\(\int \frac {x^2 (A+B x^2+C x^4+D x^6)}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [76]

Optimal result

Integrand size = 44, antiderivative size = 676 \[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=-\frac {\left (15 c d^2 D-18 c C d e+24 B c e^2+16 a D e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c^2 e^3 x}+\frac {(5 d D-6 C e) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 c e^2}-\frac {D x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{6 c e}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (16 a D e^2+3 c \left (5 d^2 D-6 C d e+8 B e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 c e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (16 a D e^2+c \left (5 d^2 D-6 C d e+24 B e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 c^{3/2} e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (4 a e^2 (d D-2 C e)+c \left (5 d^3 D-6 C d^2 e+8 B d e^2-16 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 c e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/48*(24*B*c*e^2-18*C*c*d*e+16*D*a*e^2+15*D*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^ 
4+a)^(1/2)/c^2/e^3/x+1/24*(-6*C*e+5*D*d)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2 
)/c/e^2-1/6*D*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e-1/48*(d+a^(1/2)*e/c 
^(1/2))*(16*a*D*e^2+3*c*(8*B*e^2-6*C*d*e+5*D*d^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))/c/e^ 
3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/48*a^(1/2)*(16*a*D*e^2+c*(24*B*e^2-6* 
C*d*e+5*D*d^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)*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))/c^(3/2)/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^ 
(1/2)-1/16*(4*a*e^2*(-2*C*e+D*d)+c*(-16*A*e^3+8*B*d*e^2-6*C*d^2*e+5*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 
)*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^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(x^2*(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.

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

\(\Big \downarrow \) 2251

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

Input:

Int[(x^2*(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 2251

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

Maple [F]

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

Input:

int(x^2*(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(x^2*(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(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int((x^2*(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((x^2*(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]

\[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {-6 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c e x +5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, d^{2} x -4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, d e \,x^{3}+16 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a d \,e^{2}+24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c \,e^{2}-18 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} d e +15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c \,d^{3}+36 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,d^{2} e +6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e -5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,d^{3}}{24 c \,e^{2}} \] Input:

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

Output:

( - 6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*e*x + 5*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*d**2*x - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*d*e*x**3 + 16*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)*a*d*e**2 + 24*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)*b*c*e**2 - 18*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c**2*d*e + 1 
5*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**3 + 36*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*c*e**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**2*e 
 + 6*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*c*d*e - 5*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**3)/(24*c*e**2)