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

Optimal. Leaf size=15 \[ \frac{\sqrt{a+b x^2}}{b} \]

[Out]

Sqrt[a + b*x^2]/b

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Rubi [A]  time = 0.012049, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*x^2],x]

[Out]

Sqrt[a + b*x^2]/b

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Rubi in Sympy [A]  time = 2.18028, size = 10, normalized size = 0.67 \[ \frac{\sqrt{a + b x^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x**2)/b

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Mathematica [A]  time = 0.00373068, size = 15, normalized size = 1. \[ \frac{\sqrt{a+b x^2}}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b*x^2],x]

[Out]

Sqrt[a + b*x^2]/b

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Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \[{\frac{1}{b}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a)^(1/2),x)

[Out]

(b*x^2+a)^(1/2)/b

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Maxima [A]  time = 1.35436, size = 18, normalized size = 1.2 \[ \frac{\sqrt{b x^{2} + a}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)/b

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Fricas [A]  time = 0.228136, size = 18, normalized size = 1.2 \[ \frac{\sqrt{b x^{2} + a}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)/b

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Sympy [A]  time = 1.45599, size = 20, normalized size = 1.33 \[ \begin{cases} \frac{\sqrt{a + b x^{2}}}{b} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((sqrt(a + b*x**2)/b, Ne(b, 0)), (x**2/(2*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.201714, size = 18, normalized size = 1.2 \[ \frac{\sqrt{b x^{2} + a}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2 + a)/b