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

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

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Rubi [A]  time = 0.0017315, 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} \[ -\frac{2 \sqrt{a-b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[a - b*x]),x]

[Out]

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

Mathematica [A]  time = 0.0106507, size = 20, normalized size = 1. \[ -\frac{2 \sqrt{a-b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

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Maple [A]  time = 0.004, size = 17, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{-bx+a}}{a\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.996761, size = 22, normalized size = 1.1 \begin{align*} -\frac{2 \, \sqrt{-b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

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Fricas [A]  time = 2.08596, size = 42, normalized size = 2.1 \begin{align*} -\frac{2 \, \sqrt{-b x + a}}{a \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

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Sympy [A]  time = 1.23314, size = 49, normalized size = 2.45 \begin{align*} \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{a} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.08071, size = 47, normalized size = 2.35 \begin{align*} -\frac{2 \, \sqrt{-b x + a} b^{2}}{\sqrt{{\left (b x - a\right )} b + a b} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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