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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = 0.00, 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])

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 9, normalized size = 0.53 \[ 2 \, \sqrt {x} {\left (\log \relax (x) - 2\right )} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 13, normalized size = 0.76 \[ 2 \, \sqrt {x} \log \relax (x) - 4 \, \sqrt {x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 14, normalized size = 0.82 \[ 2 \sqrt {x}\, \ln \relax (x )-4 \sqrt {x} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 13, normalized size = 0.76 \[ 2 \, \sqrt {x} \log \relax (x) - 4 \, \sqrt {x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 9, normalized size = 0.53 \[ 2\,\sqrt {x}\,\left (\ln \relax (x)-2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^(1/2)*(log(x) - 2)

________________________________________________________________________________________

sympy [A]  time = 1.74, size = 60, normalized size = 3.53 \[ \begin {cases} 2 \sqrt {x} \log {\relax (x )} - 4 \sqrt {x} & \text {for}\: \left |{x}\right | < 1 \\- 2 \sqrt {x} \log {\left (\frac {1}{x} \right )} - 4 \sqrt {x} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {3}{2}, \frac {3}{2} \\\frac {1}{2}, \frac {1}{2} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {3}{2}, \frac {3}{2}, 1 & \\ & \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________