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]

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

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 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]))

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]

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ x+4 \sqrt {x}+2 \log \left (\sqrt {x}+4\right ) \]

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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fricas [A]  time = 0.45, size = 15, normalized size = 0.79 \[ x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]

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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giac [A]  time = 0.39, size = 15, normalized size = 0.79 \[ x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]

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)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ x +2 \ln \left (\sqrt {x}+4\right )+4 \sqrt {x} \]

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

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maxima [A]  time = 0.88, size = 15, normalized size = 0.79 \[ x + 4 \, \sqrt {x} + 2 \, \log \left (\sqrt {x} + 4\right ) \]

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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mupad [B]  time = 3.04, size = 15, normalized size = 0.79 \[ x+2\,\ln \left (\sqrt {x}+4\right )+4\,\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*log(x^(1/2) + 4) + 4*x^(1/2)

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sympy [A]  time = 0.17, size = 17, normalized size = 0.89 \[ 4 \sqrt {x} + x + 2 \log {\left (\sqrt {x} + 4 \right )} \]

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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