\(\int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [74]

Optimal result

Integrand size = 39, antiderivative size = 662 \[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{9 a d x^9}-\frac {(9 B d-8 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{63 a d^2 x^7}-\frac {\left (49 A c d^2+63 a C d^2-54 a B d e+48 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a^2 d^3 x^5}-\frac {\left (75 B c d^3-62 A c d^2 e-84 a C d^2 e+72 a B d e^2-64 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a^2 d^4 x^3}+\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (A \left (147 c^2 d^4+108 a c d^2 e^2+128 a^2 e^4\right )+3 a d \left (c d^2 (63 C d-44 B e)+8 a e^2 (7 C d-6 B e)\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^3 d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (75 B c^2 d^5-111 A c^2 d^4 e-147 a c C d^4 e+96 a B c d^3 e^2-76 a A c d^2 e^3-168 a^2 C d^2 e^3+144 a^2 B d e^4-128 a^2 A e^5\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^{5/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/9*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^9-1/63*(-8*A*e+9*B*d)*(e*x^2 
+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^7-1/315*(48*A*a*e^2+49*A*c*d^2-54*B*a*d 
*e+63*C*a*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^3/x^5-1/315*(-64*A*a 
*e^3-62*A*c*d^2*e+72*B*a*d*e^2+75*B*c*d^3-84*C*a*d^2*e)*(e*x^2+d)^(1/2)*(- 
c*x^4+a)^(1/2)/a^2/d^4/x^3+1/315*c*(d+a^(1/2)*e/c^(1/2))*(A*(128*a^2*e^4+1 
08*a*c*d^2*e^2+147*c^2*d^4)+3*a*d*(c*d^2*(-44*B*e+63*C*d)+8*a*e^2*(-6*B*e+ 
7*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)*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^3/d^5/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/315 
*c^(1/2)*(-128*A*a^2*e^5-76*A*a*c*d^2*e^3-111*A*c^2*d^4*e+144*B*a^2*d*e^4+ 
96*B*a*c*d^3*e^2+75*B*c^2*d^5-168*C*a^2*d^2*e^3-147*C*a*c*d^4*e)*(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]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx \] Input:

Integrate[(A + B*x^2 + C*x^4)/(x^10*Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4)/(x^10*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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {a-c x^4} \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {a-c x^4} \sqrt {d+e x^2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4)/(x^10*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]

\[\int \frac {C \,x^{4}+B \,x^{2}+A}{x^{10} \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]

Input:

int((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorit 
hm="fricas")
 

Output:

integral(-(C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c*e*x^16 + 
 c*d*x^14 - a*e*x^12 - a*d*x^10), x)
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x^{10} \sqrt {a - c x^{4}} \sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

Integral((A + B*x**2 + C*x**4)/(x**10*sqrt(a - c*x**4)*sqrt(d + e*x**2)), 
x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorit 
hm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {-c x^{4} + a} \sqrt {e x^{2} + d} x^{10}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorit 
hm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^{10}\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/(x^10*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/(x^10*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{x^{10} \sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\text {too large to display} \] Input:

int((C*x^4+B*x^2+A)/x^10/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - 2880*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*d*e**6 - 8640*sqrt(d + e*x 
**2)*sqrt(a - c*x**4)*a**4*b*d*e**6*x**2 + 10368*sqrt(d + e*x**2)*sqrt(a - 
 c*x**4)*a**4*b*e**7*x**4 - 3360*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c* 
d**2*e**5*x**2 - 5184*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d*e**6*x**4 
 - 10800*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d**3*e**4*x**2 + 12960 
*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d**2*e**5*x**4 - 1728*sqrt(d + 
 e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d*e**6*x**6 + 20*sqrt(d + e*x**2)*sqrt( 
a - c*x**4)*a**3*c**2*d**5*e**2 - 2800*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a 
**3*c**2*d**4*e**3*x**2 + 3360*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2 
*d**3*e**4*x**4 - 14496*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**2*e 
**5*x**6 - 9540*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**5*e**2*x* 
*2 + 11448*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**4*e**3*x**4 - 
2160*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*c**2*d**3*e**4*x**6 - 210*sq 
rt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**3*d**6*e*x**2 + 316*sqrt(d + e*x** 
2)*sqrt(a - c*x**4)*a**2*c**3*d**5*e**2*x**4 - 560*sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*a**2*c**3*d**4*e**3*x**6 - 4725*sqrt(d + e*x**2)*sqrt(a - c*x** 
4)*a*b*c**3*d**7*x**2 + 5670*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**3*d* 
*6*e*x**4 - 1908*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**3*d**5*e**2*x**6 
 + 54*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**4*d**6*e*x**6 - 945*sqrt(d + 
e*x**2)*sqrt(a - c*x**4)*b*c**4*d**7*x**6 - 3317760*int((sqrt(d + e*x**...