\(\int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [80]

Optimal result

Integrand size = 44, antiderivative size = 452 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{5 a d x^5}-\frac {(5 B d-4 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{15 a d^2 x^3}+\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (5 a d (3 C d-2 B e)+A \left (9 c d^2+8 a e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 a^2 d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (5 B c d^3+15 a d^3 D-7 A c d^2 e-15 a C d^2 e+10 a B d e^2-8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 a^{3/2} d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/5*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^5-1/15*(-4*A*e+5*B*d)*(e*x^2 
+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^3+1/15*c*(d+a^(1/2)*e/c^(1/2))*(5*a*d*( 
-2*B*e+3*C*d)+A*(8*a*e^2+9*c*d^2))*(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^2/d^3/(e*x^2+d)^(1 
/2)/(-c*x^4+a)^(1/2)+1/15*c^(1/2)*(-8*A*a*e^3-7*A*c*d^2*e+10*B*a*d*e^2+5*B 
*c*d^3-15*C*a*d^2*e+15*D*a*d^3)*(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^(3/2)/d^3/(e*x^2+d)^( 
1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Fricas [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(1/2)/(-c*x^4+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^12 + c*d*x^10 - a*e*x^8 - a*d*x^6), x)
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

( - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c*d*e**2 - 40*sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*a**2*d**2*e**2*x**2 + 80*sqrt(d + e*x**2)*sqrt(a - c*x* 
*4)*a**2*d*e**3*x**4 - 20*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d*e**2*x 
**2 + 40*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*e**3*x**4 - 6*sqrt(d + e* 
x**2)*sqrt(a - c*x**4)*a*c**2*d**2*e*x**2 - 20*sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*a*c**2*d*e**2*x**4 - 45*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**4* 
x**2 + 90*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**3*e*x**4 - 15*sqrt(d + 
e*x**2)*sqrt(a - c*x**4)*b*c**2*d**3*x**2 + 30*sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*b*c**2*d**2*e*x**4 + 160*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)*a**2*c*d*e**4*x**5 + 80*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)*a*b*c**2*e**4*x**5 - 40*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)*a*c**3*d*e**3*x**5 + 180*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)*a*c**2*d**3*e**2*x**5 + 60*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)*b*c**3*d**2*e**2*x**5 + 8 
0*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 
- c*e*x**6),x)*a**2*c*d**2*e**3*x**5 + 40*int((sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c**2*d*e**3*x** 
5 - 20*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c...