3.109 \(\int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (e+f x^2)^{3/2}} \, dx\)

Optimal. Leaf size=319 \[ \frac{c^{3/2} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right ),-\frac{e (b c-a d)}{a (d e-c f)}\right )}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) - (Sqrt[c]*Sqrt[d*e - c*f]*Sqr
t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)
))])/(e*(b*e - a*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[a + b*x^2]*Elliptic
F[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*Sqrt[d*e -
c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.46667, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {554, 422, 418, 492, 411} \[ \frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} (d e-c f)}{e \sqrt{c+d x^2} \sqrt{e+f x^2} (b e-a f)}-\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e \sqrt{c+d x^2} (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]

[Out]

((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) - (Sqrt[c]*Sqrt[d*e - c*f]*Sqr
t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)
))])/(e*(b*e - a*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*Sqrt[a + b*x^2]*Elliptic
F[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*Sqrt[d*e -
c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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 422

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

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

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

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-d e+c f) x^2}{c}}}{\sqrt{1-\frac{(-b e+a f) x^2}{a}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\left ((-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{(d e-c f) x \sqrt{a+b x^2}}{e (b e-a f) \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\left (a (-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-b e+a f) x^2}{a}}}{\left (1-\frac{(-d e+c f) x^2}{c}\right )^{3/2}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e (b e-a f) \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=\frac{(d e-c f) x \sqrt{a+b x^2}}{e (b e-a f) \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d e-c f} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e (b e-a f) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0987182, size = 148, normalized size = 0.46 \[ \frac{\sqrt{a} \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{a f-b e} x}{\sqrt{a} \sqrt{f x^2+e}}\right )|\frac{a (c f-d e)}{c (a f-b e)}\right )}{e \sqrt{a+b x^2} \sqrt{a f-b e} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}} \left ( f{x}^{2}+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x)

[Out]

int((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c}}{\sqrt{b x^{2} + a}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}{b f^{2} x^{6} +{\left (2 \, b e f + a f^{2}\right )} x^{4} + a e^{2} +{\left (b e^{2} + 2 \, a e f\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b*f^2*x^6 + (2*b*e*f + a*f^2)*x^4 + a*e^2 + (b*e^2 +
2*a*e*f)*x^2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{2}}}{\sqrt{a + b x^{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**(1/2)/(b*x**2+a)**(1/2)/(f*x**2+e)**(3/2),x)

[Out]

Integral(sqrt(c + d*x**2)/(sqrt(a + b*x**2)*(e + f*x**2)**(3/2)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c}}{\sqrt{b x^{2} + a}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^2 + c)/(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)), x)