3.289 \(\int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx\)

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

[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))])

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Rubi [A]  time = 0.0431241, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

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))])

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Rubi in Sympy [A]  time = 8.12714, size = 53, normalized size = 0.87 \[ \frac{2 \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{x}{2} \right )}\middle | 1 - \frac{4 d}{c}\right )}{c \sqrt{\frac{4 c + 4 d x^{2}}{c \left (x^{2} + 4\right )}} \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

2*sqrt(c + d*x**2)*elliptic_f(atan(x/2), 1 - 4*d/c)/(c*sqrt((4*c + 4*d*x**2)/(c*
(x**2 + 4)))*sqrt(x**2 + 4))

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Mathematica [C]  time = 0.0497062, size = 47, normalized size = 0.77 \[ -\frac{i \sqrt{\frac{c+d x^2}{c}} F\left (i \sinh ^{-1}\left (\frac{x}{2}\right )|\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]

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 = 0.028, size = 53, normalized size = 0.9 \[{\frac{1}{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{1}{2}\sqrt{{\frac{c}{d}}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

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)

[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., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(x^2 + 4)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(x^2 + 4)),x, algorithm="fricas")

[Out]

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

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x^2 + c)*sqrt(x^2 + 4)),x, algorithm="giac")

[Out]

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