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]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ x-8 \sqrt {x}+32 \log \left (\sqrt {x}+4\right ) \]

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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fricas [A]  time = 0.43, size = 15, normalized size = 0.79 \[ x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \]

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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giac [A]  time = 0.39, size = 15, normalized size = 0.79 \[ x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \]

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)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ x +32 \ln \left (\sqrt {x}+4\right )-8 \sqrt {x} \]

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 = 0.87, size = 15, normalized size = 0.79 \[ x - 8 \, \sqrt {x} + 32 \, \log \left (\sqrt {x} + 4\right ) \]

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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mupad [B]  time = 0.04, size = 15, normalized size = 0.79 \[ x+32\,\ln \left (\sqrt {x}+4\right )-8\,\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x + 4*x^(1/2)),x)

[Out]

x + 32*log(x^(1/2) + 4) - 8*x^(1/2)

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sympy [A]  time = 0.17, size = 17, normalized size = 0.89 \[ - 8 \sqrt {x} + x + 32 \log {\left (\sqrt {x} + 4 \right )} \]

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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