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

Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a-b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0142779, antiderivative size = 46, 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, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a-b x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(-a + b*x)^2),x]

[Out]

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

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 208

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} (-a+b x)^2} \, dx &=\frac{\sqrt{x}}{a (a-b x)}-\frac{\int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{2 a}\\ &=\frac{\sqrt{x}}{a (a-b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{\sqrt{x}}{a (a-b x)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0209973, size = 46, normalized size = 1. \[ \frac{\sqrt{x}}{a^2-a b x}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(-a + b*x)^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{a \left ( bx-a \right ) }\sqrt{x}}+{\frac{1}{a}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.71234, size = 277, normalized size = 6.02 \begin{align*} \left [-\frac{2 \, a b \sqrt{x} - \sqrt{a b}{\left (b x - a\right )} \log \left (\frac{b x + a + 2 \, \sqrt{a b} \sqrt{x}}{b x - a}\right )}{2 \,{\left (a^{2} b^{2} x - a^{3} b\right )}}, -\frac{a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )} \arctan \left (\frac{\sqrt{-a b}}{b \sqrt{x}}\right )}{a^{2} b^{2} x - a^{3} b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*a*b*sqrt(x) - sqrt(a*b)*(b*x - a)*log((b*x + a + 2*sqrt(a*b)*sqrt(x))/(b*x - a)))/(a^2*b^2*x - a^3*b)
, -(a*b*sqrt(x) + sqrt(-a*b)*(b*x - a)*arctan(sqrt(-a*b)/(b*sqrt(x))))/(a^2*b^2*x - a^3*b)]

________________________________________________________________________________________

Sympy [A]  time = 17.7792, size = 303, normalized size = 6.59 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{2} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b**2*x**(3/2)), Eq(a, 0)), (2*sqrt(x)/a**2, Eq(b, 0)), (
-2*sqrt(a)*b*sqrt(x)*sqrt(1/b)/(-2*a**(5/2)*b*sqrt(1/b) + 2*a**(3/2)*b**2*x*sqrt(1/b)) + a*log(-sqrt(a)*sqrt(1
/b) + sqrt(x))/(-2*a**(5/2)*b*sqrt(1/b) + 2*a**(3/2)*b**2*x*sqrt(1/b)) - a*log(sqrt(a)*sqrt(1/b) + sqrt(x))/(-
2*a**(5/2)*b*sqrt(1/b) + 2*a**(3/2)*b**2*x*sqrt(1/b)) - b*x*log(-sqrt(a)*sqrt(1/b) + sqrt(x))/(-2*a**(5/2)*b*s
qrt(1/b) + 2*a**(3/2)*b**2*x*sqrt(1/b)) + b*x*log(sqrt(a)*sqrt(1/b) + sqrt(x))/(-2*a**(5/2)*b*sqrt(1/b) + 2*a*
*(3/2)*b**2*x*sqrt(1/b)), True))

________________________________________________________________________________________

Giac [A]  time = 1.17384, size = 55, normalized size = 1.2 \begin{align*} -\frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} - \frac{\sqrt{x}}{{\left (b x - a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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