\(\int \frac {A+B x^2+C x^4+D x^6}{(d+e x^2)^{3/2} \sqrt {a-c x^4}} \, dx\) [85]

Optimal result

Integrand size = 41, antiderivative size = 658 \[ \int \frac {A+B x^2+C x^4+D x^6}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {\left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x \sqrt {a-c x^4}}{d e \left (c d^2-a e^2\right ) \sqrt {d+e x^2}}+\frac {\left (a d D e^2-c \left (3 d^3 D-2 C d^2 e+2 B d e^2-2 A e^3\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 c d e^2 \left (c d^2-a e^2\right ) x}+\frac {\left (a d D e^2-c \left (3 d^3 D-2 C d^2 e+2 B d e^2-2 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {c} d e^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(a d D+2 A c e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {a} \sqrt {c} d e \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {(3 d D-2 C e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/((d + e*x^2)^(3/2)*Sqrt[a - c*x^4]), 
x]
 

Output:

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/((d + e*x^2)^(3/2)*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.

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {A+B x^2+C x^4+D x^6}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\)

Input:

Int[(A + B*x^2 + C*x^4 + D*x^6)/((d + e*x^2)^(3/2)*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]

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

Input:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-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^2*x^8 + 2*c*d*e*x^6 - 2*a*d*e*x^2 + (c*d^2 - a*e^2)*x^4 - a*d^2), x)
 

Sympy [F]

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

integrate((D*x**6+C*x**4+B*x**2+A)/(e*x**2+d)**(3/2)/(-c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4 + D*x**6)/(sqrt(a - c*x**4)*(d + e*x**2)**(3 
/2)), x)
 

Maxima [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-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)*(e*x^2 + d)^(3/2)) 
, x)
 

Giac [F]

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

integrate((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-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)*(e*x^2 + d)^(3/2)) 
, x)
 

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4 + x^6*D)/((a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)),x)
 

Output:

int((A + B*x^2 + C*x^4 + x^6*D)/((a - c*x^4)^(1/2)*(d + e*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4+D x^6}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) c d e -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) c \,e^{2} x^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) d^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{6}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) d^{2} e \,x^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b \,d^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) b d e \,x^{2}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a \,d^{2}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{2} x^{8}-2 c d e \,x^{6}+a \,e^{2} x^{4}-c \,d^{2} x^{4}+2 a d e \,x^{2}+a \,d^{2}}d x \right ) a d e \,x^{2}}{3 d \left (e \,x^{2}+d \right )} \] Input:

int((D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*x - 2*int((sqrt(d + e*x**2)*sqrt(a - 
 c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d* 
e*x**6 - c*e**2*x**8),x)*c*d*e - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)* 
x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - 
c*e**2*x**8),x)*c*e**2*x**2 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x** 
6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e 
**2*x**8),x)*d**3 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 
 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8), 
x)*d**2*e*x**2 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d**2 + 
2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)* 
b*d**2 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d**2 + 2*a*d*e* 
x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*b*d*e*x* 
*2 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2 + 2*a*d*e*x**2 + a* 
e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*d**2 + 4*int((s 
qrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c 
*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*d*e*x**2)/(3*d*(d + e*x**2))