3.1 \(\int \frac{1}{\sqrt{1-a x}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{2 \sqrt{1-a x}}{a} \]

[Out]

(-2*Sqrt[1 - a*x])/a

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Rubi [A]  time = 0.0014936, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {32} \[ -\frac{2 \sqrt{1-a x}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - a*x],x]

[Out]

(-2*Sqrt[1 - a*x])/a

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-a x}} \, dx &=-\frac{2 \sqrt{1-a x}}{a}\\ \end{align*}

Mathematica [A]  time = 0.004261, size = 15, normalized size = 1. \[ -\frac{2 \sqrt{1-a x}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - a*x],x]

[Out]

(-2*Sqrt[1 - a*x])/a

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{-ax+1}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.924617, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, \sqrt{-a x + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a*x + 1)/a

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Fricas [A]  time = 2.0903, size = 28, normalized size = 1.87 \begin{align*} -\frac{2 \, \sqrt{-a x + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a*x + 1)/a

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Sympy [A]  time = 0.055326, size = 12, normalized size = 0.8 \begin{align*} - \frac{2 \sqrt{- a x + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a*x + 1)/a

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Giac [A]  time = 1.07191, size = 18, normalized size = 1.2 \begin{align*} -\frac{2 \, \sqrt{-a x + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-a*x + 1)/a