\(\int (a+b x) \log (x) \, dx\) [611]

Optimal result

Integrand size = 8, antiderivative size = 28 \[ \int (a+b x) \log (x) \, dx=-a x-\frac {b x^2}{4}+a x \log (x)+\frac {1}{2} b x^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2350} \[ \int (a+b x) \log (x) \, dx=-a x+a x \log (x)-\frac {b x^2}{4}+\frac {1}{2} b x^2 \log (x) \]

[In]

Int[(a + b*x)*Log[x],x]

[Out]

-(a*x) - (b*x^2)/4 + a*x*Log[x] + (b*x^2*Log[x])/2

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b
, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = a x \log (x)+\frac {1}{2} b x^2 \log (x)-\int \left (a+\frac {b x}{2}\right ) \, dx \\ & = -a x-\frac {b x^2}{4}+a x \log (x)+\frac {1}{2} b x^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int (a+b x) \log (x) \, dx=-a x-\frac {b x^2}{4}+a x \log (x)+\frac {1}{2} b x^2 \log (x) \]

[In]

Integrate[(a + b*x)*Log[x],x]

[Out]

-(a*x) - (b*x^2)/4 + a*x*Log[x] + (b*x^2*Log[x])/2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
norman	\(-a x -\frac {x^{2} b}{4}+a x \ln \left (x \right )+\frac {b \,x^{2} \ln \left (x \right )}{2}\)	\(25\)
risch	\(\left (\frac {1}{2} x^{2} b +a x \right ) \ln \left (x \right )-\frac {x^{2} b}{4}-a x\)	\(25\)
parallelrisch	\(-a x -\frac {x^{2} b}{4}+a x \ln \left (x \right )+\frac {b \,x^{2} \ln \left (x \right )}{2}\)	\(25\)
parts	\(-a x -\frac {x^{2} b}{4}+a x \ln \left (x \right )+\frac {b \,x^{2} \ln \left (x \right )}{2}\)	\(25\)
default	\(b \left (-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\right )+a \left (-x +x \ln \left (x \right )\right )\)	\(27\)

[In]

int((b*x+a)*ln(x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int (a+b x) \log (x) \, dx=-\frac {1}{4} \, b x^{2} - a x + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (x\right ) \]

[In]

integrate((b*x+a)*log(x),x, algorithm="fricas")

[Out]

-1/4*b*x^2 - a*x + 1/2*(b*x^2 + 2*a*x)*log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int (a+b x) \log (x) \, dx=- a x - \frac {b x^{2}}{4} + \left (a x + \frac {b x^{2}}{2}\right ) \log {\left (x \right )} \]

[In]

integrate((b*x+a)*ln(x),x)

[Out]

-a*x - b*x**2/4 + (a*x + b*x**2/2)*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int (a+b x) \log (x) \, dx=-\frac {1}{4} \, b x^{2} - a x + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (x\right ) \]

[In]

integrate((b*x+a)*log(x),x, algorithm="maxima")

[Out]

-1/4*b*x^2 - a*x + 1/2*(b*x^2 + 2*a*x)*log(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int (a+b x) \log (x) \, dx=\frac {1}{2} \, b x^{2} \log \left (x\right ) - \frac {1}{4} \, b x^{2} + a x \log \left (x\right ) - a x \]

[In]

integrate((b*x+a)*log(x),x, algorithm="giac")

[Out]

1/2*b*x^2*log(x) - 1/4*b*x^2 + a*x*log(x) - a*x

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int (a+b x) \log (x) \, dx=-\frac {x\,\left (4\,a+b\,x-4\,a\,\ln \left (x\right )-2\,b\,x\,\ln \left (x\right )\right )}{4} \]

[In]

int(log(x)*(a + b*x),x)

[Out]

-(x*(4*a + b*x - 4*a*log(x) - 2*b*x*log(x)))/4