\(\int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [130]

Optimal result

Integrand size = 48, antiderivative size = 722 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=\frac {D \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{2 c e x}-\frac {\sqrt {b^2-4 a c} (a d D+2 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} a c d e \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} (2 B c d-a d D-2 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} a c d \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (c d D-2 c C e+b D e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{c \left (b+\sqrt {b^2-4 a c}\right ) e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

1/2*D*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e/x-1/4*(-4*a*c+b^2)^(1/2)*( 
2*A*c*e+D*a*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*E 
llipticE(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/c/d/e 
/(-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)-1/2*(-4*a*c+b^2)^(1/2)*(-2*A*c*e+2*B*c*d-D*a*d)*(-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))*2^(1/2)/a/c/d 
/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)-2^(1/2)*(-4*a*c+b^2)^(1/2)*(-2*C*c* 
e+D*b*e+D*c*d)*(-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*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a 
*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^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))/c/(b 
+(-4*a*c+b^2)^(1/2))/e/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

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

\(\Big \downarrow \) 2250

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

Input:

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

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

Input:

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

Fricas [F]

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

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

Sympy [F]

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

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*c + int((sqrt(d + e*x**2)*sqrt 
(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e* 
x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b 
*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*b**2*d*e**2*x - 2*int((s 
qrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + 
 a*c*d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 
+ 2*b*c*d*e*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*b*c* 
*2*e**2*x + 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d 
*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x 
**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c* 
*2*d*e*x**6),x)*b*c*d**2*e*x - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c 
*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d*e 
*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x**6 + 
c**2*d**2*x**4 + c**2*d*e*x**6),x)*c**3*d*e*x + int((sqrt(d + e*x**2)*sqrt 
(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e* 
x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b 
*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*c**2*d**3*x + int((sqrt( 
d + e*x**2)*sqrt(a + b*x**2 + c*x**4))/(a*b*d*e*x**2 + a*b*e**2*x**4 + a*c 
*d**2*x**2 + a*c*d*e*x**4 + b**2*d*e*x**4 + b**2*e**2*x**6 + b*c*d**2*x**4 
 + 2*b*c*d*e*x**6 + b*c*e**2*x**8 + c**2*d**2*x**6 + c**2*d*e*x**8),x)*...