\(\int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [460]

Optimal result

Integrand size = 34, antiderivative size = 159 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{\sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{\sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 159, 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.

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

\(\Big \downarrow \) 428

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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