\(\int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [73]

Optimal result

Integrand size = 39, antiderivative size = 546 \[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{7 a d x^7}-\frac {(7 B d-6 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{35 a d^2 x^5}-\frac {\left (25 A c d^2+35 a C d^2-28 a B d e+24 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{105 a^2 d^3 x^3}-\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (44 A c d^2 e+70 a C d^2 e+48 a A e^3-7 B \left (9 c d^3+8 a d e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^2 d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (A \left (25 c^2 d^4+32 a c d^2 e^2+48 a^2 e^4\right )+7 a d \left (c d^2 (5 C d-7 B e)+2 a e^2 (5 C d-4 B e)\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{105 a^{5/2} d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/7*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^7-1/35*(-6*A*e+7*B*d)*(e*x^2 
+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^5-1/105*(24*A*a*e^2+25*A*c*d^2-28*B*a*d 
*e+35*C*a*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^3/x^3-1/105*c*(d+a^( 
1/2)*e/c^(1/2))*(44*A*c*d^2*e+70*C*a*d^2*e+48*A*a*e^3-7*B*(8*a*d*e^2+9*c*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)*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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/105*c^( 
1/2)*(A*(48*a^2*e^4+32*a*c*d^2*e^2+25*c^2*d^4)+7*a*d*(c*d^2*(-7*B*e+5*C*d) 
+2*a*e^2*(-4*B*e+5*C*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^(5/2)/d^4/(e*x^2+d)^(1/2)/( 
-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4)/(x^8*Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),x]
 

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

\[\int \frac {C \,x^{4}+B \,x^{2}+A}{x^{8} \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{8}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorith 
m="fricas")
 

Output:

integral(-(C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c*e*x^14 + 
 c*d*x^12 - a*e*x^10 - a*d*x^8), x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x^{8} \sqrt {a - c x^{4}} \sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

Integral((A + B*x**2 + C*x**4)/(x**8*sqrt(a - c*x**4)*sqrt(d + e*x**2)), x 
)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{8}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{8}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^8\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

( - 48*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*d*e**4 + 4*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*a**3*c*d**3*e**2 + 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)* 
a**3*c*d**2*e**3*x**2 - 240*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*d*e** 
4*x**4 + 288*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c*e**5*x**6 - 84*sqrt( 
d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c*d**3*e**2*x**2 + 168*sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*a**2*b*c*d**2*e**3*x**4 - 336*sqrt(d + e*x**2)*sqrt(a - 
 c*x**4)*a**2*b*c*d*e**4*x**6 - 10*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2* 
c**2*d**4*e*x**2 + 20*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**3*e** 
2*x**4 + 336*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**2*d**2*e**3*x**6 - 
105*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**2*d**5*x**2 + 210*sqrt(d + e* 
x**2)*sqrt(a - c*x**4)*a*b*c**2*d**4*e*x**4 - 672*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*a*b*c**2*d**3*e**2*x**6 - 30*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a 
*c**3*d**4*e*x**6 - 315*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**3*d**5*x**6 
 + 6912*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12* 
a**2*e**3*x**2 - a*c*d**3 - a*c*d**2*e*x**2 - 12*a*c*d*e**2*x**4 - 12*a*c* 
e**3*x**6 + c**2*d**3*x**4 + c**2*d**2*e*x**6),x)*a**4*c**2*e**8*x**7 - 80 
64*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12*a**2* 
e**3*x**2 - a*c*d**3 - a*c*d**2*e*x**2 - 12*a*c*d*e**2*x**4 - 12*a*c*e**3* 
x**6 + c**2*d**3*x**4 + c**2*d**2*e*x**6),x)*a**3*b*c**2*d*e**7*x**7 + 748 
8*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(12*a**2*d*e**2 + 12*a**...