\(\int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx\) [384]

Optimal result

Integrand size = 34, antiderivative size = 533 \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d \sqrt {a+b x^2}}-\frac {\sqrt {b c-a d} e \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 \sqrt {c} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a \sqrt {b c-a d} f \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt {c} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {a (b d e-b c f+a d f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt {c} d \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/2*b*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/(b*x^2+a)^(1/2)-1/2*(-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)*EllipticE((-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))/c^(1/ 
2)/d/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/2*a*(-a*d+b*c)^(1/2 
)*f*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-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/c^(1/2 
)/d/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/2*a*(a*d*f-b*c*f+b*d 
*e)*(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))/b/c^(1/2)/d/(-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 = 4.06 (sec) , antiderivative size = 509, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|\frac {(-b c+a d) e}{a (d e-c f)}\right )}{d \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {(b c-2 a d) \sqrt {e} (-d e+c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {b c e-a d e}{b c e-a c f}\right )}{c d^2 \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {c \sqrt {e} (b d e-b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),\frac {(-b c+a d) e}{a (d e-c f)}\right )}{a d^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{2 \sqrt {e+f x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {430, 427, 321, 428, 412, 429, 327}

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.

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

\(\Big \downarrow \) 430

\(\displaystyle \frac {a f (b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{2 b d}+\frac {(a d f-b c f+b d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{2 b d}-\frac {a (b c-a d) \int \frac {\sqrt {f x^2+e}}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{2 d}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 427

\(\displaystyle \frac {(a d f-b c f+b d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{2 b d}-\frac {a (b c-a d) \int \frac {\sqrt {f x^2+e}}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{2 d}+\frac {a f \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \int \frac {1}{\sqrt {1-\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}} \sqrt {1-\frac {(b e-a f) x^2}{e \left (b x^2+a\right )}}}d\frac {x}{\sqrt {b x^2+a}}}{2 b c d \sqrt {e+f x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 428

\(\displaystyle -\frac {a (b c-a d) \int \frac {\sqrt {f x^2+e}}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{2 d}+\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b c f+b d e) \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right ) \sqrt {1-\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}} \sqrt {1-\frac {(b e-a f) x^2}{e \left (b x^2+a\right )}}}d\frac {x}{\sqrt {b x^2+a}}}{2 b c d \sqrt {e+f x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a \sqrt {e} f \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 b c d \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 d \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(b*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*d*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqr 
t[b*c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]*Ellipti 
cE[ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/ 
((b*c - a*d)*e)])/(2*d*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2) 
)]) + (a*(b*c - a*d)*Sqrt[e]*f*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a 
+ b*x^2))]*EllipticF[ArcSin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])] 
, ((b*c - a*d)*e)/(c*(b*e - a*f))])/(2*b*c*d*Sqrt[b*e - a*f]*Sqrt[(a*(c + 
d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]) + (a*(b*d*e - b*c*f + a*d*f)*Sqr 
t[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)])/(2*b*Sqrt[c]*d*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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

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 427

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_. 
)*(x_)^2]), x_Symbol] :> Simp[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/(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]
 

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]
 

rule 429

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_. 
)*(x_)^2]), x_Symbol] :> Simp[Sqrt[c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + 
b*x^2)))]/(a*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))]))   Subs 
t[Int[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]
 

rule 430

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/Sqrt[(e_) + (f_.) 
*(x_)^2], x_Symbol] :> Simp[d*x*Sqrt[a + b*x^2]*(Sqrt[e + f*x^2]/(2*f*Sqrt[ 
c + d*x^2])), x] + (-Simp[c*((d*e - c*f)/(2*f))   Int[Sqrt[a + b*x^2]/((c + 
 d*x^2)^(3/2)*Sqrt[e + f*x^2]), x], x] - Simp[(b*d*e - b*c*f - a*d*f)/(2*d* 
f)   Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[b 
*c*((d*e - c*f)/(2*d*f))   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + 
f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[(d*e - c*f)/c]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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