\(\int \frac {x^4 (A+B x^2) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx\) [52]

Optimal result

Integrand size = 34, antiderivative size = 738 \[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=-\frac {\left (105 B c d^3-120 A c d^2 e-44 a B d e^2+64 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 c e^4 x}+\frac {\left (35 B c d^2-40 A c d e-12 a B e^2\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{192 c e^3}-\frac {(7 B d-8 A e) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 e^2}+\frac {B x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 e}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (105 B c d^3-120 A c d^2 e-44 a B d e^2+64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (35 B c d^3-40 A c d^2 e-20 a B d e^2+64 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 \sqrt {c} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (8 A c d e \left (5 c d^2-4 a e^2\right )-B \left (35 c^2 d^4-24 a c d^2 e^2-16 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 c e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/384*(64*A*a*e^3-120*A*c*d^2*e-44*B*a*d*e^2+105*B*c*d^3)*(e*x^2+d)^(1/2) 
*(-c*x^4+a)^(1/2)/c/e^4/x+1/192*(-40*A*c*d*e-12*B*a*e^2+35*B*c*d^2)*x*(e*x 
^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^3-1/48*(-8*A*e+7*B*d)*x^3*(e*x^2+d)^(1/2) 
*(-c*x^4+a)^(1/2)/e^2+1/8*B*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e-1/384*( 
d+a^(1/2)*e/c^(1/2))*(64*A*a*e^3-120*A*c*d^2*e-44*B*a*d*e^2+105*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)*E 
llipticE(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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/384*a^(1/2)*(64*A 
*a*e^3-40*A*c*d^2*e-20*B*a*d*e^2+35*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^ 
3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/128*(8*A*c*d*e*(-4*a*e^2+5*c*d^2)-B*( 
-16*a^2*e^4-24*a*c*d^2*e^2+35*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^4/(e*x^2 
+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(x^4*(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.

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

\(\Big \downarrow \) 2251

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

Input:

Int[(x^4*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^4*(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^4/sqrt(e*x^2 + d), x)
 

Giac [F]

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

integrate(x^4*(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^4/sqrt(e*x^2 + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4 \left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\frac {-12 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b \,e^{2} x -40 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d e x +32 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,e^{2} x^{3}+35 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x -28 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d e \,x^{3}+24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,e^{2} x^{5}+64 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,e^{3}-44 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c d \,e^{2}-120 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{2} e +105 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{3}+24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b \,e^{3}-16 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c d \,e^{2}+14 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{2} e +12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} b d \,e^{2}+40 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c \,d^{2} e -35 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{3}}{192 c \,e^{3}} \] Input:

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

Output:

( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**2*x - 40*sqrt(d + e*x**2)* 
sqrt(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 + 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*x - 28*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 - 44*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 
- 120*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 + 105*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 - 16*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 + 14*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* 
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 + 40*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 - 35*int((s 
qrt(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**3)