\(\int \frac {2 A e+C d x^2+2 C e x^4}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [118]

Optimal result

Integrand size = 42, antiderivative size = 322 \[ \int \frac {2 A e+C d x^2+2 C e x^4}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {C \sqrt {d+e x^2} \sqrt {a-c x^4}}{c x}-\frac {C \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(2 A c+a C) e \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {a} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[(2*A*e + C*d*x^2 + 2*C*e*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.

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

\(\Big \downarrow \) 2261

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

Input:

Int[(2*A*e + C*d*x^2 + 2*C*e*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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((2*C*e*x^4+C*d*x^2+2*A*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="fricas")
 

Output:

integral(-(2*C*e*x^4 + C*d*x^2 + 2*A*e)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/( 
c*e*x^6 + c*d*x^4 - a*e*x^2 - a*d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((2*C*e*x^4+C*d*x^2+2*A*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((2*C*e*x^4+C*d*x^2+2*A*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

2*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*e + 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)*c*d + 2*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*e