Integrand size = 34, antiderivative size = 434 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{a (b c-a d) (b e-a f) \sqrt {a+b x^2} \sqrt {e+f x^2}}-\frac {c \left (a b c f^2-a^2 d f^2-b^2 e (d e-c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} (b c-a d) e (-b e+a f)^{3/2} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {(b d e-b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} (-b e+a f)^{3/2} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
b^2*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(f*x^2+e)^(1 /2)-c*(a*b*c*f^2-a^2*d*f^2-b^2*e*(-c*f+d*e))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/ c/(f*x^2+e))^(1/2)*EllipticE((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a* (-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(3/2)/(-a*d+b*c)/e/(a*f-b*e)^(3/2)/(-c*f +d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-(-a*d*f-b*c*f+b*d*e) *(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f-b*e)^(1/2) *x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(3/2)/(a*f -b*e)^(3/2)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:
Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]
Output:
Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)), 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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}dx\) |
Input:
Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]
Output:
$Aborted
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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm=" fricas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^2*d*f^2*x^10 + (2*b^2*d*e*f + (b^2*c + 2*a*b*d)*f^2)*x^8 + (b^2*d*e^2 + 2*(b^2*c + 2*a*b *d)*e*f + (2*a*b*c + a^2*d)*f^2)*x^6 + a^2*c*e^2 + (a^2*c*f^2 + (b^2*c + 2 *a*b*d)*e^2 + 2*(2*a*b*c + a^2*d)*e*f)*x^4 + (2*a^2*c*e*f + (2*a*b*c + a^2 *d)*e^2)*x^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(3/2),x)
Output:
Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)**(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm=" maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm=" giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)