3.360 \(\int \frac{1}{x^2 \sqrt{-a+b x}} \, dx\)

Optimal. Leaf size=44 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b x-a}}{a x} \]

[Out]

Sqrt[-a + b*x]/(a*x) + (b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.0108631, antiderivative size = 44, 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 = {51, 63, 205} \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b x-a}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[-a + b*x]),x]

[Out]

Sqrt[-a + b*x]/(a*x) + (b*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/a^(3/2)

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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{1}{x^2 \sqrt{-a+b x}} \, dx &=\frac{\sqrt{-a+b x}}{a x}+\frac{b \int \frac{1}{x \sqrt{-a+b x}} \, dx}{2 a}\\ &=\frac{\sqrt{-a+b x}}{a x}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{a}\\ &=\frac{\sqrt{-a+b x}}{a x}+\frac{b \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.137626, size = 53, normalized size = 1.2 \[ \frac{b \sqrt{b x-a} \left (\frac{a}{b x}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{b x}{a}}\right )}{\sqrt{1-\frac{b x}{a}}}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[-a + b*x]),x]

[Out]

(b*Sqrt[-a + b*x]*(a/(b*x) + ArcTanh[Sqrt[1 - (b*x)/a]]/Sqrt[1 - (b*x)/a]))/a^2

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Maple [A]  time = 0.007, size = 37, normalized size = 0.8 \begin{align*}{b\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}+{\frac{1}{ax}\sqrt{bx-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*arctan((b*x-a)^(1/2)/a^(1/2))/a^(3/2)+(b*x-a)^(1/2)/a/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(1/x^2/(b*x-a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52628, size = 230, normalized size = 5.23 \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^{2} x}, \frac{\sqrt{a} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + \sqrt{b x - a} a}{a^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x-a)^(1/2),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^2*x), (sqrt(a)*b*x*a
rctan(sqrt(b*x - a)/sqrt(a)) + sqrt(b*x - a)*a)/(a^2*x)]

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Sympy [B]  time = 3.02319, size = 124, normalized size = 2.82 \begin{align*} \begin{cases} \frac{i \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{a \sqrt{x}} + \frac{i b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{1}{\sqrt{b} x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{\sqrt{b}}{a \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} - \frac{b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19898, size = 58, normalized size = 1.32 \begin{align*} \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} + \frac{\sqrt{b x - a} b}{a x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(3/2) + sqrt(b*x - a)*b/(a*x))/b