\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^{3/2} \sqrt {e+f x^2}} \, dx\) [466]

Optimal result

Integrand size = 34, antiderivative size = 493 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=-\frac {(b c-a d) \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{c d \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {b (b c-a d) \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a d^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a d^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-(-a*d+b*c)*e^(1/2)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*Ellipt 
icE((-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))/c/d/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1 
/2)-b*(-a*d+b*c)*e^(1/2)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*E 
llipticF((-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/d^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)+b^2*c*e^(1/2)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*E 
llipticPi((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),d*e/(-c*f+d*e),(-(-a* 
d+b*c)*e/a/(-c*f+d*e))^(1/2))/a/d^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)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F(-1)]

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

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="fr 
icas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

int((a + b*x^2)^(3/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{d^{2} f \,x^{6}+2 c d f \,x^{4}+d^{2} e \,x^{4}+c^{2} f \,x^{2}+2 c d e \,x^{2}+c^{2} e}d x \right ) b +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{2} f \,x^{6}+2 c d f \,x^{4}+d^{2} e \,x^{4}+c^{2} f \,x^{2}+2 c d e \,x^{2}+c^{2} e}d x \right ) a \] Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2),x)
 

Output:

int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(c**2*e + c* 
*2*f*x**2 + 2*c*d*e*x**2 + 2*c*d*f*x**4 + d**2*e*x**4 + d**2*f*x**6),x)*b 
+ int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**2*e + c**2* 
f*x**2 + 2*c*d*e*x**2 + 2*c*d*f*x**4 + d**2*e*x**4 + d**2*f*x**6),x)*a