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]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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]

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 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]]

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*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ \frac {\log (x)}{a}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a} \]

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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fricas [A]  time = 1.02, size = 20, normalized size = 0.91 \[ -\frac {2 \, {\left (\log \left (b \sqrt {x} + a\right ) - \log \left (\sqrt {x}\right )\right )}}{a} \]

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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giac [A]  time = 0.17, size = 22, normalized size = 1.00 \[ -\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]

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

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maple [A]  time = 0.01, size = 21, normalized size = 0.95 \[ \frac {\ln \relax (x )}{a}-\frac {2 \ln \left (b \sqrt {x}+a \right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.89, size = 20, normalized size = 0.91 \[ -\frac {2 \, \log \left (b \sqrt {x} + a\right )}{a} + \frac {\log \relax (x)}{a} \]

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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mupad [B]  time = 0.04, size = 17, normalized size = 0.77 \[ -\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^(1/2))),x)

[Out]

-(4*atanh((2*b*x^(1/2))/a + 1))/a

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sympy [A]  time = 0.38, size = 37, normalized size = 1.68 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{a} - \frac {2 \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a} & \text {otherwise} \end {cases} \]

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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