3.576 \(\int \frac{x}{4 \sqrt{x}+x} \, dx\)

Optimal. Leaf size=19 \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

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Rubi [A]  time = 0.0143723, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 266, 43} \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(4*Sqrt[x] + x),x]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{4 \sqrt{x}+x} \, dx &=\int \frac{\sqrt{x}}{4+\sqrt{x}} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{x^2}{4+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-4+x+\frac{16}{4+x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-8 \sqrt{x}+x+32 \log \left (4+\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.008548, size = 19, normalized size = 1. \[ x-8 \sqrt{x}+32 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(4*Sqrt[x] + x),x]

[Out]

-8*Sqrt[x] + x + 32*Log[4 + Sqrt[x]]

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Maple [A]  time = 0.001, size = 16, normalized size = 0.8 \begin{align*} x+32\,\ln \left ( 4+\sqrt{x} \right ) -8\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x+4*x^(1/2)),x)

[Out]

x+32*ln(4+x^(1/2))-8*x^(1/2)

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Maxima [A]  time = 1.29466, size = 20, normalized size = 1.05 \begin{align*} x - 8 \, \sqrt{x} + 32 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+4*x^(1/2)),x, algorithm="maxima")

[Out]

x - 8*sqrt(x) + 32*log(sqrt(x) + 4)

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Fricas [A]  time = 1.28512, size = 50, normalized size = 2.63 \begin{align*} x - 8 \, \sqrt{x} + 32 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+4*x^(1/2)),x, algorithm="fricas")

[Out]

x - 8*sqrt(x) + 32*log(sqrt(x) + 4)

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Sympy [A]  time = 0.143043, size = 17, normalized size = 0.89 \begin{align*} - 8 \sqrt{x} + x + 32 \log{\left (\sqrt{x} + 4 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+4*x**(1/2)),x)

[Out]

-8*sqrt(x) + x + 32*log(sqrt(x) + 4)

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Giac [A]  time = 1.08843, size = 20, normalized size = 1.05 \begin{align*} x - 8 \, \sqrt{x} + 32 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+4*x^(1/2)),x, algorithm="giac")

[Out]

x - 8*sqrt(x) + 32*log(sqrt(x) + 4)