3.633 \(\int \frac{1}{x^{3/2} \sqrt{2-b x}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Rubi [A]  time = 0.001366, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {37} \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} \sqrt{2-b x}} \, dx &=-\frac{\sqrt{2-b x}}{\sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0038564, size = 17, normalized size = 1. \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\sqrt{-bx+2}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00026, size = 18, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-b x + 2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Fricas [A]  time = 1.54185, size = 34, normalized size = 2. \begin{align*} -\frac{\sqrt{-b x + 2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Sympy [A]  time = 1.17606, size = 39, normalized size = 2.29 \begin{align*} \begin{cases} - \sqrt{b} \sqrt{-1 + \frac{2}{b x}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- i \sqrt{b} \sqrt{1 - \frac{2}{b x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.06825, size = 41, normalized size = 2.41 \begin{align*} -\frac{\sqrt{-b x + 2} b^{2}}{\sqrt{{\left (b x - 2\right )} b + 2 \, b}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x + 2)*b^2/(sqrt((b*x - 2)*b + 2*b)*abs(b))