3.2192 \(\int \frac{1}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]

[Out]

(2*Sqrt[x])/b - (2*a*Log[a + b*Sqrt[x]])/b^2

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Rubi [A]  time = 0.0393953, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^(-1),x]

[Out]

(2*Sqrt[x])/b - (2*a*Log[a + b*Sqrt[x]])/b^2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \log{\left (a + b \sqrt{x} \right )}}{b^{2}} + 2 \int ^{\sqrt{x}} \frac{1}{b}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*log(a + b*sqrt(x))/b**2 + 2*Integral(1/b, (x, sqrt(x)))

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Mathematica [A]  time = 0.0108455, size = 27, normalized size = 1. \[ \frac{2 \sqrt{x}}{b}-\frac{2 a \log \left (a+b \sqrt{x}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^(-1),x]

[Out]

(2*Sqrt[x])/b - (2*a*Log[a + b*Sqrt[x]])/b^2

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Maple [B]  time = 0.013, size = 57, normalized size = 2.1 \[ 2\,{\frac{\sqrt{x}}{b}}+{\frac{a}{{b}^{2}}\ln \left ( b\sqrt{x}-a \right ) }-{\frac{a}{{b}^{2}}\ln \left ( a+b\sqrt{x} \right ) }-{\frac{a\ln \left ({b}^{2}x-{a}^{2} \right ) }{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.44789, size = 36, normalized size = 1.33 \[ -\frac{2 \, a \log \left (b \sqrt{x} + a\right )}{b^{2}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*log(b*sqrt(x) + a)/b^2 + 2*(b*sqrt(x) + a)/b^2

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Fricas [A]  time = 0.234246, size = 30, normalized size = 1.11 \[ -\frac{2 \,{\left (a \log \left (b \sqrt{x} + a\right ) - b \sqrt{x}\right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a*log(b*sqrt(x) + a) - b*sqrt(x))/b^2

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Sympy [A]  time = 0.424704, size = 27, normalized size = 1. \[ \begin{cases} - \frac{2 a \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{2}} + \frac{2 \sqrt{x}}{b} & \text{for}\: b \neq 0 \\\frac{x}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**(1/2)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215655, size = 32, normalized size = 1.19 \[ -\frac{2 \, a{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{2}} + \frac{2 \, \sqrt{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*ln(abs(b*sqrt(x) + a))/b^2 + 2*sqrt(x)/b