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

Optimal. Leaf size=47 \[ \frac{\sqrt{x}}{b (a-b x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0147455, antiderivative size = 47, 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, 208} \[ \frac{\sqrt{x}}{b (a-b x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-a + b*x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0152502, size = 61, normalized size = 1.3 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x}+(b x-a) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2} (a-b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 40, normalized size = 0.9 \begin{align*} -{\frac{1}{b \left ( bx-a \right ) }\sqrt{x}}-{\frac{1}{b}{\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(x^(1/2)/(b*x-a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65146, size = 277, normalized size = 5.89 \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 b^{3} x - a^{2} b^{2}\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 b^{3} x - a^{2} b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [A]  time = 9.04806, size = 311, normalized size = 6.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b^{2} \sqrt{x}} & \text{for}\: a = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a^{2}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{- 2 a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} + 2 \sqrt{a} b^{3} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20035, size = 54, normalized size = 1.15 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} - \frac{\sqrt{x}}{{\left (b x - a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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