\(\int \frac {(e+f x^2)^{3/2}}{(a+b x^2)^{3/2} (c+d x^2)^{3/2}} \, dx\) [450]

Optimal result

Integrand size = 34, antiderivative size = 401 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b e-a f) x \sqrt {e+f x^2}}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} (b c e+a d e-2 a c f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a c (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {2 \sqrt {e} (b e-a f) \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

(-a*f+b*e)*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+ 
e^(1/2)*(-c*f+d*e)^(1/2)*(-2*a*c*f+a*d*e+b*c*e)*(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)-2*e^(1/2)*(-a*f+b*e)*(-c*f+d*e)^(1/2)*(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*( 
b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx \] Input:

Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
 

Output:

Integrate[(e + f*x^2)^(3/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}dx\)

Input:

Int[(e + f*x^2)^(3/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

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]
 

Maple [F]

\[\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (x^{2} d +c \right )^{\frac {3}{2}}}d x\]

Input:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
 

Output:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(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)*(f*x^2 + e)^(3/2)/(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)
 

Sympy [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((f*x**2+e)**(3/2)/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2),x)
 

Output:

Integral((e + f*x**2)**(3/2)/((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((f*x^2 + e)^(3/2)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int((e + f*x^2)^(3/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x)
 

Output:

int((e + f*x^2)^(3/2)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}d x \] Input:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
 

Output:

int((f*x^2+e)^(3/2)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)