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

Optimal. Leaf size=35 \[ \frac{(a+b x)^2 \log (a+b x)}{2 b}-\frac{(a+b x)^2}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0146922, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2390, 2304} \[ \frac{(a+b x)^2 \log (a+b x)}{2 b}-\frac{(a+b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Log[a + b*x],x]

[Out]

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int (a+b x) \log (a+b x) \, dx &=\frac{\operatorname{Subst}(\int x \log (x) \, dx,x,a+b x)}{b}\\ &=-\frac{(a+b x)^2}{4 b}+\frac{(a+b x)^2 \log (a+b x)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0207654, size = 33, normalized size = 0.94 \[ \frac{(a+b x)^2 \log (a+b x)}{2 b}-\frac{1}{4} x (2 a+b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*Log[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 55, normalized size = 1.6 \begin{align*}{\frac{b\ln \left ( bx+a \right ){x}^{2}}{2}}+\ln \left ( bx+a \right ) xa+{\frac{\ln \left ( bx+a \right ){a}^{2}}{2\,b}}-{\frac{b{x}^{2}}{4}}-{\frac{ax}{2}}-{\frac{{a}^{2}}{4\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09095, size = 70, normalized size = 2. \begin{align*} \frac{1}{4} \, b{\left (\frac{2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac{b x^{2} + 2 \, a x}{b}\right )} + \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \log \left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.84343, size = 96, normalized size = 2.74 \begin{align*} -\frac{b^{2} x^{2} + 2 \, a b x - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.339864, size = 41, normalized size = 1.17 \begin{align*} \frac{a^{2} \log{\left (a + b x \right )}}{2 b} - \frac{a x}{2} - \frac{b x^{2}}{4} + \left (a x + \frac{b x^{2}}{2}\right ) \log{\left (a + b x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3148, size = 42, normalized size = 1.2 \begin{align*} \frac{{\left (b x + a\right )}^{2} \log \left (b x + a\right )}{2 \, b} - \frac{{\left (b x + a\right )}^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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