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

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

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Rubi [A]  time = 0.01, 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*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \[ -a x+a x \log (x)-\frac {b x^2}{4}+\frac {1}{2} b x^2 \log (x) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int (a+b x) \log (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*Log[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.05, size = 25, normalized size = 0.89 \[ -\frac {1}{4} \, b x^{2} - a x + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \relax (x) \]

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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giac [A]  time = 0.58, size = 24, normalized size = 0.86 \[ \frac {1}{2} \, b x^{2} \log \relax (x) - \frac {1}{4} \, b x^{2} + a x \log \relax (x) - a x \]

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

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maple [A]  time = 0.02, size = 25, normalized size = 0.89

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.42, size = 25, normalized size = 0.89 \[ -\frac {1}{4} \, b x^{2} - a x + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \relax (x) \]

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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mupad [B]  time = 0.33, size = 21, normalized size = 0.75 \[ -\frac {x\,\left (4\,a+b\,x-4\,a\,\ln \relax (x)-2\,b\,x\,\ln \relax (x)\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)*(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.11, size = 22, normalized size = 0.79 \[ - a x - \frac {b x^{2}}{4} + \left (a x + \frac {b x^{2}}{2}\right ) \log {\relax (x )} \]

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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