3.151 \(\int \frac{\sqrt{x+x^2}}{x} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{x^2+x}+\tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

[Out]

Sqrt[x + x^2] + ArcTanh[x/Sqrt[x + x^2]]

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Rubi [A]  time = 0.0070825, antiderivative size = 22, 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 = {664, 620, 206} \[ \sqrt{x^2+x}+\tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^2]/x,x]

[Out]

Sqrt[x + x^2] + ArcTanh[x/Sqrt[x + x^2]]

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0137368, size = 31, normalized size = 1.41 \[ \sqrt{x (x+1)} \left (\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x} \sqrt{x+1}}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^2]/x,x]

[Out]

Sqrt[x*(1 + x)]*(1 + ArcSinh[Sqrt[x]]/(Sqrt[x]*Sqrt[1 + x]))

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Maple [A]  time = 0.004, size = 22, normalized size = 1. \begin{align*} \sqrt{{x}^{2}+x}+{\frac{1}{2}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2+x)^(1/2)+1/2*ln(1/2+x+(x^2+x)^(1/2))

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Maxima [A]  time = 0.94717, size = 34, normalized size = 1.55 \begin{align*} \sqrt{x^{2} + x} + \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} + x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + x) + 1/2*log(2*x + 2*sqrt(x^2 + x) + 1)

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Fricas [A]  time = 1.07853, size = 72, normalized size = 3.27 \begin{align*} \sqrt{x^{2} + x} - \frac{1}{2} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + x) - 1/2*log(-2*x + 2*sqrt(x^2 + x) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (x + 1\right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(x + 1))/x, x)

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Giac [A]  time = 1.10562, size = 35, normalized size = 1.59 \begin{align*} \sqrt{x^{2} + x} - \frac{1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + x) - 1/2*log(abs(-2*x + 2*sqrt(x^2 + x) - 1))