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

Optimal. Leaf size=545 \[ \frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{e} \sqrt{a+b x^2}}\right ),\frac{e (b c-a d)}{c (b e-a f)}\right )}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{d x \sqrt{a+b x^2} \sqrt{e+f x^2}}{2 f \sqrt{c+d x^2}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.486577, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206, Rules used = {555, 554, 424, 552, 419, 553, 537} \[ \frac{d x \sqrt{a+b x^2} \sqrt{e+f x^2}}{2 f \sqrt{c+d x^2}}+\frac{b \sqrt{e} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{\sqrt{e} \sqrt{a+b x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt{e+f x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[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 555

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

Rule 554

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 424

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

Rule 552

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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), 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 553

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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)
])

Rubi steps

\begin{align*} \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{(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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 1.35082, size = 503, normalized size = 0.92 \[ \frac{\frac{\sqrt{e+f x^2} (b e-2 a f) (d e-c f) \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x \sqrt{b e-a f}}{\sqrt{e} \sqrt{a+b x^2}}\right ),\frac{b c e-a d e}{b c e-a c f}\right )}{\sqrt{e} f^2 \sqrt{b e-a f} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}+\frac{e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} (a d f+b c f-b d e) \Pi \left (\frac{a f}{a f-b e};\sin ^{-1}\left (\frac{\sqrt{a f-b e} x}{\sqrt{a} \sqrt{f x^2+e}}\right )|\frac{a d e-a c f}{b c e-a c f}\right )}{\sqrt{a} f^2 \sqrt{a f-b e} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (c+d x^2\right )}{\sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{c f-d e} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{c f-d e} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|\frac{b c e-a c f}{a d e-a c f}\right )}{f \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}}{2 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[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]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{\sqrt{e + f x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{\sqrt{f x^{2} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)