\(\int \frac {A+B x^2+C x^4+D x^6}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [77]

Optimal result

Integrand size = 41, antiderivative size = 558 \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {(3 d D-4 C e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 c e^2 x}-\frac {D x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 c e}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) (3 d D-4 C e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {(a d D-8 A c e-4 a C e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 \sqrt {a} \sqrt {c} e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (4 (2 B c+a D) e^2+c d (3 d D-4 C e)\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 c e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/8*(-4*C*e+3*D*d)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^2/x-1/4*D*x*(e*x^2 
+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e+1/8*(d+a^(1/2)*e/c^(1/2))*(-4*C*e+3*D*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)*El 
lipticE(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^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/8*(-8*A*c*e-4*C*a* 
e+D*a*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)*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/(e*x^2+d)^(1/2)/(-c*x^4+a)^( 
1/2)+1/8*(4*(2*B*c+D*a)*e^2+c*d*(-4*C*e+3*D*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))/c/e^ 
2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(Sqrt[d + e*x^2]*Sqrt[a - c*x^4]),x]
 

Output:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a-c x^4} \sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a-c x^4} \sqrt {d+e x^2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/(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]

\[\int \frac {D x^{6}+C \,x^{4}+B \,x^{2}+A}{\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]

Input:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algori 
thm="fricas")
 

Output:

integral(-(D*x^6 + C*x^4 + B*x^2 + A)*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]

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

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

Maxima [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algori 
thm="maxima")
 

Output:

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

Giac [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\int \frac {A+B\,x^2+C\,x^4+x^6\,D}{\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}} \,d x \] Input:

int((A + B*x^2 + C*x^4 + x^6*D)/((a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/((a - c*x^4)^(1/2)*(d + e*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, d x +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} e -3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c \,d^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a d e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c e +\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,d^{2}}{4 c e} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*d*x + 4*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**2*e - 3*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*d**2 + 2*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)*a*d*e + 4*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*c*e + 4*int((sqr 
t(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*c*e + 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*d**2)/(4*c*e)