3.2.12 \(\int \frac {2 x+5 \log (4)}{x^2+5 x \log (4)} \, dx\)

Optimal. Leaf size=17 \[ \log \left (5 x+\frac {x^2}{\log (4)}\right )+\log (\log (27)) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.59, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {628} \begin {gather*} \log \left (x^2+5 x \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x + 5*Log[4])/(x^2 + 5*x*Log[4]),x]

[Out]

Log[x^2 + 5*x*Log[4]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (x^2+5 x \log (4)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 10, normalized size = 0.59 \begin {gather*} \log (x)+\log (x+5 \log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 5*Log[4])/(x^2 + 5*x*Log[4]),x]

[Out]

Log[x] + Log[x + 5*Log[4]]

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 10, normalized size = 0.59 \begin {gather*} \log \left (x^{2} + 10 \, x \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*log(2)+2*x)/(10*x*log(2)+x^2),x, algorithm="fricas")

[Out]

log(x^2 + 10*x*log(2))

________________________________________________________________________________________

giac [A]  time = 0.34, size = 15, normalized size = 0.88 \begin {gather*} \log \left (2 \, {\left | \frac {1}{2} \, x^{2} + 5 \, x \log \relax (2) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*log(2)+2*x)/(10*x*log(2)+x^2),x, algorithm="giac")

[Out]

log(2*abs(1/2*x^2 + 5*x*log(2)))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 10, normalized size = 0.59

method	result	size
default	\(\ln \left (x \left (10 \ln \relax (2)+x \right )\right )\)	\(10\)
derivativedivides	\(\ln \left (10 x \ln \relax (2)+x^{2}\right )\)	\(11\)
norman	\(\ln \relax (x )+\ln \left (10 \ln \relax (2)+x \right )\)	\(11\)
risch	\(\ln \left (10 x \ln \relax (2)+x^{2}\right )\)	\(11\)
meijerg	\(\ln \left (1+\frac {x}{10 \ln \relax (2)}\right )+\ln \relax (x )-\ln \relax (2)-\ln \relax (5)-\ln \left (\ln \relax (2)\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*ln(2)+2*x)/(10*x*ln(2)+x^2),x,method=_RETURNVERBOSE)

[Out]

ln(x*(10*ln(2)+x))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 10, normalized size = 0.59 \begin {gather*} \log \left (x^{2} + 10 \, x \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*log(2)+2*x)/(10*x*log(2)+x^2),x, algorithm="maxima")

[Out]

log(x^2 + 10*x*log(2))

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 9, normalized size = 0.53 \begin {gather*} \ln \left (x\,\left (x+10\,\ln \relax (2)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 10*log(2))/(10*x*log(2) + x^2),x)

[Out]

log(x*(x + 10*log(2)))

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 10, normalized size = 0.59 \begin {gather*} \log {\left (x^{2} + 10 x \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*ln(2)+2*x)/(10*x*ln(2)+x**2),x)

[Out]

log(x**2 + 10*x*log(2))

________________________________________________________________________________________