3.2232 \(\int \frac{1}{(2+b \sqrt{x}) x} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]

[Out]

-Log[2 + b*Sqrt[x]] + Log[x]/2

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Rubi [A]  time = 0.0082974, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + b*Sqrt[x])*x),x]

[Out]

-Log[2 + b*Sqrt[x]] + Log[x]/2

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (2+b \sqrt{x}\right ) x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (2+b x)} \, dx,x,\sqrt{x}\right )\\ &=-\left (b \operatorname{Subst}\left (\int \frac{1}{2+b x} \, dx,x,\sqrt{x}\right )\right )+\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sqrt{x}\right )\\ &=-\log \left (2+b \sqrt{x}\right )+\frac{\log (x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0045757, size = 19, normalized size = 1. \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + b*Sqrt[x])*x),x]

[Out]

-Log[2 + b*Sqrt[x]] + Log[x]/2

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-\ln \left ( 2+b\sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(2+b*x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.974452, size = 20, normalized size = 1.05 \begin{align*} -\log \left (b \sqrt{x} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^(1/2)),x, algorithm="maxima")

[Out]

-log(b*sqrt(x) + 2) + 1/2*log(x)

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Fricas [A]  time = 1.30217, size = 49, normalized size = 2.58 \begin{align*} -\log \left (b \sqrt{x} + 2\right ) + \log \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^(1/2)),x, algorithm="fricas")

[Out]

-log(b*sqrt(x) + 2) + log(sqrt(x))

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Sympy [A]  time = 0.57503, size = 19, normalized size = 1. \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} - \log{\left (\sqrt{x} + \frac{2}{b} \right )} & \text{for}\: b \neq 0 \\\frac{\log{\left (x \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x**(1/2)),x)

[Out]

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

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Giac [A]  time = 1.11872, size = 23, normalized size = 1.21 \begin{align*} -\log \left ({\left | b \sqrt{x} + 2 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^(1/2)),x, algorithm="giac")

[Out]

-log(abs(b*sqrt(x) + 2)) + 1/2*log(abs(x))