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

Optimal. Leaf size=150 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}+\frac{4 \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

[Out]

(x*Sqrt[c + d*x^2])/(d*Sqrt[4 + x^2]) - (Sqrt[c + d*x^2]*EllipticE[ArcTan[x/2],
1 - (4*d)/c])/(d*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))]) + (4*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 = 0.187302, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}+\frac{4 \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[4 + x^2]/Sqrt[c + d*x^2],x]

[Out]

(x*Sqrt[c + d*x^2])/(d*Sqrt[4 + x^2]) - (Sqrt[c + d*x^2]*EllipticE[ArcTan[x/2],
1 - (4*d)/c])/(d*Sqrt[4 + x^2]*Sqrt[(c + d*x^2)/(c*(4 + x^2))]) + (4*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 = 27.3993, size = 129, normalized size = 0.86 \[ \frac{x \sqrt{c + d x^{2}}}{d \sqrt{x^{2} + 4}} - \frac{2 \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{x}{2} \right )}\middle | 1 - \frac{4 d}{c}\right )}{d \sqrt{\frac{4 c + 4 d x^{2}}{c \left (x^{2} + 4\right )}} \sqrt{x^{2} + 4}} + \frac{8 \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{x}{2} \right )}\middle | 1 - \frac{4 d}{c}\right )}{c \sqrt{\frac{4 c + 4 d x^{2}}{c \left (x^{2} + 4\right )}} \sqrt{x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(c + d*x**2)/(d*sqrt(x**2 + 4)) - 2*sqrt(c + d*x**2)*elliptic_e(atan(x/2),
1 - 4*d/c)/(d*sqrt((4*c + 4*d*x**2)/(c*(x**2 + 4)))*sqrt(x**2 + 4)) + 8*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 [A]  time = 0.0464657, size = 60, normalized size = 0.4 \[ \frac{2 \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{c}{4 d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}} \]

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

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Maple [A]  time = 0.035, size = 53, normalized size = 0.4 \[ 2\,{\frac{1}{\sqrt{d{x}^{2}+c}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 + 4)/sqrt(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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