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

Optimal. Leaf size=19 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{\sqrt {2}}\right )}{b} \]

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {63, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{\sqrt {2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[b*x]*Sqrt[2 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[b*x]/Sqrt[2]])/b

Rule 63

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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.89 \begin {gather*} \frac {2 \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b} \sqrt {b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[b*x]*Sqrt[2 + b*x]),x]

[Out]

(2*Sqrt[x]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(Sqrt[b]*Sqrt[b*x])

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IntegrateAlgebraic [A]  time = 0.04, size = 25, normalized size = 1.32 \begin {gather*} -\frac {2 \log \left (\sqrt {b x+2}-\sqrt {b x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[b*x]*Sqrt[2 + b*x]),x]

[Out]

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

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fricas [A]  time = 0.98, size = 25, normalized size = 1.32 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + 2} \sqrt {b x} - 1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.07, size = 21, normalized size = 1.11 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 2} - \sqrt {b x}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(sqrt(b*x + 2) - sqrt(b*x))/b

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maple [B]  time = 0.01, size = 58, normalized size = 3.05 \begin {gather*} \frac {\sqrt {\left (b x +2\right ) b x}\, \ln \left (\frac {b^{2} x +b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 b x}\right )}{\sqrt {b x}\, \sqrt {b x +2}\, \sqrt {b^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 32, normalized size = 1.68 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 2 \, b x} b + 2 \, b\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.28, size = 37, normalized size = 1.95 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {b\,x+2}\right )}{\sqrt {b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*atan((b*(2^(1/2) - (b*x + 2)^(1/2)))/((b*x)^(1/2)*(-b^2)^(1/2))))/(-b^2)^(1/2)

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sympy [A]  time = 1.35, size = 20, normalized size = 1.05 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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