\(\int \frac {A+B x^2+C x^4+D x^6}{x^6 (d+e x^2)^{3/2} \sqrt {a-c x^4}} \, dx\) [88]

Optimal result

Integrand size = 44, antiderivative size = 669 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) \sqrt {a-c x^4}}{d e \left (c d^2-a e^2\right ) x^5 \sqrt {d+e x^2}}-\frac {\left (\frac {A c d}{a}-5 C d+\frac {5 d^2 D}{e}+5 B e-\frac {6 A e^2}{d}\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{5 d \left (c d^2-a e^2\right ) x^5}-\frac {\left (5 B c d^3-15 a d^3 D-9 A c d^2 e+15 a C d^2 e-20 a B d e^2+24 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{15 a d^3 \left (c d^2-a e^2\right ) x^3}+\frac {\sqrt {c} \left (3 A \left (3 c^2 d^4+8 a c d^2 e^2-16 a^2 e^4\right )+5 a d \left (c d^2 (3 C d-5 B e)+a e \left (3 d^2 D-6 C d e+8 B e^2\right )\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 a^2 d^4 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (5 B c d^3+15 a d^3 D-12 A c d^2 e-30 a C d^2 e+40 a B d e^2-48 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 a^{3/2} d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

(-A*e^3+B*d*e^2-C*d^2*e+D*d^3)*(-c*x^4+a)^(1/2)/d/e/(-a*e^2+c*d^2)/x^5/(e* 
x^2+d)^(1/2)-1/5*(A*c*d/a-5*C*d+5*d^2*D/e+5*B*e-6*A*e^2/d)*(e*x^2+d)^(1/2) 
*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/x^5-1/15*(24*A*a*e^3-9*A*c*d^2*e-20*B*a 
*d*e^2+5*B*c*d^3+15*C*a*d^2*e-15*D*a*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2) 
/a/d^3/(-a*e^2+c*d^2)/x^3+1/15*c^(1/2)*(3*A*(-16*a^2*e^4+8*a*c*d^2*e^2+3*c 
^2*d^4)+5*a*d*(c*d^2*(-5*B*e+3*C*d)+a*e*(8*B*e^2-6*C*d*e+3*D*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)*Ellipt 
icE(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^4/(c^(1/2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2 
)+1/15*c^(1/2)*(-48*A*a*e^3-12*A*c*d^2*e+40*B*a*d*e^2+5*B*c*d^3-30*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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^( 
1/2)
 

Mathematica [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(d + e*x^2)^(3/2)*Sqrt[a - c*x^ 
4]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(d + e*x^2)^(3/2)*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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^6 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^6*(d + e*x^2)^(3/2)*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]

\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(3/2)/(-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)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(3/2)/(-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^2*x^14 + 2*c*d*e*x^12 - 2*a*d*e*x^8 + (c*d^2 - a*e^2)*x^10 - a*d^2*x^6), 
 x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{x^{6} \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((D*x**6+C*x**4+B*x**2+A)/x**6/(e*x**2+d)**(3/2)/(-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)*(d + e*x**2 
)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(3/2)/(-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)*(e*x^2 + d)^(3/2)* 
x^6), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{6}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(3/2)/(-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)*(e*x^2 + d)^(3/2)* 
x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^6\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^6*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)), 
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)^(3/2)), 
 x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\text {too large to display} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^6/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*d*e**2 - 6*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a**2*c*d**2*e*x**2 - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4) 
*a**2*c*d*e**2*x**4 + 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c*e**3*x** 
6 - 15*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*d**4*x**2 - 30*sqrt(d + e*x* 
*2)*sqrt(a - c*x**4)*a**2*d**3*e*x**4 - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4 
)*a*b*c*d**3*x**2 - 10*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d**2*e*x**4 
 - 20*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d*e**2*x**6 + 6*sqrt(d + e*x 
**2)*sqrt(a - c*x**4)*a*c**2*d**2*e*x**6 + 15*sqrt(d + e*x**2)*sqrt(a - c* 
x**4)*a*c*d**4*x**6 + 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*d**3*x**6 
 + 48*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 
+ a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2*c**2*d*e 
**4*x**5 + 48*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d 
*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a**2* 
c**2*e**5*x**7 - 40*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 
 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x) 
*a*b*c**2*d**2*e**3*x**5 - 40*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6) 
/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e** 
2*x**8),x)*a*b*c**2*d*e**4*x**7 + 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4 
)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 
- c*e**2*x**8),x)*a*c**3*d**3*e**2*x**5 + 12*int((sqrt(d + e*x**2)*sqrt...