\(\int \frac {x^2 (A+B x^2) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx\) [53]

Optimal result

Integrand size = 34, antiderivative size = 629 \[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\frac {\left (15 B c d^2-18 A c d e-8 a B e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c e^3 x}-\frac {(5 B d-6 A e) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 e^2}+\frac {B x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{6 e}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (15 B c d^2-18 A c d e-8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \left (5 B c d^2-6 A c d e-8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 \sqrt {c} e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (5 B c d^3-6 A c d^2 e-4 a B d e^2+8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {x^2 \left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\frac {6 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a e x -5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b d x +4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b e \,x^{3}+8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,e^{2}+18 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e -15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c \,d^{2}+12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{2}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d e -6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} d e +5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,d^{2}}{24 e^{2}} \] Input:

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

Output:

(6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e*x - 5*sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*b*d*x + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*e*x**3 + 8*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*e**2 + 18*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*d*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*d**2 + 12*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*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*b*d*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**2*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*b*d**2)/(24*e**2)