\(\int \frac {A+B x^2+C x^4}{x^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx\) [124]

Optimal result

Integrand size = 43, antiderivative size = 634 \[ \int \frac {A+B x^2+C x^4}{x^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \, dx=-\frac {A \sqrt {b^2-4 a c} \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} a d \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (B d-A e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{a d \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} C \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/2*A*(-4*a*c+b^2)^(1/2)*(-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/d/(-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)-2^(1/2)*(-4*a*c+b^2)^(1/2)*(-A*e+B*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*El 
lipticF(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/d/(e*x^2+d)^(1 
/2)/(c*x^4+b*x^2+a)^(1/2)+2*2^(1/2)*(-4*a*c+b^2)^(1/2)*C*(-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*EllipticPi(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*( 
-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^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))/(b+(-4*a*c+b^2)^(1/2))/(e*x^2+d)^(1/ 
2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2250

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

Input:

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

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

Input:

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

Fricas [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(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^8 + (c*d + b*e)*x^6 + (b*d + a*e)*x^4 + a*d*x^2), x)
 

Sympy [F]

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

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4))/(d*x**2 + e*x**4),x)