\(\int \frac {(a+b x^2)^{3/2}}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [465]

Optimal result

Integrand size = 34, antiderivative size = 538 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d f \sqrt {a+b x^2}}-\frac {b \sqrt {b c-a d} e \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 \sqrt {c} d f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {b c-a d} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 \sqrt {c} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a (b d e+b c f-3 a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 \sqrt {c} d \sqrt {b c-a d} f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [F]

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

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

\(\Big \downarrow \) 434

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

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]

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

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]

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

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

Output:

integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d*f*x^4 + (d*e 
 + c*f)*x^2 + c*e), x)
 

Sympy [F]

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

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

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

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

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

Output:

int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(c*e + c*f*x 
**2 + d*e*x**2 + d*f*x**4),x)*b + int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*s 
qrt(a + b*x**2))/(c*e + c*f*x**2 + d*e*x**2 + d*f*x**4),x)*a