3.2195 \(\int \frac{1}{(a+b \sqrt{x}) x} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

[Out]

(-2*Log[a + b*Sqrt[x]])/a + Log[x]/a

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Rubi [A]  time = 0.0091657, antiderivative size = 22, 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} \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])*x),x]

[Out]

(-2*Log[a + b*Sqrt[x]])/a + Log[x]/a

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

Mathematica [A]  time = 0.0035301, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])*x),x]

[Out]

(-2*Log[a + b*Sqrt[x]])/a + Log[x]/a

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Maple [A]  time = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967363, size = 27, normalized size = 1.23 \begin{align*} -\frac{2 \, \log \left (b \sqrt{x} + a\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(b*sqrt(x) + a)/a + log(x)/a

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Fricas [A]  time = 1.30794, size = 57, normalized size = 2.59 \begin{align*} -\frac{2 \,{\left (\log \left (b \sqrt{x} + a\right ) - \log \left (\sqrt{x}\right )\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.354577, size = 37, normalized size = 1.68 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{2 \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09593, size = 30, normalized size = 1.36 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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