Integrand size = 34, antiderivative size = 502 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {c (-b e+a f)^{3/2} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} e f (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\sqrt {-b e+a f} (b d e-2 b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f^2 (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {b^2 e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f^2 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:
-c*(a*f-b*e)^(3/2)*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*Ellipti cE((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^(1/2)/e/f/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/ 2)+(a*f-b*e)^(1/2)*(a*d*f-2*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^(1/2)/f^2/(-c*f+d*e)/(d*x^2+c)^(1/2)/(e*(b*x ^2+a)/a/(f*x^2+e))^(1/2)+b^2*e*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^( 1/2)*EllipticPi((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),-a*f/(-a*f+b*e), (a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/f^2/(a*f-b*e)^(1/2)/(d*x^2+c)^( 1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:
Integrate[(a + b*x^2)^(3/2)/(Sqrt[c + d*x^2]*(e + f*x^2)^(3/2)),x]
Output:
Integrate[(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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 434 |
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}dx\) |
Input:
Int[(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]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]
Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Output:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(3/2),x)
Output:
Integral((a + b*x**2)**(3/2)/(sqrt(c + d*x**2)*(e + f*x**2)**(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="ma xima")
Output:
integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="gi ac")
Output:
integrate((b*x^2 + a)^(3/2)/(sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)),x)
Output:
int((a + b*x^2)^(3/2)/((c + d*x^2)^(1/2)*(e + f*x^2)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{d \,f^{2} x^{6}+c \,f^{2} x^{4}+2 d e f \,x^{4}+2 c e f \,x^{2}+d \,e^{2} x^{2}+c \,e^{2}}d x \right ) b +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d \,f^{2} x^{6}+c \,f^{2} x^{4}+2 d e f \,x^{4}+2 c e f \,x^{2}+d \,e^{2} x^{2}+c \,e^{2}}d x \right ) a \] Input:
int((b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
Output:
int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(c*e**2 + 2* c*e*f*x**2 + c*f**2*x**4 + d*e**2*x**2 + 2*d*e*f*x**4 + d*f**2*x**6),x)*b + int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c*e**2 + 2*c*e *f*x**2 + c*f**2*x**4 + d*e**2*x**2 + 2*d*e*f*x**4 + d*f**2*x**6),x)*a