\(\int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx\) [111]

Optimal result

Integrand size = 34, antiderivative size = 541 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx=\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 \sqrt {a+b x^2}}-\frac {\sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {(b c-a d) \sqrt {e} (2 b e-a f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a (a d f-b (d e+c f)) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt {c} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 541, 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 = {570, 568, 435, 567, 551, 566, 430} \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {e} \sqrt {c+d x^2} (b c-a d) (2 b e-a f) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt {c} \sqrt {e+f x^2} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {e+f x^2} \sqrt {b c-a d} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2}}{2 \sqrt {a+b x^2}} \]

[In]

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

[Out]

(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[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*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))]*S
qrt[e + f*x^2]) - (a*(a*d*f - b*(d*e + c*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])

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 570

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] + (Dist[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] - Dist[(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] + Dist[(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]

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

Mathematica [A] (verified)

Time = 3.48 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^2} \sqrt {e+f x^2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (b^2 c \sqrt {b c-a d} x \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (e+f x^2\right )-b c \sqrt {b c-a d} \sqrt {e} \sqrt {b e-a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {b c e-a d e}{b c e-a c f}\right )+\sqrt {b c-a d} (2 b c-a d) \sqrt {e} \sqrt {b e-a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {b c e-a d e}{b c e-a c f}\right )-a \sqrt {c} (a d f-b (d e+c f)) \sqrt {a+b x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b c}{b c-a d},\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {b c e-a c f}{b c e-a d e}\right )\right )}{2 a b^2 \sqrt {b c-a d} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]

[In]

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

[Out]

(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]*Sqrt[b*e - a*f]*Sqrt[a + b*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)] - 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), ArcS
in[(Sqrt[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])

Maple [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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