3.724 \(\int \frac{9+6 \sqrt{x}+x}{4 \sqrt{x}+x} \, dx\)

Optimal. Leaf size=19 \[ x+4 \sqrt{x}+2 \log \left (\sqrt{x}+4\right ) \]

[Out]

4*Sqrt[x] + x + 2*Log[4 + Sqrt[x]]

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Rubi [A]  time = 0.0207082, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {28, 1397, 771} \[ x+4 \sqrt{x}+2 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]

Int[(9 + 6*Sqrt[x] + x)/(4*Sqrt[x] + x),x]

[Out]

4*Sqrt[x] + x + 2*Log[4 + Sqrt[x]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1397

Int[((a_.) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g =
Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + b*x^(g*n) + c*x^(2*g*n))^p, x], x, x^(1/g
)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{9+6 \sqrt{x}+x}{4 \sqrt{x}+x} \, dx &=\int \frac{\left (3+\sqrt{x}\right )^2}{4 \sqrt{x}+x} \, dx\\ &=2 \operatorname{Subst}\left (\int \frac{x (3+x)^2}{4 x+x^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (2+x+\frac{1}{4+x}\right ) \, dx,x,\sqrt{x}\right )\\ &=4 \sqrt{x}+x+2 \log \left (4+\sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0113331, size = 19, normalized size = 1. \[ x+4 \sqrt{x}+2 \log \left (\sqrt{x}+4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(9 + 6*Sqrt[x] + x)/(4*Sqrt[x] + x),x]

[Out]

4*Sqrt[x] + x + 2*Log[4 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} x+2\,\ln \left ( 4+\sqrt{x} \right ) +4\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9+x+6*x^(1/2))/(x+4*x^(1/2)),x)

[Out]

x+2*ln(4+x^(1/2))+4*x^(1/2)

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Maxima [A]  time = 1.03659, size = 20, normalized size = 1.05 \begin{align*} x + 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9+x+6*x^(1/2))/(x+4*x^(1/2)),x, algorithm="maxima")

[Out]

x + 4*sqrt(x) + 2*log(sqrt(x) + 4)

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Fricas [A]  time = 1.43486, size = 49, normalized size = 2.58 \begin{align*} x + 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9+x+6*x^(1/2))/(x+4*x^(1/2)),x, algorithm="fricas")

[Out]

x + 4*sqrt(x) + 2*log(sqrt(x) + 4)

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Sympy [A]  time = 0.14789, size = 17, normalized size = 0.89 \begin{align*} 4 \sqrt{x} + x + 2 \log{\left (\sqrt{x} + 4 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9+x+6*x**(1/2))/(x+4*x**(1/2)),x)

[Out]

4*sqrt(x) + x + 2*log(sqrt(x) + 4)

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Giac [A]  time = 1.20554, size = 20, normalized size = 1.05 \begin{align*} x + 4 \, \sqrt{x} + 2 \, \log \left (\sqrt{x} + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9+x+6*x^(1/2))/(x+4*x^(1/2)),x, algorithm="giac")

[Out]

x + 4*sqrt(x) + 2*log(sqrt(x) + 4)