\(\int x^4 (A+B x^2) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx\) [104]

Optimal result

Integrand size = 38, antiderivative size = 1604 \[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Output:

1/3840*(10*A*c*e*(15*c^3*d^3+15*b^3*e^3-c^2*d*e*(-20*a*e+7*b*d)-b*c*e^2*(5 
2*a*e+7*b*d))-B*(105*c^4*d^4+105*b^4*e^4-4*c^3*d^2*e*(-23*a*e+10*b*d)-20*b 
^2*c*e^3*(23*a*e+2*b*d)-2*c^2*e^2*(-128*a^2*e^2-72*a*b*d*e+17*b^2*d^2)))*( 
e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^4/e^4/x-1/1920*(10*A*c*e*(5*c^2*d^2 
+5*b^2*e^2-2*c*e*(6*a*e+b*d))-B*(35*c^3*d^3+35*b^3*e^3-c^2*d*e*(-28*a*e+11 
*b*d)-b*c*e^2*(116*a*e+11*b*d)))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c 
^3/e^3+1/480*(10*A*c*e*(b*e+c*d)-B*(7*c^2*d^2+7*b^2*e^2-2*c*e*(8*a*e+b*d)) 
)*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e^2+1/80*(10*A*c*e+B*b*e+B 
*c*d)*x^5*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/e+1/10*B*x^7*(e*x^2+d)^( 
1/2)*(c*x^4+b*x^2+a)^(1/2)-1/7680*(-4*a*c+b^2)^(1/2)*(10*A*c*e*(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))-B*(105*c^4*d^4 
+105*b^4*e^4-4*c^3*d^2*e*(-23*a*e+10*b*d)-20*b^2*c*e^3*(23*a*e+2*b*d)-2*c^ 
2*e^2*(-128*a^2*e^2-72*a*b*d*e+17*b^2*d^2)))*(-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^4/e^4/(-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)+1/3840*(-4*a*c+b^2)^(1/2)*(10*A*c*e*(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))-B*(35*c^ 
4*d^4+105*b^4*e^4-18*c^3*d^2*e*(-2*a*e+b*d)-10*b^2*c*e^3*(46*a*e+11*b*d)-4 
*c^2*e^2*(-64*a^2*e^2-94*a*b*d*e+3*b^2*d^2)))*(-a*(c+a/x^4+b/x^2)/(-4*a...
 

Mathematica [F]

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

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

Output:

Integrate[x^4*(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.

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

\(\Big \downarrow \) 2250

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

Input:

Int[x^4*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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