\(\int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [132]

Optimal result

Integrand size = 48, antiderivative size = 626 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{5 a d x^5}-\frac {(5 a B d-4 A (b d+a e)) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{15 a^2 d^2 x^3}+\frac {\sqrt {b^2-4 a c} \left (5 a d (2 b B d-3 a C d+2 a B e)-A \left (8 b^2 d^2+7 a b d e-a \left (9 c d^2-8 a e^2\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{15 \sqrt {2} a^3 d^3 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (A \left (4 b c d^3-4 b^2 d^2 e+7 a c d^2 e-3 a b d e^2-8 a^2 e^3\right )+5 a d \left (3 a d (d D-C e)-B \left (c d^2-b d e-2 a e^2\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{15 a^3 d^3 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/5*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/d/x^5-1/15*(5*B*a*d-4*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^3+1/30*(-4*a*c+b^2 
)^(1/2)*(5*a*d*(2*B*a*e+2*B*b*d-3*C*a*d)-A*(8*b^2*d^2+7*a*b*d*e-a*(-8*a*e^ 
2+9*c*d^2)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*Ell 
ipticE(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^3/d^3/( 
-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) 
-1/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(A*(-8*a^2*e^3-3*a*b*d*e^2+7*a*c*d^2*e-4* 
b^2*d^2*e+4*b*c*d^3)+5*a*d*(3*a*d*(-C*e+D*d)-B*(-2*a*e^2-b*d*e+c*d^2)))*(- 
a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2) 
)*d-2*a*e))^(1/2)*x^3*EllipticF(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))/a^3/d^3/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

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

\(\Big \downarrow \) 2250

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

Input:

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

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

Input:

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

Fricas [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(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^12 + (c*d + b*e)*x^10 + (b*d + a*e)*x^8 + a*d*x^6), x)
 

Sympy [F]

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

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

Maxima [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(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^6), x)
 

Giac [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(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^6), x)
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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