\(\int \frac {(e+f x^2)^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [442]

Optimal result

Integrand size = 34, antiderivative size = 541 \[ \int \frac {\left (e+f x^2\right )^{3/2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b d \sqrt {e+f x^2}}-\frac {c f \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e \sqrt {-b e+a f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e (3 b d e-b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((f*x^2+e)^(3/2)/(b*x^2+a)^(1/2)/(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*c + a*d*x 
**2 + b*c*x**2 + b*d*x**4),x)*f + int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*s 
qrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*e