Integrand size = 34, antiderivative size = 395 \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {e+f x^2}}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {(b c+a d) \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a c (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (2 b d e-b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a (b c-a d)^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:
b*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+(a*d+b*c) *e^(1/2)*(-c*f+d*e)^(1/2)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)* EllipticE((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),(-(-a*d+b*c)*e/a/(-c* f+d*e))^(1/2))/a/c/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^ (1/2)-e^(1/2)*(-a*d*f-b*c*f+2*b*d*e)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2 +c))^(1/2)*EllipticF((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),(-(-a*d+b* c)*e/a/(-c*f+d*e))^(1/2))/a/(-a*d+b*c)^2/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/( d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx \] Input:
Integrate[Sqrt[e + f*x^2]/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
Output:
Integrate[Sqrt[e + f*x^2]/((a + b*x^2)^(3/2)*(c + d*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.
\(\displaystyle \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 434 |
\(\displaystyle \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}dx\) |
Input:
Int[Sqrt[e + f*x^2]/((a + b*x^2)^(3/2)*(c + d*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]
\[\int \frac {\sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (x^{2} d +c \right )^{\frac {3}{2}}}d x\]
Input:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
Output:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="fr icas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b^2*d^2*x^8 + 2* (b^2*c*d + a*b*d^2)*x^6 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(a*b*c^2 + a^2*c*d)*x^2), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((f*x**2+e)**(1/2)/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(e + f*x**2)/((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="ma xima")
Output:
integrate(sqrt(f*x^2 + e)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="gi ac")
Output:
integrate(sqrt(f*x^2 + e)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((e + f*x^2)^(1/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x)
Output:
int((e + f*x^2)^(1/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {f \,x^{2}+e}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}d x \] Input:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
Output:
int((f*x^2+e)^(1/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)