Integrand size = 39, antiderivative size = 321 \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\frac {a \sqrt {c-d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a-b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a-b x^2}}\right ),\frac {c (b e+a f)}{(b c-a d) e}\right )}{b \sqrt {c} \sqrt {-b c+a d} \sqrt {\frac {a \left (c-d x^2\right )}{c \left (a-b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {c-d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a-b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a-b x^2}}\right ),\frac {c (b e+a f)}{(b c-a d) e}\right )}{b \sqrt {c} \sqrt {-b c+a d} \sqrt {\frac {a \left (c-d x^2\right )}{c \left (a-b x^2\right )}} \sqrt {e+f x^2}} \] 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, algorit hm="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, algorit hm="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, algorit hm="giac")
Output:
integrate(x^2/(sqrt(-b*x^2 + a)*sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {x^2}{\sqrt {a-b x^2} \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {x^2}{\sqrt {a-b\,x^2}\,\sqrt {c-d\,x^2}\,\sqrt {f\,x^2+e}} \,d x \] 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)