3.3.92 \(\int \frac {1}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx\) [292]

Optimal. Leaf size=61 \[ \frac {\sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{c \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {429} \begin {gather*} \frac {\sqrt {c+d x^2} F\left (\text {ArcTan}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{c \sqrt {x^2+4} \sqrt {\frac {c+d x^2}{c \left (x^2+4\right )}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 429

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.48, size = 47, normalized size = 0.77 \begin {gather*} -\frac {i \sqrt {\frac {c+d x^2}{c}} F\left (i \sinh ^{-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]

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

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Maple [A]
time = 0.08, size = 53, normalized size = 0.87

method	result	size
default	\(\frac {\sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {\frac {c}{d}}}{2}\right )}{2 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}}\)	\(53\)
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (x^{2}+4\right )}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {-4+\frac {c +4 d}{d}}}{2}\right )}{2 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}\, \sqrt {d \,x^{4}+c \,x^{2}+4 d \,x^{2}+4 c}}\)	\(95\)

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]

1/2/(d*x^2+c)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-d/c)^(1/2),1/2*(c/d)^(1/2))/(-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 [C] Result contains complex when optimal does not.
time = 0.37, size = 15, normalized size = 0.25 \begin {gather*} -\frac {i \, {\rm ellipticF}\left (\frac {1}{2} i \, 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]

-I*ellipticF(1/2*I*x, 4*d/c)/sqrt(c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c + d x^{2}} \sqrt {x^{2} + 4}}\, dx \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]

Integral(1/(sqrt(c + d*x**2)*sqrt(x**2 + 4)), x)

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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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2+4}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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