3.2.39 \(\int \frac {1}{x \sqrt {a+b x}} \, dx\) [139]

Optimal. Leaf size=42 \[ \frac {\log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{\sqrt {a}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 0.55, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {65, 214} \begin {gather*} -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + b*x]),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.55 \begin {gather*} -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a + b*x]),x]

[Out]

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

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Maple [A]
time = 0.03, size = 18, normalized size = 0.43

method	result	size
derivativedivides	\(-\frac {2 \,\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(18\)
default	\(-\frac {2 \,\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(18\)
pseudoelliptic	\(-\frac {2 \,\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.43, size = 32, normalized size = 0.76 \begin {gather*} \frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.59, size = 56, normalized size = 1.33 \begin {gather*} \left [\frac {\log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right )}{\sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right )}{a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x)/sqrt(a), 2*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a)/a]

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Sympy [A]
time = 0.57, size = 24, normalized size = 0.57 \begin {gather*} - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a)

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Giac [A]
time = 0.45, size = 21, normalized size = 0.50 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.10, size = 17, normalized size = 0.40 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^(1/2)),x)

[Out]

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

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