Integrand size = 37, antiderivative size = 313 \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{b \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{b \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:
-a*(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/c^(1/2) /(-a*d+b*c)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+a*(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/ c^(1/2)/(-a*d+b*c)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx \] Input:
Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
Integrate[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]
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.
\(\displaystyle \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) |
\(\Big \downarrow \) 450 |
\(\displaystyle \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}dx\) |
Input:
Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
$Aborted
Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ (q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x]
\[\int \frac {x^{2}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]
Input:
int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
Output:
int(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
Timed out. \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:
integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm ="fricas")
Output:
Timed out
\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:
integrate(x**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
Output:
Integral(x**2/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm ="maxima")
Output:
integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate(x^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm ="giac")
Output:
integrate(x^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {x^2}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int(x^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
Output:
int(x^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \] Input:
int(x^2/(b*x^2+a)^(1/2)/(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)/(a*c*e + a*c *f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)