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

Optimal. Leaf size=131 \[ \frac{2 \sqrt{x^2+1} \sqrt{a x}}{x+1}+\frac{\sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{a x}}{\sqrt{a}}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}}-\frac{2 \sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{a x}}{\sqrt{a}}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.181553, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 \sqrt{x^2+1} \sqrt{a x}}{x+1}+\frac{\sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{a x}}{\sqrt{a}}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}}-\frac{2 \sqrt{a} (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt{a x}}{\sqrt{a}}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x]/Sqrt[1 + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 16.5096, size = 119, normalized size = 0.91 \[ - \frac{2 \sqrt{a} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{a x}}{\sqrt{a}} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{2} + 1}} + \frac{\sqrt{a} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{a x}}{\sqrt{a}} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{2} + 1}} + \frac{2 \sqrt{a x} \sqrt{x^{2} + 1}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*sqrt((x**2 + 1)/(x + 1)**2)*(x + 1)*elliptic_e(2*atan(sqrt(a*x)/sqrt(
a)), 1/2)/sqrt(x**2 + 1) + sqrt(a)*sqrt((x**2 + 1)/(x + 1)**2)*(x + 1)*elliptic_
f(2*atan(sqrt(a*x)/sqrt(a)), 1/2)/sqrt(x**2 + 1) + 2*sqrt(a*x)*sqrt(x**2 + 1)/(x
+ 1)

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Mathematica [C]  time = 0.0521742, size = 58, normalized size = 0.44 \[ \frac{2 (-1)^{3/4} \sqrt{a x} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{x}\right )\right |-1\right )-E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{x}\right )\right |-1\right )\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1)^(3/4)*Sqrt[a*x]*(-EllipticE[I*ArcSinh[(-1)^(1/4)*Sqrt[x]], -1] + Ellipti
cF[I*ArcSinh[(-1)^(1/4)*Sqrt[x]], -1]))/Sqrt[x]

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Maple [C]  time = 0.024, size = 81, normalized size = 0.6 \[{\frac{\sqrt{2}}{x}\sqrt{ax}\sqrt{-i \left ( x+i \right ) }\sqrt{-i \left ( -x+i \right ) }\sqrt{ix} \left ( 2\,{\it EllipticE} \left ( \sqrt{-i \left ( x+i \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{-i \left ( x+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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. \[{\rm integral}\left (\frac{\sqrt{a x}}{\sqrt{x^{2} + 1}}, x\right ) \]

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 [A]  time = 3.48374, size = 36, normalized size = 0.27 \[ \frac{\sqrt{a} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{x^{2} e^{i \pi }} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**2*exp_polar(I*pi))/(2*g
amma(7/4))

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

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)