*a^2*e^4-108*a*c*d^2*e^2+ 
147*c^2*d^4))-3*a*d*(48*b^3*B*d^3-8*a*b^2*d^2*(-5*B*e+7*C*d)+a*b*d*(7*a*d* 
(-7*C*e+10*D*d)-8*B*(-5*a*e^2+13*c*d^2))+a^2*(c*d^2*(-44*B*e+63*C*d)+2*a*e 
*(24*B*e^2-28*C*d*e+35*D*d^2))))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x 
*(e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^ 
(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2 
))*2^(1/2)/a^5/d^5/(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/( 
c*x^4+b*x^2+a)^(1/2)+1/315*2^(1/2)*(-4*a*c+b^2)^(1/2)*(A*(64*b^4*d^4*e-3*a 
*b^2*d^2*e*(-13*a*e^2+80*c*d^2)-b^3*(-44*a*d^3*e^2+64*c*d^5)+2*a*b*d*(20*a 
^2*e^4-63*a*c*d^2*e^2+66*c^2*d^4)+a^2*e*(128*a^2*e^4-76*a*c*d^2*e^2+111*c^ 
2*d^4))-3*a*d*(24*b^3*B*d^3*e-b^2*d^2*(-17*B*a*e^2+24*B*c*d^2+28*C*a*d*e)+ 
a*b*d*(2*c*d^2*(-33*B*e+14*C*d)+a*e*(16*B*e^2-21*C*d*e+35*D*d^2))+a*(B*...
 

Mathematica [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 
 + c*x^4]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 
 + c*x^4]), 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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}dx\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + c*x 
^4]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ 
q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol 
yQ[Px, x]
 

Maple [F]

\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{10} \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}d x\]

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2) 
,x, algorithm="fricas")
 

Output:

integral((D*x^6 + C*x^4 + B*x^2 + A)*sqrt(c*x^4 + b*x^2 + a)*sqrt(e*x^2 + 
d)/(c*e*x^16 + (c*d + b*e)*x^14 + (b*d + a*e)*x^12 + a*d*x^10), x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{x^{10} \sqrt {d + e x^{2}} \sqrt {a + b x^{2} + c x^{4}}}\, dx \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**10/(e*x**2+d)**(1/2)/(c*x**4+b*x**2+ 
a)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4 + D*x**6)/(x**10*sqrt(d + e*x**2)*sqrt(a + b 
*x**2 + c*x**4)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2) 
,x, algorithm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(e*x^2 
+ d)*x^10), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + b x^{2} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2) 
,x, algorithm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*sqrt(e*x^2 
+ d)*x^10), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^{10}\,\sqrt {e\,x^2+d}\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4 
)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4 
)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{10} \sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}d x \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int((D*x^6+C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x)