3.286 \(\int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

[Out]

-((Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])
/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])) + (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip
ticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])

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Rubi [A]  time = 0.120646, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {423, 427, 426, 424, 421, 419} \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - d*x^2]/Sqrt[a + b*x^2],x]

[Out]

-((Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])
/(b*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])) + (Sqrt[c]*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*Ellip
ticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])

Rule 423

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

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

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 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

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)])

Rubi steps

\begin{align*} \int \frac{\sqrt{c-d x^2}}{\sqrt{a+b x^2}} \, dx &=-\frac{d \int \frac{\sqrt{a+b x^2}}{\sqrt{c-d x^2}} \, dx}{b}+\frac{(b c+a d) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx}{b}\\ &=-\frac{\left (d \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{a+b x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{c-d x^2}}+\frac{\left ((b c+a d) \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{c-d x^2}}\\ &=-\frac{\left (d \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{1+\frac{b x^2}{a}}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{1+\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{\left ((b c+a d) \sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{a+b x^2} \sqrt{c-d x^2}}\\ &=-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{1+\frac{b x^2}{a}} \sqrt{c-d x^2}}+\frac{\sqrt{c} (b c+a d) \sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0497271, size = 89, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - d*x^2]/Sqrt[a + b*x^2],x]

[Out]

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

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Maple [A]  time = 0.011, size = 164, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac \right ) b} \left ( -ad{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) -c{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) b+ad{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \right ) \sqrt{-d{x}^{2}+c}\sqrt{b{x}^{2}+a}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \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),x)

[Out]

(-a*d*EllipticF(x*(d/c)^(1/2),(-b*c/a/d)^(1/2))-c*EllipticF(x*(d/c)^(1/2),(-b*c/a/d)^(1/2))*b+a*d*EllipticE(x*
(d/c)^(1/2),(-b*c/a/d)^(1/2)))*(-d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)*(-(d*x^2-c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)/(b*d*
x^4+a*d*x^2-b*c*x^2-a*c)/(d/c)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c}}{\sqrt{b x^{2} + a}}\,{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),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-d x^{2} + c}}{\sqrt{b x^{2} + a}}, 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),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c}}{\sqrt{b x^{2} + a}}\,{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),x, algorithm="giac")

[Out]

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