3.2.82 \(\int \frac {\sqrt {4-x^2}}{\sqrt {c+d x^2}} \, dx\) [182]

Optimal. Leaf size=91 \[ -\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}+\frac {(c+4 d) \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {434, 437, 435, 432, 430} \begin {gather*} \frac {(c+4 d) \sqrt {\frac {d x^2}{c}+1} F\left (\text {ArcSin}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}-\frac {\sqrt {c+d x^2} E\left (\text {ArcSin}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[4 - x^2]/Sqrt[c + d*x^2],x]

[Out]

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

Rule 434

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 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 437

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.63, size = 60, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {\frac {c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c}{4 d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 78, normalized size = 0.86

method	result	size
default	\(\frac {\left (c \EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+4 \EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right ) d -c \EllipticE \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}}{\sqrt {d \,x^{2}+c}\, d}\)	\(78\)
elliptic	\(\frac {\sqrt {-\left (d \,x^{2}+c \right ) \left (x^{2}-4\right )}\, \left (\frac {4 \sqrt {-x^{2}+4}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )}{\sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}}+\frac {c \sqrt {-x^{2}+4}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )-\EllipticE \left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )\right )}{\sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}\, d}\right )}{\sqrt {-x^{2}+4}\, \sqrt {d \,x^{2}+c}}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 2\right ) \left (x + 2\right )}}{\sqrt {c + d x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\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((4 - x^2)^(1/2)/(c + d*x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________