3.1.1 \(\int \frac {1}{\sqrt {1-a x}} \, dx\) [1]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {32} \begin {gather*} -\frac {2 \sqrt {1-a x}}{a} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1-a x}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - a*x],x]

[Out]

(-2*Sqrt[1 - a*x])/a

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Mathics [A]
time = 1.59, size = 13, normalized size = 0.87 \begin {gather*} \frac {-2 \sqrt {1-a x}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/Sqrt[1 - a*x],x]')

[Out]

-2 Sqrt[1 - a x] / a

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Maple [A]
time = 0.09, size = 14, normalized size = 0.93

method	result	size
gosper	\(-\frac {2 \sqrt {-a x +1}}{a}\)	\(14\)
derivativedivides	\(-\frac {2 \sqrt {-a x +1}}{a}\)	\(14\)
default	\(-\frac {2 \sqrt {-a x +1}}{a}\)	\(14\)
trager	\(-\frac {2 \sqrt {-a x +1}}{a}\)	\(14\)
risch	\(\frac {2 a x -2}{a \sqrt {-a x +1}}\)	\(19\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a x +1}}{\sqrt {\pi }\, a}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.24, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2 \, \sqrt {-a x + 1}}{a} \end {gather*}

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 = 0.30, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2 \, \sqrt {-a x + 1}}{a} \end {gather*}

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 = 0.03, size = 12, normalized size = 0.80 \begin {gather*} - \frac {2 \sqrt {- a x + 1}}{a} \end {gather*}

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 = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-a x+1}}{a} \end {gather*}

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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Mupad [B]
time = 0.03, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2\,\sqrt {1-a\,x}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - a*x)^(1/2),x)

[Out]

-(2*(1 - a*x)^(1/2))/a

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