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

Optimal. Leaf size=42 \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

(-2*Sqrt[2 - b*x])/Sqrt[x] - 2*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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Rubi [A]  time = 0.0091793, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {47, 54, 216} \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

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]*ArcSin[(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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[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} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.014115, size = 42, normalized size = 1. \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

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

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Maple [B]  time = 0.024, size = 90, normalized size = 2.1 \begin{align*} 2\,{\frac{ \left ( bx-2 \right ) \sqrt{ \left ( -bx+2 \right ) x}}{\sqrt{-x \left ( bx-2 \right ) }\sqrt{x}\sqrt{-bx+2}}}-{\sqrt{b}\arctan \left ({\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\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)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/x^(1/2)/(-b*x+2)^(1/2)-b^(1/2)*arctan(b^(1/2)*(x-1/b)/(-b*x^2+
2*x)^(1/2))*((-b*x+2)*x)^(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 = 1.54535, size = 240, normalized size = 5.71 \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} \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*arcta
n(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - sqrt(-b*x + 2)*sqrt(x))/x]

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Sympy [C]  time = 1.74417, size = 136, normalized size = 3.24 \begin{align*} \begin{cases} - 2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}} - i \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 i \sqrt{b} \log{\left (\frac{1}{\sqrt{b} \sqrt{x}} \right )} - 2 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- 2 i \sqrt{b} \sqrt{1 - \frac{2}{b x}} - i \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 i \sqrt{b} \log{\left (\sqrt{1 - \frac{2}{b x}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(b)*sqrt(-1 + 2/(b*x)) - I*sqrt(b)*log(1/(b*x)) + 2*I*sqrt(b)*log(1/(sqrt(b)*sqrt(x))) - 2*s
qrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2), 2/Abs(b*x) > 1), (-2*I*sqrt(b)*sqrt(1 - 2/(b*x)) - I*sqrt(b)*log(1/(b*
x)) + 2*I*sqrt(b)*log(sqrt(1 - 2/(b*x)) + 1), True))

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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