\(\int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx\) [403]

Optimal result

Integrand size = 34, antiderivative size = 537 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 b \sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 b \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {c \sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {c \sqrt {e} (b d e+b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a b d \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 4.65 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (b^2 c \sqrt {b c-a d} x \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (e+f x^2\right )-b c \sqrt {b c-a d} \sqrt {e} \sqrt {b e-a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {b c e-a d e}{b c e-a c f}\right )+\sqrt {b c-a d} (2 b c-a d) \sqrt {e} \sqrt {b e-a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {b c e-a d e}{b c e-a c f}\right )-a \sqrt {c} (a d f-b (d e+c f)) \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {b c e-a c f}{b c e-a d e}\right )\right )}{2 a b^2 \sqrt {b c-a d} \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {431, 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.

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

\(\Big \downarrow \) 431

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

\(\Big \downarrow \) 427

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 428

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 429

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqrt[b* 
c - a*d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]*EllipticE[A 
rcSin[(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*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) 
+ ((b*c - a*d)*Sqrt[e]*(2*b*e - a*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^2*c*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)*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)])/(2*b^2*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 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 431

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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