\(\int \frac {5+5 x \log (4)}{x \log (4)} \, dx\) [4290]

Optimal result

Integrand size = 15, antiderivative size = 19 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=3+5 x-\log (3)+\log (5)+\frac {5 \log (x)}{\log (4)} \]

[Out]

5*x-ln(3)+5/2*ln(x)/ln(2)+3+ln(5)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 45} \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=5 x+\frac {5 \log (x)}{\log (4)} \]

[In]

Int[(5 + 5*x*Log[4])/(x*Log[4]),x]

[Out]

5*x + (5*Log[x])/Log[4]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {5+5 x \log (4)}{x} \, dx}{\log (4)} \\ & = \frac {\int \left (\frac {5}{x}+5 \log (4)\right ) \, dx}{\log (4)} \\ & = 5 x+\frac {5 \log (x)}{\log (4)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=5 x+\frac {5 \log (x)}{\log (4)} \]

[In]

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

[Out]

5*x + (5*Log[x])/Log[4]

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68

method	result	size
norman	\(5 x +\frac {5 \ln \left (x \right )}{2 \ln \left (2\right )}\)	\(13\)
risch	\(5 x +\frac {5 \ln \left (x \right )}{2 \ln \left (2\right )}\)	\(13\)
default	\(\frac {5 x \ln \left (2\right )+\frac {5 \ln \left (x \right )}{2}}{\ln \left (2\right )}\)	\(15\)
parallelrisch	\(\frac {10 x \ln \left (2\right )+5 \ln \left (x \right )}{2 \ln \left (2\right )}\)	\(17\)

[In]

int(1/2*(10*x*ln(2)+5)/x/ln(2),x,method=_RETURNVERBOSE)

[Out]

5*x+5/2*ln(x)/ln(2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=\frac {5 \, {\left (2 \, x \log \left (2\right ) + \log \left (x\right )\right )}}{2 \, \log \left (2\right )} \]

[In]

integrate(1/2*(10*x*log(2)+5)/x/log(2),x, algorithm="fricas")

[Out]

5/2*(2*x*log(2) + log(x))/log(2)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=\frac {10 x \log {\left (2 \right )} + 5 \log {\left (x \right )}}{2 \log {\left (2 \right )}} \]

[In]

integrate(1/2*(10*x*ln(2)+5)/x/ln(2),x)

[Out]

(10*x*log(2) + 5*log(x))/(2*log(2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=\frac {5 \, {\left (2 \, x \log \left (2\right ) + \log \left (x\right )\right )}}{2 \, \log \left (2\right )} \]

[In]

integrate(1/2*(10*x*log(2)+5)/x/log(2),x, algorithm="maxima")

[Out]

5/2*(2*x*log(2) + log(x))/log(2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=\frac {5 \, {\left (2 \, x \log \left (2\right ) + \log \left ({\left | x \right |}\right )\right )}}{2 \, \log \left (2\right )} \]

[In]

integrate(1/2*(10*x*log(2)+5)/x/log(2),x, algorithm="giac")

[Out]

5/2*(2*x*log(2) + log(abs(x)))/log(2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {5+5 x \log (4)}{x \log (4)} \, dx=5\,x+\frac {5\,\ln \left (x\right )}{2\,\ln \left (2\right )} \]

[In]

int((5*x*log(2) + 5/2)/(x*log(2)),x)

[Out]

5*x + (5*log(x))/(2*log(2))