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

Optimal. Leaf size=39 \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0105244, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 63, 205} \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-a + b*x]/x,x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0204661, size = 39, normalized size = 1. \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-a + b*x]/x,x]

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -2\,\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{bx-a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52038, size = 186, normalized size = 4.77 \begin{align*} \left [\sqrt{-a} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \, \sqrt{b x - a}, -2 \, \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \, \sqrt{b x - a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.16425, size = 151, normalized size = 3.87 \begin{align*} \begin{cases} - 2 i \sqrt{a} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 i a}{\sqrt{b} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{2 i \sqrt{b} \sqrt{x}}{\sqrt{\frac{a}{b x} - 1}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\2 \sqrt{a} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{2 a}{\sqrt{b} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{2 \sqrt{b} \sqrt{x}}{\sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18125, size = 42, normalized size = 1.08 \begin{align*} -2 \, \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \, \sqrt{b x - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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