\(\int \frac {(e+f x^2)^{3/2}}{(a+b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [449]

Optimal result

Integrand size = 34, antiderivative size = 488 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c} (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{a b \sqrt {b c-a d} \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {\sqrt {c} f (b e-a f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{b^2 \sqrt {b c-a d} e \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {a \sqrt {c} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{b^2 \sqrt {b c-a d} e \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \] Output:

c^(1/2)*(-a*f+b*e)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*(f*x^2+e)^(1/2)*Ellipti 
cE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^ 
(1/2))/a/b/(-a*d+b*c)^(1/2)/(d*x^2+c)^(1/2)/(a*(f*x^2+e)/e/(b*x^2+a))^(1/2 
)+c^(1/2)*f*(-a*f+b*e)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*(f*x^2+e)^(1/2)*Ell 
ipticF((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c) 
/e)^(1/2))/b^2/(-a*d+b*c)^(1/2)/e/(d*x^2+c)^(1/2)/(a*(f*x^2+e)/e/(b*x^2+a) 
)^(1/2)+a*c^(1/2)*f^2*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*(f*x^2+e)^(1/2)*Elli 
pticPi((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2),b*c/(-a*d+b*c),(c*(-a*f+ 
b*e)/(-a*d+b*c)/e)^(1/2))/b^2/(-a*d+b*c)^(1/2)/e/(d*x^2+c)^(1/2)/(a*(f*x^2 
+e)/e/(b*x^2+a))^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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