\(\int (A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4} \, dx\) [114]

Optimal result

Integrand size = 35, antiderivative size = 1241 \[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Output:

1/384*(8*A*c*e*(3*c^2*d^2-3*b^2*e^2+8*c*e*(a*e+b*d))-B*(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)))*(e*x^2+d)^(1/2)*(c*x 
^4+b*x^2+a)^(1/2)/c^3/e^2/x+1/192*(8*A*c*e*(b*e+7*c*d)+B*(3*c^2*d^2-5*b^2* 
e^2+2*c*e*(6*a*e+5*b*d)))*x*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e+1/ 
48*(8*A*c*e+B*b*e+9*B*c*d)*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c+1/8 
*B*e*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+8*c*e*(a*e+b*d))-B*(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)))*(-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^2/(-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*(31*c^ 
2*d^2+3*b^2*e^2-2*c*e*(4*a*e+5*b*d))+B*(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)))*(-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/(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*(c^3*d^3-b^3*e^ 
3-3*c^2*d*e*(4*a*e+b*d)+b*c*e^2*(4*a*e+3*b*d))-B*(3*c^4*d^4-5*b^4*e^4-4*c^ 
3*d^2*e*(-6*a*e+b*d)+12*b^2*c*e^3*(2*a*e+b*d)-2*c^2*e^2*(8*a^2*e^2+24*a...
 

Mathematica [F]

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

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

Output:

Integrate[(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.

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

\(\Big \downarrow \) 2260

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

Input:

Int[(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 2260

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((B*x^2+A)*(e*x^2+d)^(3/2)*(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 + 56*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 + 10*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 
+ 3*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d**2*x + 36*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)*s 
qrt(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 + 148*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 - 31*int((sqr 
t(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 + 9*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...