\(\int \frac {2 A e+C d x^2+2 C e x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx\) [122]

Optimal result

Integrand size = 41, antiderivative size = 569 \[ \int \frac {2 A e+C d x^2+2 C e x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=-\frac {\sqrt {a} C d \left (c+\frac {a}{x^4}\right ) \left (e+\frac {d}{x^2}\right ) x^3}{c \left (\sqrt {c d^2+a e^2}+\sqrt {a} \left (e+\frac {d}{x^2}\right )\right ) \sqrt {d+e x^2} \sqrt {a+c x^4}}+\frac {C \sqrt {d+e x^2} \sqrt {a+c x^4}}{c x}+\frac {\sqrt [4]{a} C \sqrt [4]{c d^2+a e^2} \left (\sqrt {c d^2+a e^2}+\sqrt {a} \left (e+\frac {d}{x^2}\right )\right ) \sqrt {\frac {d^2 \left (c+\frac {a}{x^4}\right )}{\left (\sqrt {c d^2+a e^2}+\sqrt {a} \left (e+\frac {d}{x^2}\right )\right )^2}} \sqrt {e+\frac {d}{x^2}} x^3 E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {e+\frac {d}{x^2}}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {a} e}{\sqrt {c d^2+a e^2}}\right )\right )}{c d \sqrt {d+e x^2} \sqrt {a+c x^4}}-\frac {\sqrt [4]{c d^2+a e^2} \left (2 A c e-a C e+\sqrt {a} C \sqrt {c d^2+a e^2}\right ) \left (1+\frac {\sqrt {a} \left (e+\frac {d}{x^2}\right )}{\sqrt {c d^2+a e^2}}\right ) \sqrt {\frac {d^2 \left (c+\frac {a}{x^4}\right )}{\left (c d^2+a e^2\right ) \left (1+\frac {\sqrt {a} \left (e+\frac {d}{x^2}\right )}{\sqrt {c d^2+a e^2}}\right )^2}} \sqrt {e+\frac {d}{x^2}} x^3 \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {e+\frac {d}{x^2}}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {a} e}{\sqrt {c d^2+a e^2}}\right )\right )}{2 \sqrt [4]{a} c d \sqrt {d+e x^2} \sqrt {a+c x^4}} \] Output:

-a^(1/2)*C*d*(c+a/x^4)*(e+d/x^2)*x^3/c/((a*e^2+c*d^2)^(1/2)+a^(1/2)*(e+d/x 
^2))/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)+C*(e*x^2+d)^(1/2)*(c*x^4+a)^(1/2)/c/x 
+a^(1/4)*C*(a*e^2+c*d^2)^(1/4)*((a*e^2+c*d^2)^(1/2)+a^(1/2)*(e+d/x^2))*(d^ 
2*(c+a/x^4)/((a*e^2+c*d^2)^(1/2)+a^(1/2)*(e+d/x^2))^2)^(1/2)*(e+d/x^2)^(1/ 
2)*x^3*EllipticE(sin(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4)) 
),1/2*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^(1/2))/c/d/(e*x^2+d)^(1/2)/(c*x^ 
4+a)^(1/2)-1/2*(a*e^2+c*d^2)^(1/4)*(2*A*c*e-C*a*e+a^(1/2)*C*(a*e^2+c*d^2)^ 
(1/2))*(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))*(d^2*(c+a/x^4)/(a*e^2+c*d 
^2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))^2)^(1/2)*(e+d/x^2)^(1/2)*x^3 
*InverseJacobiAM(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4)),1/2 
*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^(1/2))/a^(1/4)/c/d/(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, alg 
orithm="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, alg 
orithm="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, alg 
orithm="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. \[ \int \frac {2 A e+C d x^2+2 C e x^4}{\sqrt {d+e x^2} \sqrt {a+c x^4}} \, dx=\int \frac {2\,C\,e\,x^4+C\,d\,x^2+2\,A\,e}{\sqrt {c\,x^4+a}\,\sqrt {e\,x^2+d}} \,d x \] 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