\(\int \frac {\log (x)}{\sqrt {x}} \, dx\) [223]

Optimal result

Integrand size = 8, antiderivative size = 17 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=-4 \sqrt {x}+2 \sqrt {x} \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2 \sqrt {x} \log (x)-4 \sqrt {x} \]

[In]

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

[Out]

-4*Sqrt[x] + 2*Sqrt[x]*Log[x]

Rule 2341

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*} \text {integral}& = -4 \sqrt {x}+2 \sqrt {x} \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2 \sqrt {x} (-2+\log (x)) \]

[In]

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

[Out]

2*Sqrt[x]*(-2 + Log[x])

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(-4 \sqrt {x}+2 \ln \left (x \right ) \sqrt {x}\)	\(14\)
default	\(-4 \sqrt {x}+2 \ln \left (x \right ) \sqrt {x}\)	\(14\)
risch	\(-4 \sqrt {x}+2 \ln \left (x \right ) \sqrt {x}\)	\(14\)

[In]

int(ln(x)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.53 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2 \, \sqrt {x} {\left (\log \left (x\right ) - 2\right )} \]

[In]

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

[Out]

2*sqrt(x)*(log(x) - 2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (15) = 30\).

Time = 0.81 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.53 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=\begin {cases} - 2 \sqrt {x} \log {\left (\frac {1}{x} \right )} + 2 \sqrt {x} \log {\left (x \right )} - 8 \sqrt {x} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\2 \sqrt {x} \log {\left (x \right )} - 4 \sqrt {x} & \text {for}\: \left |{x}\right | < 1 \\- 2 \sqrt {x} \log {\left (\frac {1}{x} \right )} - 4 \sqrt {x} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{3, 3}^{2, 1}\left (\begin {matrix} 1 & \frac {3}{2}, \frac {3}{2} \\\frac {1}{2}, \frac {1}{2} & 0 \end {matrix} \middle | {x} \right )} + {G_{3, 3}^{0, 3}\left (\begin {matrix} \frac {3}{2}, \frac {3}{2}, 1 & \\ & \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {x} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*sqrt(x)*log(1/x) + 2*sqrt(x)*log(x) - 8*sqrt(x), (Abs(x) < 1) & (1/Abs(x) < 1)), (2*sqrt(x)*log(
x) - 4*sqrt(x), Abs(x) < 1), (-2*sqrt(x)*log(1/x) - 4*sqrt(x), 1/Abs(x) < 1), (-meijerg(((1,), (3/2, 3/2)), ((
1/2, 1/2), (0,)), x) + meijerg(((3/2, 3/2, 1), ()), ((), (1/2, 1/2, 0)), x), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2 \, \sqrt {x} \log \left (x\right ) - 4 \, \sqrt {x} \]

[In]

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

[Out]

2*sqrt(x)*log(x) - 4*sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2 \, \sqrt {x} \log \left (x\right ) - 4 \, \sqrt {x} \]

[In]

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

[Out]

2*sqrt(x)*log(x) - 4*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.53 \[ \int \frac {\log (x)}{\sqrt {x}} \, dx=2\,\sqrt {x}\,\left (\ln \left (x\right )-2\right ) \]

[In]

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

[Out]

2*x^(1/2)*(log(x) - 2)