3.3.91 \(\int \frac {1}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx\) [291]

Optimal. Leaf size=39 \[ \frac {\sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{\sqrt {c+d x^2}} \]

[Out]

EllipticF(1/2*x,2*(-d/c)^(1/2))*(1+d*x^2/c)^(1/2)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430} \begin {gather*} \frac {\sqrt {\frac {d x^2}{c}+1} F\left (\text {ArcSin}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{\sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*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 432

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx &=\frac {\sqrt {1+\frac {d x^2}{c}} \int \frac {1}{\sqrt {4-x^2} \sqrt {1+\frac {d x^2}{c}}} \, dx}{\sqrt {c+d x^2}}\\ &=\frac {\sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{\sqrt {c+d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 40, normalized size = 1.03 \begin {gather*} \frac {\sqrt {\frac {c+d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{\sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[(c + d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/Sqrt[c + d*x^2]

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Maple [A]
time = 0.09, size = 38, normalized size = 0.97

method	result	size
default	\(\frac {\sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )}{\sqrt {d \,x^{2}+c}}\)	\(38\)
elliptic	\(\frac {\sqrt {-\left (d \,x^{2}+c \right ) \left (x^{2}-4\right )}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(d*x^2+c)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(1/2*x,2*(-d/c)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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Fricas [A]
time = 0.50, size = 14, normalized size = 0.36 \begin {gather*} \frac {{\rm ellipticF}\left (\frac {1}{2} \, x, -\frac {4 \, d}{c}\right )}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

ellipticF(1/2*x, -4*d/c)/sqrt(c)

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Sympy [A]
time = 1.16, size = 20, normalized size = 0.51 \begin {gather*} \begin {cases} \frac {F\left (\operatorname {asin}{\left (\frac {x}{2} \right )}\middle | - \frac {4 d}{c}\right )}{\sqrt {c}} & \text {for}\: x > -2 \wedge x < 2 \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Piecewise((elliptic_f(asin(x/2), -4*d/c)/sqrt(c), (x > -2) & (x < 2)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {4-x^2}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((4 - x^2)^(1/2)*(c + d*x^2)^(1/2)),x)

[Out]

int(1/((4 - x^2)^(1/2)*(c + d*x^2)^(1/2)), x)

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