\(\int \sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)^{3/2} \, dx\) [419]

Optimal result

Integrand size = 34, antiderivative size = 1146 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2} \, dx =\text {Too large to display} \] Output:

1/384*(15*a^3*d^2*f^2/b-31*a^2*d*f*(c*f+d*e)+a*b*(9*c^2*f^2+86*c*d*e*f+9*d 
^2*e^2)+b^2*(33*c*d*e^2-9*d^2*e^3/f+33*c^2*e*f-9*c^3*f^2/d))*x*(d*x^2+c)^( 
1/2)*(f*x^2+e)^(1/2)/b/d/f/(b*x^2+a)^(1/2)+1/192*(66*c*e+3*d*e^2/f+3*c^2*f 
/d-5*a^2*d*f/b^2+10*a*(c*f+d*e)/b)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^ 
2+e)^(1/2)+1/48*(a*d*f+9*b*(c*f+d*e))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)* 
(f*x^2+e)^(1/2)/b+1/8*d*f*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1 
/2)-1/384*(-a*d+b*c)^(1/2)*e*(15*a^3*d^3*f^3-31*a^2*b*d^2*f^2*(c*f+d*e)+a* 
b^2*d*f*(9*c^2*f^2+86*c*d*e*f+9*d^2*e^2)-b^3*(9*c^3*f^3-33*c^2*d*e*f^2-33* 
c*d^2*e^2*f+9*d^3*e^3))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*El 
lipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c 
)/e)^(1/2))/b^3/c^(1/2)/d^2/f^2/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^ 
(1/2)-1/384*a*(-a*d+b*c)^(1/2)*(15*a^3*d^3*f^3-3*a^2*b*d^2*f^2*(7*c*f+17*d 
*e)+a*b^2*d*f*(-3*c^2*f^2+98*c*d*e*f+49*d^2*e^2)+3*b^3*(3*c^3*f^3-13*c^2*d 
*e*f^2-55*c*d^2*e^2*f+d^3*e^3))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^ 
(1/2)*EllipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/( 
-a*d+b*c)/e)^(1/2))/b^4/c^(1/2)/d^2/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x 
^2+e)^(1/2)-1/128*a*(5*a^4*d^4*f^4-3*b^4*(-c*f+d*e)^4-12*a^3*b*d^3*f^3*(c* 
f+d*e)+6*a^2*b^2*d^2*f^2*(c^2*f^2+6*c*d*e*f+d^2*e^2)+4*a*b^3*d*f*(c^3*f^3- 
9*c^2*d*e*f^2-9*c*d^2*e^2*f+d^3*e^3))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^ 
2+a))^(1/2)*EllipticPi((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),b*c/(...
 

Mathematica [F]

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

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

Output:

Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2), 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.

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

\(\Big \downarrow \) 434

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

Input:

Int[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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