3.509 \(\int \frac{\sqrt{2+b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.009401, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {47, 54, 215} \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + b*x]/x^(3/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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{\sqrt{2+b x}}{x^{3/2}} \, dx &=-\frac{2 \sqrt{2+b x}}{\sqrt{x}}+b \int \frac{1}{\sqrt{x} \sqrt{2+b x}} \, dx\\ &=-\frac{2 \sqrt{2+b x}}{\sqrt{x}}+(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \sqrt{2+b x}}{\sqrt{x}}+2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0138621, size = 41, normalized size = 1. \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + b*x]/x^(3/2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 59, normalized size = 1.4 \begin{align*} -2\,{\frac{\sqrt{bx+2}}{\sqrt{x}}}+{\sqrt{b}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+2)^(1/2)/x^(3/2),x)

[Out]

-2*(b*x+2)^(1/2)/x^(1/2)+b^(1/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*x)^(1/2))*(x*(b*x+2))^(1/2)/x^(1/2)/(b*x+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((b*x+2)^(1/2)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50655, size = 238, normalized size = 5.8 \begin{align*} \left [\frac{\sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) - 2 \, \sqrt{b x + 2} \sqrt{x}}{x}, -\frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) + \sqrt{b x + 2} \sqrt{x}\right )}}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(1/2)/x^(3/2),x, algorithm="fricas")

[Out]

[(sqrt(b)*x*log(b*x + sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 1) - 2*sqrt(b*x + 2)*sqrt(x))/x, -2*(sqrt(-b)*x*arctan(s
qrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))) + sqrt(b*x + 2)*sqrt(x))/x]

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Sympy [A]  time = 1.58143, size = 48, normalized size = 1.17 \begin{align*} - 2 \sqrt{b} \sqrt{1 + \frac{2}{b x}} - \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 \sqrt{b} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)**(1/2)/x**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(1/2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError