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

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

[Out]

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

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Rubi [A]  time = 0.0109207, 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} \[ \frac{1}{2} \log (x) \left (a x^2+2 b x\right )-\frac{a x^2}{4}-b x \]

Antiderivative was successfully verified.

[In]

Int[(b + a*x)*Log[x],x]

[Out]

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

Mathematica [A]  time = 0.001392, size = 28, normalized size = 1. \[ -\frac{a x^2}{4}+\frac{1}{2} a x^2 \log (x)-b x+b x \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(b + a*x)*Log[x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.9 \begin{align*} -bx-{\frac{a{x}^{2}}{4}}+bx\ln \left ( x \right ) +{\frac{a{x}^{2}\ln \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.950142, size = 34, normalized size = 1.21 \begin{align*} -\frac{1}{4} \, a x^{2} - b x + \frac{1}{2} \,{\left (a x^{2} + 2 \, b x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9649, size = 63, normalized size = 2.25 \begin{align*} -\frac{1}{4} \, a x^{2} - b x + \frac{1}{2} \,{\left (a x^{2} + 2 \, b x\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.103396, size = 22, normalized size = 0.79 \begin{align*} - \frac{a x^{2}}{4} - b x + \left (\frac{a x^{2}}{2} + b x\right ) \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08335, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{2} \, a x^{2} \log \left (x\right ) - \frac{1}{4} \, a x^{2} + b x \log \left (x\right ) - b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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