3.73.24 \(\int (2+4 x+2 \log (x)) \, dx\)

Optimal. Leaf size=16 \[ 3+2 x \left (-\frac {5}{7 x}+x+\log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.69, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2295} \begin {gather*} 2 x^2+2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + 4*x + 2*Log[x],x]

[Out]

2*x^2 + 2*x*Log[x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+2 x^2+2 \int \log (x) \, dx\\ &=2 x^2+2 x \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 11, normalized size = 0.69 \begin {gather*} 2 x^2+2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + 4*x + 2*Log[x],x]

[Out]

2*x^2 + 2*x*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 11, normalized size = 0.69 \begin {gather*} 2 \, x^{2} + 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(x)+4*x+2,x, algorithm="fricas")

[Out]

2*x^2 + 2*x*log(x)

________________________________________________________________________________________

giac [A]  time = 0.11, size = 11, normalized size = 0.69 \begin {gather*} 2 \, x^{2} + 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(x)+4*x+2,x, algorithm="giac")

[Out]

2*x^2 + 2*x*log(x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 12, normalized size = 0.75

method	result	size
default	\(2 x \ln \relax (x )+2 x^{2}\)	\(12\)
norman	\(2 x \ln \relax (x )+2 x^{2}\)	\(12\)
risch	\(2 x \ln \relax (x )+2 x^{2}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*ln(x)+4*x+2,x,method=_RETURNVERBOSE)

[Out]

2*x*ln(x)+2*x^2

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 11, normalized size = 0.69 \begin {gather*} 2 \, x^{2} + 2 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(x)+4*x+2,x, algorithm="maxima")

[Out]

2*x^2 + 2*x*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.29, size = 7, normalized size = 0.44 \begin {gather*} 2\,x\,\left (x+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x + 2*log(x) + 2,x)

[Out]

2*x*(x + log(x))

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 10, normalized size = 0.62 \begin {gather*} 2 x^{2} + 2 x \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*ln(x)+4*x+2,x)

[Out]

2*x**2 + 2*x*log(x)

________________________________________________________________________________________