\(\int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [404]

Optimal result

Integrand size = 34, antiderivative size = 160 \[ \int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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)/(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 [A] (verified)

Time = 3.10 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (\frac {a f}{-b e+a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (-d e+c f)}{c (-b e+a f)}\right )}{\sqrt {a} \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Input:

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

Output:

(e*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*EllipticPi[(a*f)/ 
(-(b*e) + a*f), ArcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], 
(a*(-(d*e) + c*f))/(c*(-(b*e) + a*f))])/(Sqrt[a]*Sqrt[-(b*e) + a*f]*Sqrt[c 
 + d*x^2]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {428, 412}

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.

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

\(\Big \downarrow \) 428

\(\displaystyle \frac {e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \int \frac {1}{\left (1-\frac {f x^2}{f x^2+e}\right ) \sqrt {\frac {(b e-a f) x^2}{a \left (f x^2+e\right )}+1} \sqrt {\frac {(d e-c f) x^2}{c \left (f x^2+e\right )}+1}}d\frac {x}{\sqrt {f x^2+e}}}{a \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {a f-b e} x}{\sqrt {a} \sqrt {f x^2+e}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} \sqrt {c+d x^2} \sqrt {a f-b e} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}\)

Input:

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

Output:

(e*Sqrt[a + b*x^2]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*EllipticPi[-((a*f 
)/(b*e - a*f)), ArcSin[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])], 
(a*(d*e - c*f))/(c*(b*e - a*f))])/(Sqrt[a]*Sqrt[-(b*e) + a*f]*Sqrt[c + d*x 
^2]*Sqrt[(e*(a + b*x^2))/(a*(e + f*x^2))])
 

Defintions of rubi rules used

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 428

Int[Sqrt[(a_) + (b_.)*(x_)^2]/(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* 
(x_)^2]), x_Symbol] :> Simp[a*Sqrt[c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + 
b*x^2)))]/(c*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))]))   Subs 
t[Int[1/((1 - b*x^2)*Sqrt[1 - (b*c - a*d)*(x^2/c)]*Sqrt[1 - (b*e - a*f)*(x^ 
2/e)]), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
 

Maple [F]

\[\int \frac {\sqrt {f \,x^{2}+e}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {e + f x^{2}}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {e+f x^2}}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {f x^{2} + e}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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