\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [69]

Optimal result

Integrand size = 36, antiderivative size = 476 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {C \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 c e x}-\frac {C \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(2 A c+a C) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {(C d-2 B e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2261

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorithm="f 
ricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - 
c*e*x**6),x)*c + int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x 
**2 - c*d*x**4 - c*e*x**6),x)*b + int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/ 
(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a