3.48 \(\int \frac{1}{\sqrt{x} (b+a x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.013797, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(b + a*x)),x]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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

Mathematica [A]  time = 0.0074179, size = 29, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(b + a*x)),x]

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.008, size = 19, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01815, size = 163, normalized size = 5.62 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right )}{a b}, -\frac{2 \, \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.58298, size = 94, normalized size = 3.24 \begin{align*} \begin{cases} \tilde{\infty } \sqrt{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: b = 0 \\\frac{2 \sqrt{x}}{b} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} + \frac{i \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a \sqrt{b} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05216, size = 24, normalized size = 0.83 \begin{align*} \frac{2 \, \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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