3.611 \(\int (a+b x) \log (x) \, dx\)

Optimal. Leaf size=28 \[ -a x+a x \log (x)-\frac{b x^2}{4}+\frac{1}{2} b x^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0112043, antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2313} \[ \frac{1}{2} \log (x) \left (2 a x+b x^2\right )-a x-\frac{b x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Log[x],x]

[Out]

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

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), 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*} \int (a+b x) \log (x) \, dx &=\frac{1}{2} \left (2 a x+b x^2\right ) \log (x)-\int \left (a+\frac{b x}{2}\right ) \, dx\\ &=-a x-\frac{b x^2}{4}+\frac{1}{2} \left (2 a x+b x^2\right ) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.001755, size = 28, normalized size = 1. \[ -a x+a x \log (x)-\frac{b x^2}{4}+\frac{1}{2} b x^2 \log (x) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.9 \begin{align*} -ax-{\frac{b{x}^{2}}{4}}+ax\ln \left ( x \right ) +{\frac{b{x}^{2}\ln \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989507, size = 34, normalized size = 1.21 \begin{align*} -\frac{1}{4} \, b x^{2} - a x + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.04165, size = 63, normalized size = 2.25 \begin{align*} -\frac{1}{4} \, b x^{2} - a x + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.10186, size = 22, normalized size = 0.79 \begin{align*} - a x - \frac{b x^{2}}{4} + \left (a x + \frac{b x^{2}}{2}\right ) \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09703, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{2} \, b x^{2} \log \left (x\right ) - \frac{1}{4} \, b x^{2} + a x \log \left (x\right ) - a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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