3.359 \(\int \frac{1}{x \sqrt{-a+b x}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

(2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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Rubi [A]  time = 0.0064869, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-a + b*x]),x]

[Out]

(2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{-a+b x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{b}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.004967, size = 25, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-a + b*x]),x]

[Out]

(2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/Sqrt[a]

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Maple [A]  time = 0.005, size = 20, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{\sqrt{a}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59789, size = 140, normalized size = 5.6 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right )}{a}, \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x)/a, 2*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a)]

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Sympy [A]  time = 1.55866, size = 58, normalized size = 2.32 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22039, size = 26, normalized size = 1.04 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(b*x - a)/sqrt(a))/sqrt(a)