\(\int x^4 (A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx\) [34]

Optimal result

Integrand size = 34, antiderivative size = 874 \[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {\left (50 A c d e \left (3 c d^2-4 a e^2\right )-B \left (105 c^2 d^4-92 a c d^2 e^2+256 a^2 e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3840 c^2 e^4 x}+\frac {\left (35 B c d^3-50 A c d^2 e-28 a B d e^2-120 a A e^3\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{1920 c e^3}-\frac {\left (7 B c d^2-10 A c d e+16 a B e^2\right ) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{480 c e^2}+\frac {(B d+10 A e) x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{80 e}+\frac {1}{10} B x^7 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (50 A c d e \left (3 c d^2-4 a e^2\right )-B \left (105 c^2 d^4-92 a c d^2 e^2+256 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}} 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 )}{3840 c e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \left (10 A c d e \left (5 c d^2-44 a e^2\right )-B \left (35 c^2 d^4-36 a c d^2 e^2+256 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 {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 )}{3840 c^{3/2} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (2 A e \left (5 c^2 d^4-8 a c d^2 e^2+16 a^2 e^4\right )-B \left (7 c^2 d^5-8 a c d^3 e^2-16 a^2 d 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 )}{256 c e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/3840*(50*A*c*d*e*(-4*a*e^2+3*c*d^2)-B*(256*a^2*e^4-92*a*c*d^2*e^2+105*c^ 
2*d^4))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c^2/e^4/x+1/1920*(-120*A*a*e^3-50 
*A*c*d^2*e-28*B*a*d*e^2+35*B*c*d^3)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e 
^3-1/480*(-10*A*c*d*e+16*B*a*e^2+7*B*c*d^2)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a) 
^(1/2)/c/e^2+1/80*(10*A*e+B*d)*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e+1/10 
*B*x^7*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/3840*(d+a^(1/2)*e/c^(1/2))*(50*A 
*c*d*e*(-4*a*e^2+3*c*d^2)-B*(256*a^2*e^4-92*a*c*d^2*e^2+105*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)*Elli 
pticE(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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/3840*a^(1/2)*(10*A 
*c*d*e*(-44*a*e^2+5*c*d^2)-B*(256*a^2*e^4-36*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)*Elli 
pticF(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^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/256*(2*A*e*( 
16*a^2*e^4-8*a*c*d^2*e^2+5*c^2*d^4)-B*(-16*a^2*d*e^4-8*a*c*d^3*e^2+7*c^2*d 
^5))*(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]

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

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

Output:

Integrate[x^4*(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.

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

\(\Big \downarrow \) 2251

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

Input:

Int[x^4*(A + B*x^2)*Sqrt[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

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

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

Output:

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

Maxima [F]

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

integrate(x^4*(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^4, x)
 

Giac [F]

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

integrate(x^4*(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^4, x)
 

Mupad [F(-1)]

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

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

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

Output:

( - 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**3*x - 28*sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*a*b*d*e**2*x - 64*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b 
*e**3*x**3 - 50*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x + 40*sqrt(d 
 + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**3 + 240*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*a*c*e**3*x**5 + 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**3*x 
- 28*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**3 + 24*sqrt(d + e*x** 
2)*sqrt(a - c*x**4)*b*c*d*e**2*x**5 + 192*sqrt(d + e*x**2)*sqrt(a - c*x**4 
)*b*c*e**3*x**7 + 256*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*b*e**4 + 200*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*d 
*e**3 - 92*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**2*e**2 - 150*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**3*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**4 + 240*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**3*e**4 + 248*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*d*e**3 - 20*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**2*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)*...