\(\int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [78]

Optimal result

Integrand size = 44, antiderivative size = 501 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {D \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 c e x}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) (a d D-2 A c e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a d e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(2 B c d+a d D-2 A c e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {a} \sqrt {c} d \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {(d D-2 C e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/2*D*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e/x-1/2*(d+a^(1/2)*e/c^(1/2))*(- 
2*A*c*e+D*a*d)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2) 
*e)/x^2)^(1/2)*EllipticE(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2) 
*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/a/d/e/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1 
/2*(-2*A*c*e+2*B*c*d+D*a*d)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1 
/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticF(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^ 
(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/a^(1/2)/c^(1/2)/d/(e*x^2+d) 
^(1/2)/(-c*x^4+a)^(1/2)-1/2*(-2*C*e+D*d)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e 
*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a^(1/2)/c^(1/2) 
/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/e/(e*x^2+d) 
^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

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

\(\Big \downarrow \) 2251

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2251

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

Maple [F]

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

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4) - 2*int((sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c*e*x + int((sqrt(d 
 + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x 
)*d**2*x + int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x 
**4 - c*e*x**6),x)*b*d*x)/(d*x)