\(\int x^2 (A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx\) [35]

Optimal result

Integrand size = 34, antiderivative size = 737 \[ \int x^2 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {\left (15 B c d^3-24 A c d^2 e-20 a B d e^2-64 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 c e^3 x}-\frac {\left (5 B c d^2-8 A c d e+12 a B e^2\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{192 c e^2}+\frac {(B d+8 A e) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 e}+\frac {1}{8} B x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (8 A e \left (3 c d^2+8 a e^2\right )-5 B \left (3 c d^3-4 a d e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \left (5 B c d^3-8 A c d^2 e-44 a B d e^2-64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 \sqrt {c} e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (8 A c d e \left (c d^2-4 a e^2\right )-B \left (5 c^2 d^4-8 a c d^2 e^2+16 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 c e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/384*(-64*A*a*e^3-24*A*c*d^2*e-20*B*a*d*e^2+15*B*c*d^3)*(e*x^2+d)^(1/2)*( 
-c*x^4+a)^(1/2)/c/e^3/x-1/192*(-8*A*c*d*e+12*B*a*e^2+5*B*c*d^2)*x*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/c/e^2+1/48*(8*A*e+B*d)*x^3*(e*x^2+d)^(1/2)*(-c*x^ 
4+a)^(1/2)/e+1/8*B*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/384*(d+a^(1/2)*e 
/c^(1/2))*(8*A*e*(8*a*e^2+3*c*d^2)-5*B*(-4*a*d*e^2+3*c*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))/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/384*a^(1/2)*(-64*A*a*e^3-8*A* 
c*d^2*e-44*B*a*d*e^2+5*B*c*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)*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^(1/2)/e^2/(e*x^2+d)^( 
1/2)/(-c*x^4+a)^(1/2)-1/128*(8*A*c*d*e*(-4*a*e^2+c*d^2)-B*(16*a^2*e^4-8*a* 
c*d^2*e^2+5*c^2*d^4))*(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]

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

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

Output:

Integrate[x^2*(A + B*x^2)*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.

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

\(\Big \downarrow \) 2251

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

Input:

Int[x^2*(A + B*x^2)*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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F]

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

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

Maxima [F]

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

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {-12 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b \,e^{2} x +8 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d e x +32 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,e^{2} x^{3}-5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x +4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d e \,x^{3}+24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,e^{2} x^{5}+64 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,e^{3}+20 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c d \,e^{2}+24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{2} e -15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{3}+24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b \,e^{3}+80 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c d \,e^{2}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{2} e +12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b d \,e^{2}-8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,d^{2} e +5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{3}}{192 c \,e^{2}} \] Input:

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

Output:

( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2*x + 8*sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*a*c*d*e*x + 32*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e**2* 
x**3 - 5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*x + 4*sqrt(d + e*x**2) 
*sqrt(a - c*x**4)*b*c*d*e*x**3 + 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c* 
e**2*x**5 + 64*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**2*c*e**3 + 20*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*b*c*d*e**2 + 2 
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)*a*c**2*d**2*e - 15*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**2*d**3 + 24*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**2*b*e**3 + 80*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**2*c*d*e**2 - 2*int((sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d**2 
*e + 12*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**2*b*d*e**2 - 8*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**2*c*d**2*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*b*c* 
d**3)/(192*c*e**2)