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

Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0138221, antiderivative size = 40, 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, 208} \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0099759, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-a + b*x),x]

[Out]

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

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Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{b}}-2\,{\frac{a}{b\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36639, size = 188, normalized size = 4.7 \begin{align*} \left [\frac{\sqrt{\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{\frac{a}{b}} + a}{b x - a}\right ) + 2 \, \sqrt{x}}{b}, \frac{2 \,{\left (\sqrt{-\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{-\frac{a}{b}}}{a}\right ) + \sqrt{x}\right )}}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.16132, size = 87, normalized size = 2.17 \begin{align*} \begin{cases} \frac{\sqrt{a} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} - \frac{\sqrt{a} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{2} \sqrt{\frac{1}{b}}} + \frac{2 \sqrt{x}}{b} & \text{for}\: b \neq 0 \\- \frac{2 x^{\frac{3}{2}}}{3 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(a)*log(-sqrt(a)*sqrt(1/b) + sqrt(x))/(b**2*sqrt(1/b)) - sqrt(a)*log(sqrt(a)*sqrt(1/b) + sqrt(x
))/(b**2*sqrt(1/b)) + 2*sqrt(x)/b, Ne(b, 0)), (-2*x**(3/2)/(3*a), True))

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Giac [A]  time = 1.15392, size = 45, normalized size = 1.12 \begin{align*} \frac{2 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} + \frac{2 \, \sqrt{x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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