3.1532 \(\int \frac{1}{\sqrt{b x} \sqrt{2+b x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.004641, antiderivative size = 19, normalized size of antiderivative = 1., 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} \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{\sqrt{2}}\right )}{b} \]

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*}

Mathematica [A]  time = 0.0078384, size = 36, normalized size = 1.89 \[ \frac{2 \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{b x}} \]

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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Maple [B]  time = 0.005, size = 58, normalized size = 3.1 \begin{align*}{\sqrt{bx \left ( bx+2 \right ) }\ln \left ({({b}^{2}x+b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,bx} \right ){\frac{1}{\sqrt{bx}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*(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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.96446, size = 59, normalized size = 3.11 \begin{align*} -\frac{\log \left (-b x + \sqrt{b x + 2} \sqrt{b x} - 1\right )}{b} \end{align*}

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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Sympy [A]  time = 1.4215, size = 20, normalized size = 1.05 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} \end{align*}

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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Giac [A]  time = 1.13851, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b x + 2} + \sqrt{b x} \right |}\right )}{b} \end{align*}

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(abs(-sqrt(b*x + 2) + sqrt(b*x)))/b