3.220 \(\int \frac{\sqrt{\frac{a}{x}}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{\sqrt{x} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{\frac{a}{x}} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

[Out]

(Sqrt[a/x]*Sqrt[x]*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]]
, 1/2])/Sqrt[1 + x^2]

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Rubi [A]  time = 0.042399, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\sqrt{x} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{\frac{a}{x}} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a/x]*Sqrt[x]*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]]
, 1/2])/Sqrt[1 + x^2]

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Rubi in Sympy [A]  time = 8.72322, size = 48, normalized size = 0.89 \[ \frac{\sqrt{x} \sqrt{\frac{a}{x}} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt{x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a/x)**(1/2)/(x**2+1)**(1/2),x)

[Out]

sqrt(x)*sqrt(a/x)*sqrt((x**2 + 1)/(x + 1)**2)*(x + 1)*elliptic_f(2*atan(sqrt(x))
, 1/2)/sqrt(x**2 + 1)

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Mathematica [C]  time = 0.02865, size = 57, normalized size = 1.06 \[ \frac{2 \sqrt [4]{-1} \sqrt{x^2+1} \sqrt{\frac{a}{x}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{1}{x^2}+1} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x]/Sqrt[1 + x^2],x]

[Out]

(2*(-1)^(1/4)*Sqrt[a/x]*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[(-1)^(1/4)/Sqrt[x]], -
1])/(Sqrt[1 + x^(-2)]*Sqrt[x])

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Maple [C]  time = 0.038, size = 62, normalized size = 1.2 \[{i\sqrt{2}\sqrt{{\frac{a}{x}}}\sqrt{-i \left ( x+i \right ) }\sqrt{-i \left ( -x+i \right ) }\sqrt{ix}{\it EllipticF} \left ( \sqrt{-i \left ( x+i \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a/x)^(1/2)/(x^2+1)^(1/2),x)

[Out]

I*(a/x)^(1/2)/(x^2+1)^(1/2)*(-I*(x+I))^(1/2)*2^(1/2)*(-I*(-x+I))^(1/2)*(I*x)^(1/
2)*EllipticF((-I*(x+I))^(1/2),1/2*2^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x}}}{\sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x)/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a/x)/sqrt(x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x)/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

integral(sqrt(a/x)/sqrt(x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a/x)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a/x)/sqrt(x**2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x)/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

integrate(sqrt(a/x)/sqrt(x^2 + 1), x)