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

Optimal. Leaf size=20 \[ \frac{2 \sqrt{x}}{a \sqrt{a-b x}} \]

[Out]

(2*Sqrt[x])/(a*Sqrt[a - b*x])

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Rubi [A]  time = 0.0017072, antiderivative size = 20, 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} \[ \frac{2 \sqrt{x}}{a \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a - b*x)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0045051, size = 20, normalized size = 1. \[ \frac{2 \sqrt{x}}{a \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a - b*x])

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Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{x}}{a\sqrt{-bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02052, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \, \sqrt{x}}{\sqrt{-b x + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)/(sqrt(-b*x + a)*a)

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Fricas [A]  time = 1.72294, size = 55, normalized size = 2.75 \begin{align*} -\frac{2 \, \sqrt{-b x + a} \sqrt{x}}{a b x - a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.14351, size = 48, normalized size = 2.4 \begin{align*} \begin{cases} \frac{2}{a \sqrt{b} \sqrt{\frac{a}{b x} - 1}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i}{a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.07288, size = 72, normalized size = 3.6 \begin{align*} -\frac{4 \, \sqrt{-b} b}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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