\(\int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [127]

Optimal result

Integrand size = 43, antiderivative size = 851 \[ \int \frac {A+B x^2+C x^4}{x^8 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{7 a d x^7}-\frac {(7 a B d-6 A (b d+a e)) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{35 a^2 d^2 x^5}+\frac {\left (7 a d (4 b B d-5 a C d+4 a B e)-A \left (24 b^2 d^2+23 a b d e-a \left (25 c d^2-24 a e^2\right )\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{105 a^3 d^3 x^3}+\frac {\sqrt {b^2-4 a c} \left (4 A \left (12 b^3 d^3+10 a b^2 d^2 e-a^2 e \left (11 c d^2-12 a e^2\right )-2 a b d \left (13 c d^2-5 a e^2\right )\right )-7 a d \left (8 b^2 B d^2-a b d (10 C d-7 B e)-a \left (9 B c d^2+10 a C d e-8 a B e^2\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{105 \sqrt {2} a^4 d^4 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (A \left (24 b^3 d^3 e-2 a b d e \left (33 c d^2-8 a e^2\right )-b^2 \left (24 c d^4-17 a d^2 e^2\right )+a \left (25 c^2 d^4-32 a c d^2 e^2+48 a^2 e^4\right )\right )-7 a d \left (4 b^2 B d^2 e-b d \left (4 B c d^2+5 a C d e-3 a B e^2\right )+a \left (c d^2 (5 C d-7 B e)-2 a e^2 (5 C d-4 B e)\right )\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{105 a^4 d^4 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/7*A*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a/d/x^7-1/35*(7*B*a*d-6*A*(a* 
e+b*d))*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^2/d^2/x^5+1/105*(7*a*d*(4* 
B*a*e+4*B*b*d-5*C*a*d)-A*(24*b^2*d^2+23*a*b*d*e-a*(-24*a*e^2+25*c*d^2)))*( 
e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^3/d^3/x^3+1/210*(-4*a*c+b^2)^(1/2)* 
(4*A*(12*b^3*d^3+10*a*b^2*d^2*e-a^2*e*(-12*a*e^2+11*c*d^2)-2*a*b*d*(-5*a*e 
^2+13*c*d^2))-7*a*d*(8*b^2*B*d^2-a*b*d*(-7*B*e+10*C*d)-a*(-8*B*a*e^2+9*B*c 
*d^2+10*C*a*d*e)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/ 
2)*EllipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)* 
((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/a^4 
/d^4/(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a) 
^(1/2)-1/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(A*(24*b^3*d^3*e-2*a*b*d*e*(-8*a*e 
^2+33*c*d^2)-b^2*(-17*a*d^2*e^2+24*c*d^4)+a*(48*a^2*e^4-32*a*c*d^2*e^2+25* 
c^2*d^4))-7*a*d*(4*b^2*B*d^2*e-b*d*(-3*B*a*e^2+4*B*c*d^2+5*C*a*d*e)+a*(c*d 
^2*(-7*B*e+5*C*d)-2*a*e^2*(-4*B*e+5*C*d))))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^ 
2))^(1/2)*(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*Ellipt 
icF(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+ 
b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))/a^4/d^4/(e*x^2+d)^(1 
/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2250

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) 
^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^ 
q*(a + b*x^2 + c*x^4)^p, x] /; FreeQ[{a, b, c, d, e, f, m, p, q}, x] && Pol 
yQ[Px, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^4+B*x^2+A)/x^8/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2),x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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