\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx\) [104]

Optimal result

Integrand size = 34, antiderivative size = 545 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}-\frac {\sqrt {e} \sqrt {d e-c f} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b \sqrt {e} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {c \sqrt {e} (b d e-b c f-a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {569, 568, 435, 567, 551, 566, 430} \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\frac {b \sqrt {e} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {e} \sqrt {a+b x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt {e+f x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt {e+f x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}} \]

[In]

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

[Out]

(d*x*Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(2*f*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)))])/(2*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2]) + (b*Sqrt[e]*(d*e - c*f)*Sq
rt[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*d*f*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e +
f*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)))]
)/(2*a*d*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 566

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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)))])),
Subst[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] /; Fr
eeQ[{a, b, c, d, e, f}, x]

Rule 567

Int[Sqrt[(a_) + (b_.)*(x_)^2]/(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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)))])),
Subst[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 568

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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)))])),
Subst[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 569

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

Rubi steps \begin{align*} \text {integral}& = \frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}-\frac {(c (d e-c f)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx}{2 f}+\frac {(b c (d e-c f)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{2 d f}-\frac {(b d e-b c f-a d f) \int \frac {\sqrt {c+d x^2}}{\sqrt {a+b x^2} \sqrt {e+f x^2}} \, dx}{2 d f} \\ & = \frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}+\frac {\left (b (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 )}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {(b c-a d) x^2}{c}} \sqrt {1-\frac {(b e-a f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 d f \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\left ((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 )}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {(-b c+a d) x^2}{a}}}{\sqrt {1-\frac {(d e-c f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (c (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 )}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-d x^2\right ) \sqrt {1-\frac {(-b c+a d) x^2}{a}} \sqrt {1-\frac {(d e-c f) x^2}{e}}} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a d f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \\ & = \frac {d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \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 (\sin ^{-1}\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 f \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b \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 )}} F\left (\sin ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b 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 )}} \Pi \left (\frac {d e}{d e-c f};\sin ^{-1}\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 f \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.28 (sec) , antiderivative size = 503, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{\sqrt {e+f x^2}} \, dx=\frac {\frac {x \sqrt {a+b x^2} \left (c+d x^2\right )}{\sqrt {e+f x^2}}-\frac {\sqrt {c} \sqrt {-d e+c f} \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-d e+c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|\frac {b c e-a c f}{a d e-a c f}\right )}{f \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b e-2 a f) (d e-c f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {b c e-a d e}{b c e-a c f}\right )}{\sqrt {e} f^2 \sqrt {b e-a f} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {e (-b d e+b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (\frac {a f}{-b e+a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a d e-a c f}{b c e-a c f}\right )}{\sqrt {a} f^2 \sqrt {-b e+a f} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}}{2 \sqrt {c+d x^2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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