\(\int x^2 (A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4} \, dx\) [113]

Optimal result

Integrand size = 38, antiderivative size = 1576 \[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Output:

-1/3840*(10*A*c*e*(9*c^3*d^3-15*b^3*e^3-3*c^2*d*e*(28*a*e+3*b*d)+b*c*e^2*( 
52*a*e+31*b*d))-B*(45*c^4*d^4-105*b^4*e^4-6*c^3*d^2*e*(-18*a*e+5*b*d)+10*b 
^2*c*e^3*(46*a*e+19*b*d)-4*c^2*e^2*(64*a^2*e^2+166*a*b*d*e+9*b^2*d^2)))*(e 
*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^4/e^3/x+1/1920*(10*A*c*e*(3*c^2*d^2- 
5*b^2*e^2+2*c*e*(6*a*e+5*b*d))-B*(15*c^3*d^3-35*b^3*e^3+b*c*e^2*(116*a*e+6 
1*b*d)-c^2*d*e*(148*a*e+9*b*d)))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c 
^3/e^2+1/480*(10*A*c*e*(b*e+9*c*d)+B*(3*c^2*d^2-7*b^2*e^2+4*c*e*(4*a*e+3*b 
*d)))*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e+1/80*(10*A*c*e+B*b*e 
+11*B*c*d)*x^5*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c+1/10*B*e*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*(9*c^3 
*d^3-15*b^3*e^3-3*c^2*d*e*(28*a*e+3*b*d)+b*c*e^2*(52*a*e+31*b*d))-B*(45*c^ 
4*d^4-105*b^4*e^4-6*c^3*d^2*e*(-18*a*e+5*b*d)+10*b^2*c*e^3*(46*a*e+19*b*d) 
-4*c^2*e^2*(64*a^2*e^2+166*a*b*d*e+9*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^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)-1/3840*(-4*a*c+b^2)^(1/2)*(10*A*c*e*(3 
*c^3*d^3-15*b^3*e^3+b*c*e^2*(52*a*e+41*b*d)-c^2*d*e*(108*a*e+29*b*d))-B*(1 
5*c^4*d^4-105*b^4*e^4-4*c^3*d^2*e*(-101*a*e+3*b*d)+20*b^2*c*e^3*(23*a*e+13 
*b*d)-2*c^2*e^2*(128*a^2*e^2+448*a*b*d*e+79*b^2*d^2)))*(-a*(c+a/x^4+b/x...
 

Mathematica [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx \] Input:

Integrate[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a + b*x^2 + c*x^4],x]
 

Output:

Integrate[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*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.

\(\displaystyle \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx\)

\(\Big \downarrow \) 2250

\(\displaystyle \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}dx\)

Input:

Int[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*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]

\[\int x^{2} \left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}d x\]

Input:

int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int x^{2} \left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}} \sqrt {a + b x^{2} + c x^{4}}\, dx \] Input:

integrate(x**2*(B*x**2+A)*(e*x**2+d)**(3/2)*(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

Integral(x**2*(A + B*x**2)*(d + e*x**2)**(3/2)*sqrt(a + b*x**2 + c*x**4), 
x)
 

Maxima [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(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)*(e*x^2 + d)^(3/2)*x^2, x)
 

Giac [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:

integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(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)*(e*x^2 + d)^(3/2)*x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int x^2\,\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^{3/2}\,\sqrt {c\,x^4+b\,x^2+a} \,d x \] Input:

int(x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2),x)
 

Output:

int(x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*(a + b*x^2 + c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx=\int x^{2} \left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c \,x^{4}+b \,x^{2}+a}d x \] Input:

int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2),x)