Integrand size = 34, antiderivative size = 614 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {b^2 x}{a (b c-a d) (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {d \left (a b d^2 e-a^2 d^2 f+b^2 c (d e-c f)\right ) x \sqrt {a+b x^2}}{a c (b c-a d)^2 (b e-a f) (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\left (b^3 c e (d e-c f)^2+a^3 d^2 f^2 (d e+c f)-2 a^2 b d f \left (d^2 e^2+c^2 f^2\right )+a b^2 \left (d^3 e^3+c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} (b c-a d)^2 e (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (2 a^2 d^2 f^2-b^2 (d e-c f)^2-a b d f (d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{a^{3/2} (b c-a d) (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
b^2*x/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1 /2)+d*(a*b*d^2*e-a^2*d^2*f+b^2*c*(-c*f+d*e))*x*(b*x^2+a)^(1/2)/a/c/(-a*d+b *c)^2/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)+(b^3*c*e*(-c*f +d*e)^2+a^3*d^2*f^2*(c*f+d*e)-2*a^2*b*d*f*(c^2*f^2+d^2*e^2)+a*b^2*(c^3*f^3 +d^3*e^3))*(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)^2/e/(a*f-b*e)^(3/2)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^ 2+a)/a/(f*x^2+e))^(1/2)+(2*a^2*d^2*f^2-b^2*(-c*f+d*e)^2-a*b*d*f*(c*f+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* d+b*c)/(a*f-b*e)^(3/2)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+ e))^(1/2)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:
Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]
Output:
Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}dx\) |
Input:
Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(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]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (x^{2} d +c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(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^2*f^2*x^12 + 2*(b^2*d^2*e*f + (b^2*c*d + a*b*d^2)*f^2)*x^10 + (b^2*d^2*e^2 + 4*(b^2* c*d + a*b*d^2)*e*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2)*x^8 + 2*((b^2*c* d + a*b*d^2)*e^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e*f + (a*b*c^2 + a^2*c* d)*f^2)*x^6 + a^2*c^2*e^2 + (a^2*c^2*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2) *e^2 + 4*(a*b*c^2 + a^2*c*d)*e*f)*x^4 + 2*(a^2*c^2*e*f + (a*b*c^2 + a^2*c* d)*e^2)*x^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**(3/2),x)
Output:
Integral(1/((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)*(e + f*x**2)**(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x, algorithm=" maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x, algorithm=" giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x)