\(\int \frac {A+B x^2+C x^4+D x^6}{x^8 (d+e x^2)^{3/2} \sqrt {a-c x^4}} \, dx\) [89]

Optimal result

Integrand size = 44, antiderivative size = 840 \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) \sqrt {a-c x^4}}{d e \left (c d^2-a e^2\right ) x^7 \sqrt {d+e x^2}}-\frac {\left (\frac {A c d}{a}-7 C d+\frac {7 d^2 D}{e}+7 B e-\frac {8 A e^2}{d}\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{7 d \left (c d^2-a e^2\right ) x^7}-\frac {\left (7 B c d^3-35 a d^3 D-13 A c d^2 e+35 a C d^2 e-42 a B d e^2+48 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{35 a d^3 \left (c d^2-a e^2\right ) x^5}-\frac {\left (A \left (25 c^2 d^4+62 a c d^2 e^2-192 a^2 e^4\right )+7 a d \left (c d^2 (5 C d-9 B e)+a e \left (15 d^2 D-20 C d e+24 B e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{105 a^2 d^4 \left (c d^2-a e^2\right ) x^3}-\frac {\sqrt {c} \left (107 A c^2 d^4 e+2 a^2 e^2 \left (105 d^3 D-140 C d^2 e-192 A e^3\right )-a c d^2 \left (105 d^3 D-175 C d^2 e-172 A e^3\right )-21 B \left (3 c^2 d^5+8 a c d^3 e^2-16 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^2 d^5 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (A \left (25 c^2 d^4+116 a c d^2 e^2+384 a^2 e^4\right )+7 a d \left (c d^2 (5 C d-12 B e)-2 a e \left (15 d^2 D-20 C d e+24 B e^2\right )\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^{5/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] 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^7/(e* 
x^2+d)^(1/2)-1/7*(A*c*d/a-7*C*d+7*d^2*D/e+7*B*e-8*A*e^2/d)*(e*x^2+d)^(1/2) 
*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/x^7-1/35*(48*A*a*e^3-13*A*c*d^2*e-42*B* 
a*d*e^2+7*B*c*d^3+35*C*a*d^2*e-35*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^5-1/105*(A*(-192*a^2*e^4+62*a*c*d^2*e^2+25*c^2*d^ 
4)+7*a*d*(c*d^2*(-9*B*e+5*C*d)+a*e*(24*B*e^2-20*C*d*e+15*D*d^2)))*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^4/(-a*e^2+c*d^2)/x^3-1/105*c^(1/2)*(107*A*c 
^2*d^4*e+2*a^2*e^2*(-192*A*e^3-140*C*d^2*e+105*D*d^3)-a*c*d^2*(-172*A*e^3- 
175*C*d^2*e+105*D*d^3)-21*B*(-16*a^2*d*e^4+8*a*c*d^3*e^2+3*c^2*d^5))*(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)*Ellip 
ticE(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^5/(c^(1/2)*d-a^(1/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/ 
2)+1/105*c^(1/2)*(A*(384*a^2*e^4+116*a*c*d^2*e^2+25*c^2*d^4)+7*a*d*(c*d^2* 
(-12*B*e+5*C*d)-2*a*e*(24*B*e^2-20*C*d*e+15*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)*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^ 
(5/2)/d^5/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(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^8*(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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^8 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{x^8 \sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^8*(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]

\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{x^{8} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

int((D*x^6+C*x^4+B*x^2+A)/x^8/(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^8/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^8/(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^16 + 2*c*d*e*x^14 - 2*a*d*e*x^10 + (c*d^2 - a*e^2)*x^12 - a*d^2*x^8) 
, x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4} + D x^{6}}{x^{8} \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Maxima [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^8/(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^8), x)
 

Giac [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int { \frac {D x^{6} + C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{8}} \,d x } \] Input:

integrate((D*x^6+C*x^4+B*x^2+A)/x^8/(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^8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^8\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/(x^8*(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^8*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)), 
 x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{x^8 \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\text {too large to display} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - 5184*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*c*d*e**6 - 5376*sqrt(d + e 
*x**2)*sqrt(a - c*x**4)*a**5*d**2*e**6*x**2 + 10752*sqrt(d + e*x**2)*sqrt( 
a - c*x**4)*a**5*d*e**7*x**4 - 4032*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4 
*b*c*d*e**6*x**2 + 8064*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*c*e**7*x* 
*4 - 1152*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c**2*d**3*e**4 - 4320*sqr 
t(d + e*x**2)*sqrt(a - c*x**4)*a**4*c**2*d**2*e**5*x**2 - 12096*sqrt(d + e 
*x**2)*sqrt(a - c*x**4)*a**4*c**2*d*e**6*x**4 - 2800*sqrt(d + e*x**2)*sqrt 
(a - c*x**4)*a**4*c*d**4*e**4*x**2 + 5600*sqrt(d + e*x**2)*sqrt(a - c*x**4 
)*a**4*c*d**3*e**5*x**4 - 5376*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d* 
*2*e**6*x**6 + 1680*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c**2*d**3*e** 
4*x**2 - 3360*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c**2*d**2*e**5*x**4 
 - 4032*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c**2*d*e**6*x**6 + 36*sqr 
t(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**3*d**5*e**2 + 1200*sqrt(d + e*x**2) 
*sqrt(a - c*x**4)*a**3*c**3*d**4*e**3*x**2 - 7008*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*a**3*c**3*d**3*e**4*x**4 - 14688*sqrt(d + e*x**2)*sqrt(a - c*x** 
4)*a**3*c**3*d**2*e**5*x**6 + 1736*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3* 
c**2*d**6*e**2*x**2 - 3472*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d** 
5*e**3*x**4 - 2800*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**4*e**4*x 
**6 + 252*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**3*d**5*e**2*x**2 - 5 
04*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**3*d**4*e**3*x**4 + 1680*...