### 3.223 $$\int \frac{\log (x)}{\sqrt{x}} \, dx$$

Optimal. Leaf size=17 $2 \sqrt{x} \log (x)-4 \sqrt{x}$

[Out]

-4*Sqrt[x] + 2*Sqrt[x]*Log[x]

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Rubi [A]  time = 0.0066523, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {2304} $2 \sqrt{x} \log (x)-4 \sqrt{x}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x]/Sqrt[x],x]

[Out]

-4*Sqrt[x] + 2*Sqrt[x]*Log[x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{\log (x)}{\sqrt{x}} \, dx &=-4 \sqrt{x}+2 \sqrt{x} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0018617, size = 11, normalized size = 0.65 $2 \sqrt{x} (\log (x)-2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x]/Sqrt[x],x]

[Out]

2*Sqrt[x]*(-2 + Log[x])

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Maple [A]  time = 0.007, size = 14, normalized size = 0.8 \begin{align*} -4\,\sqrt{x}+2\,\ln \left ( x \right ) \sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02922, size = 18, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \log \left (x\right ) - 4 \, \sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*log(x) - 4*sqrt(x)

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Fricas [A]  time = 1.81478, size = 32, normalized size = 1.88 \begin{align*} 2 \, \sqrt{x}{\left (\log \left (x\right ) - 2\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*(log(x) - 2)

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Sympy [A]  time = 1.8802, size = 60, normalized size = 3.53 \begin{align*} \begin{cases} 2 \sqrt{x} \log{\left (x \right )} - 4 \sqrt{x} & \text{for}\: \left |{x}\right | < 1 \\- 2 \sqrt{x} \log{\left (\frac{1}{x} \right )} - 4 \sqrt{x} & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{3, 3}^{2, 1}\left (\begin{matrix} 1 & \frac{3}{2}, \frac{3}{2} \\\frac{1}{2}, \frac{1}{2} & 0 \end{matrix} \middle |{x} \right )} +{G_{3, 3}^{0, 3}\left (\begin{matrix} \frac{3}{2}, \frac{3}{2}, 1 & \\ & \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{x} \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*sqrt(x)*log(x) - 4*sqrt(x), Abs(x) < 1), (-2*sqrt(x)*log(1/x) - 4*sqrt(x), 1/Abs(x) < 1), (-meije
rg(((1,), (3/2, 3/2)), ((1/2, 1/2), (0,)), x) + meijerg(((3/2, 3/2, 1), ()), ((), (1/2, 1/2, 0)), x), True))

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Giac [A]  time = 1.31282, size = 18, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \log \left (x\right ) - 4 \, \sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*log(x) - 4*sqrt(x)