### 3.231 $$\int \frac{\sqrt{x}}{x+x^2} \, dx$$

Optimal. Leaf size=8 $2 \tan ^{-1}\left (\sqrt{x}\right )$

[Out]

2*ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0049472, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {647, 63, 203} $2 \tan ^{-1}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[x]/(x + x^2),x]

[Out]

2*ArcTan[Sqrt[x]]

Rule 647

Int[((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e*x)^(m + p)*(b + c*x)
^p, x], x] /; FreeQ[{b, c, e, m}, x] && IntegerQ[p]

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0016925, size = 8, normalized size = 1. $2 \tan ^{-1}\left (\sqrt{x}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[x]/(x + x^2),x]

[Out]

2*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \begin{align*} 2\,\arctan \left ( \sqrt{x} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x^2+x),x)

[Out]

2*arctan(x^(1/2))

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Maxima [A]  time = 1.41381, size = 8, normalized size = 1. \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(x^2+x),x, algorithm="maxima")

[Out]

2*arctan(sqrt(x))

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Fricas [A]  time = 2.29497, size = 26, normalized size = 3.25 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(x))

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Sympy [A]  time = 0.29128, size = 7, normalized size = 0.88 \begin{align*} 2 \operatorname{atan}{\left (\sqrt{x} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan(sqrt(x))

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Giac [A]  time = 1.05882, size = 8, normalized size = 1. \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(x))