3.326 \(\int \frac{\sqrt{-a+b x}}{x^2} \, dx\)

Optimal. Leaf size=42 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0103159, antiderivative size = 42, 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 = {47, 63, 205} \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-a + b*x]/x^2,x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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^2} \, dx &=-\frac{\sqrt{-a+b x}}{x}+\frac{1}{2} b \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=-\frac{\sqrt{-a+b x}}{x}+\operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )\\ &=-\frac{\sqrt{-a+b x}}{x}+\frac{b \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0477617, size = 52, normalized size = 1.24 \[ \frac{-b x \sqrt{1-\frac{b x}{a}} \tanh ^{-1}\left (\sqrt{1-\frac{b x}{a}}\right )+a-b x}{x \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-a + b*x]/x^2,x]

[Out]

(a - b*x - b*x*Sqrt[1 - (b*x)/a]*ArcTanh[Sqrt[1 - (b*x)/a]])/(x*Sqrt[-a + b*x])

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Maple [A]  time = 0.008, size = 35, normalized size = 0.8 \begin{align*}{b\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{x}\sqrt{bx-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53313, size = 224, normalized size = 5.33 \begin{align*} \left [-\frac{\sqrt{-a} b x \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \, \sqrt{b x - a} a}{2 \, a x}, \frac{\sqrt{a} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) - \sqrt{b x - a} a}{a x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(-a)*b*x*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*sqrt(b*x - a)*a)/(a*x), (sqrt(a)*b*x*arc
tan(sqrt(b*x - a)/sqrt(a)) - sqrt(b*x - a)*a)/(a*x)]

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Sympy [A]  time = 2.73324, size = 124, normalized size = 2.95 \begin{align*} \begin{cases} - \frac{i a}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b \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{\sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \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((b*x-a)**(1/2)/x**2,x)

[Out]

Piecewise((-I*a/(sqrt(b)*x**(3/2)*sqrt(a/(b*x) - 1)) + I*sqrt(b)/(sqrt(x)*sqrt(a/(b*x) - 1)) + I*b*acosh(sqrt(
a)/(sqrt(b)*sqrt(x)))/sqrt(a), Abs(a)/(Abs(b)*Abs(x)) > 1), (-sqrt(b)*sqrt(-a/(b*x) + 1)/sqrt(x) - b*asin(sqrt
(a)/(sqrt(b)*sqrt(x)))/sqrt(a), True))

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Giac [A]  time = 1.1966, size = 55, normalized size = 1.31 \begin{align*} \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{\sqrt{b x - a} b}{x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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