\(\int x^2 (A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx\) [105]

Optimal result

Integrand size = 38, antiderivative size = 1229 \[ \int x^2 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Output:

-1/384*(8*A*c*e*(3*c^2*d^2+3*b^2*e^2-2*c*e*(4*a*e+b*d))-B*(15*c^3*d^3+15*b 
^3*e^3-c^2*d*e*(-20*a*e+7*b*d)-b*c*e^2*(52*a*e+7*b*d)))*(e*x^2+d)^(1/2)*(c 
*x^4+b*x^2+a)^(1/2)/c^3/e^3/x+1/192*(8*A*c*e*(b*e+c*d)-B*(5*c^2*d^2+5*b^2* 
e^2-2*c*e*(6*a*e+b*d)))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e^2+1/ 
48*(8*A*c*e+B*b*e+B*c*d)*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e+1/8 
*B*x^5*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)+1/768*(-4*a*c+b^2)^(1/2)*(8*A 
*c*e*(3*c^2*d^2+3*b^2*e^2-2*c*e*(4*a*e+b*d))-B*(15*c^3*d^3+15*b^3*e^3-c^2* 
d*e*(-20*a*e+7*b*d)-b*c*e^2*(52*a*e+7*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b 
^2))^(1/2)*x*(e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/ 
2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d- 
2*a*e))^(1/2))*2^(1/2)/c^3/e^3/(-a*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a 
*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/384*(-4*a*c+b^2)^(1/2)*(8*A*c*e*(c^2*d^ 
2+3*b^2*e^2-4*c*e*(2*a*e+b*d))-B*(5*c^3*d^3+15*b^3*e^3-c^2*d*e*(-44*a*e+3* 
b*d)-b*c*e^2*(52*a*e+17*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a 
*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticF(1/2*(1+(b 
+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/ 
(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/c^3/e^2/(e*x^2+d)^(1/2)/( 
c*x^4+b*x^2+a)^(1/2)+1/64*(-4*a*c+b^2)^(1/2)*(8*A*c*e*(-b*e+c*d)*(4*a*c*e^ 
2-b^2*e^2+c^2*d^2)-B*(5*c^4*d^4+5*b^4*e^4-4*c^3*d^2*e*(-2*a*e+b*d)-4*b^2*c 
*e^3*(6*a*e+b*d)-2*c^2*e^2*(-8*a^2*e^2-8*a*b*d*e+b^2*d^2)))*(-a*(c+a/x^...
 

Mathematica [F]

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

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

Output:

Integrate[x^2*(A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a + b*x^2 + 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.

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

\(\Big \downarrow \) 2250

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(20*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*e**2*x + 8*sqrt(d + e 
*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c**2*d*e*x + 32*sqrt(d + e*x**2)*sqrt(a 
 + b*x**2 + c*x**4)*a*c**2*e**2*x**3 - 5*sqrt(d + e*x**2)*sqrt(a + b*x**2 
+ c*x**4)*b**3*e**2*x + 2*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2* 
c*d*e*x + 4*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c*e**2*x**3 - 
5*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d**2*x + 4*sqrt(d + e* 
x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d*e*x**3 + 24*sqrt(d + e*x**2)*sqrt 
(a + b*x**2 + c*x**4)*b*c**2*e**2*x**5 + 64*int((sqrt(d + e*x**2)*sqrt(a + 
 b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + 
 c*e*x**6),x)*a**2*c**2*e**3 - 76*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + 
c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6) 
,x)*a*b**2*c*e**3 + 36*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x** 
4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*b*c** 
2*d*e**2 - 24*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + 
 a*e*x**2 + b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*a*c**3*d**2*e + 
15*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + 
 b*d*x**2 + b*e*x**4 + c*d*x**4 + c*e*x**6),x)*b**4*e**3 - 7*int((sqrt(d + 
 e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e* 
x**4 + c*d*x**4 + c*e*x**6),x)*b**3*c*d*e**2 - 7*int((sqrt(d + e*x**2)*sqr 
t(a + b*x**2 + c*x**4)*x**4)/(a*d + a*e*x**2 + b*d*x**2 + b*e*x**4 + c*...