\(\int (a+b x^2)^{3/2} \sqrt {c+d x^2} (e+f x^2)^{3/2} \, dx\) [427]

Optimal result

Integrand size = 34, antiderivative size = 1147 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2} \, dx =\text {Too large to display} \] Output:

-1/384*(9*a^3*d^3*f^3-3*a^2*b*d^2*f^2*(3*c*f+11*d*e)-a*b^2*d*f*(-31*c^2*f^ 
2+86*c*d*e*f+33*d^2*e^2)+b^3*(-15*c^3*f^3+31*c^2*d*e*f^2-9*c*d^2*e^2*f+9*d 
^3*e^3))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b/d^3/f^2/(b*x^2+a)^(1/2)+1/192 
*(66*a*e+10*b*c*e/d+3*b*e^2/f+3*a^2*f/b-5*b*c^2*f/d^2+10*a*c*f/d)*x*(b*x^2 
+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)+1/48*(9*a*d*f+b*c*f+9*b*d*e)*x^3 
*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d+1/8*b*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*(9*a^3*d^3*f 
^3-3*a^2*b*d^2*f^2*(3*c*f+11*d*e)-a*b^2*d*f*(-31*c^2*f^2+86*c*d*e*f+33*d^2 
*e^2)+b^3*(-15*c^3*f^3+31*c^2*d*e*f^2-9*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)*EllipticE((-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^2/c^(1/2)/d^3/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)*(9* 
a^3*d^3*f^3-3*a^2*b*d^2*f^2*(c*f+15*d*e)+3*a*b^2*d*f*(-7*c^2*f^2+26*c*d*e* 
f+29*d^2*e^2)-b^3*(-15*c^3*f^3+41*c^2*d*e*f^2-29*c*d^2*e^2*f+3*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^3/c^(1/2)/d 
^3/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/128*a*(3*a^4*d^4*f^ 
4-12*a*b^3*d*f*(-c*f+d*e)^3-4*a^3*b*d^3*f^3*(c*f+3*d*e)+b^4*(-c*f+d*e)^3*( 
5*c*f+3*d*e)+6*a^2*b^2*d^2*f^2*(-c^2*f^2+6*c*d*e*f+3*d^2*e^2))*(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...
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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