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

Optimal. Leaf size=16 \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

[Out]

Sqrt[x]/Sqrt[2 - b*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0013493, antiderivative size = 16, 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} \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 - b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 - b*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}{\sqrt{x} (2-b x)^{3/2}} \, dx &=\frac{\sqrt{x}}{\sqrt{2-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0037654, size = 16, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/Sqrt[2 - b*x]

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\sqrt{x}{\frac{1}{\sqrt{-bx+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01769, size = 16, normalized size = 1. \begin{align*} \frac{\sqrt{x}}{\sqrt{-b x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(-b*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.5107, size = 47, normalized size = 2.94 \begin{align*} -\frac{\sqrt{-b x + 2} \sqrt{x}}{b x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x + 2)*sqrt(x)/(b*x - 2)

________________________________________________________________________________________

Sympy [A]  time = 1.12369, size = 39, normalized size = 2.44 \begin{align*} \begin{cases} \frac{1}{\sqrt{b} \sqrt{-1 + \frac{2}{b x}}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{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/(-b*x+2)**(3/2)/x**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.08857, size = 68, normalized size = 4.25 \begin{align*} -\frac{4 \, \sqrt{-b} b}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(-b)*b/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*abs(b))