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

Optimal. Leaf size=19 \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]]

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Rubi [A]  time = 0.10589, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x + Sqrt[a^2 + x^2]]/Sqrt[a^2 + x^2],x]

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]]

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Rubi in Sympy [A]  time = 3.64378, size = 15, normalized size = 0.79 \[ 2 \sqrt{x + \sqrt{a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + sqrt(a**2 + x**2))

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Mathematica [A]  time = 0.0264594, size = 19, normalized size = 1. \[ 2 \sqrt{\sqrt{a^2+x^2}+x} \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[x + Sqrt[a^2 + x^2]]

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{1\sqrt{x+\sqrt{{a}^{2}+{x}^{2}}}{\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((x+(a^2+x^2)^(1/2))^(1/2)/(a^2+x^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + sqrt(a^2 + x^2))/sqrt(a^2 + x^2), x)

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Fricas [A]  time = 0.218391, size = 20, normalized size = 1.05 \[ 2 \, \sqrt{x + \sqrt{a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + sqrt(a^2 + x^2))

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Sympy [A]  time = 0.282316, size = 15, normalized size = 0.79 \[ 2 \sqrt{x + \sqrt{a^{2} + x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + sqrt(a**2 + x**2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + sqrt(a^2 + x^2))/sqrt(a^2 + x^2), x)