\(\int (a+b x^2)^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx\) [389]

Optimal result

Integrand size = 34, antiderivative size = 859 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {\left (3 a^2 d^2 f^2+8 a b d f (d e+c f)-b^2 \left (3 d^2 e^2-2 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{48 d^2 f^2 \sqrt {a+b x^2}}+\frac {(b d e+b c f+7 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{24 d f}+\frac {1}{6} b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}-\frac {\sqrt {b c-a d} e \left (3 a^2 d^2 f^2+8 a b d f (d e+c f)-b^2 \left (3 d^2 e^2-2 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b \sqrt {c} d^2 f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {b c-a d} \left (3 a^2 d^2 f^2-6 a b d f (2 d e+c f)+b^2 \left (d^2 e^2-4 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{48 b^2 \sqrt {c} d^2 f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \left (a^3 d^3 f^3+3 a b^2 d f (d e-c f)^2-3 a^2 b d^2 f^2 (d e+c f)-b^3 (d e-c f)^2 (d e+c f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{16 b^2 \sqrt {c} d^2 \sqrt {b c-a d} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/48*(3*a^2*d^2*f^2+8*a*b*d*f*(c*f+d*e)-b^2*(3*c^2*f^2-2*c*d*e*f+3*d^2*e^2 
))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d^2/f^2/(b*x^2+a)^(1/2)+1/24*(7*a*d*f 
+b*c*f+b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/f+1/6*b* 
x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)-1/48*(-a*d+b*c)^(1/2)* 
e*(3*a^2*d^2*f^2+8*a*b*d*f*(c*f+d*e)-b^2*(3*c^2*f^2-2*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)*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/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)-1/48*a*(-a*d+b*c)^(1 
/2)*(3*a^2*d^2*f^2-6*a*b*d*f*(c*f+2*d*e)+b^2*(3*c^2*f^2-4*c*d*e*f+d^2*e^2) 
)*(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^2/c^(1/2 
)/d^2/f/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/16*a*(a^3*d^3*f^ 
3+3*a*b^2*d*f*(-c*f+d*e)^2-3*a^2*b*d^2*f^2*(c*f+d*e)-b^3*(-c*f+d*e)^2*(c*f 
+d*e))*(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/(-a*d+b*c),(c*(-a*f+b*e)/(-a*d+b*c) 
/e)^(1/2))/b^2/c^(1/2)/d^2/(-a*d+b*c)^(1/2)/f^2/(a*(d*x^2+c)/c/(b*x^2+a))^ 
(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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