3.1.64 \(\int \log (a+b x) \, dx\) [64]

Optimal. Leaf size=19 \[ -x+\frac {(a+b x) \log (a+b x)}{b} \]

[Out]

-x+(b*x+a)*ln(b*x+a)/b

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2436, 2332} \begin {gather*} \frac {(a+b x) \log (a+b x)}{b}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} -x+\frac {(a+b x) \log (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

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Maple [A]
time = 0.10, size = 25, normalized size = 1.32

method	result	size
norman	\(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\)	\(24\)
risch	\(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\)	\(24\)
derivativedivides	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\)	\(25\)
default	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*((b*x+a)*ln(b*x+a)-b*x-a)

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Maxima [A]
time = 0.27, size = 23, normalized size = 1.21 \begin {gather*} -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b

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Fricas [A]
time = 0.35, size = 22, normalized size = 1.16 \begin {gather*} -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a))/b

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Sympy [A]
time = 0.05, size = 24, normalized size = 1.26 \begin {gather*} - b \left (- \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b}\right ) + x \log {\left (a + b x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.93, size = 23, normalized size = 1.21 \begin {gather*} -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b

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Mupad [B]
time = 0.07, size = 23, normalized size = 1.21 \begin {gather*} x\,\ln \left (a+b\,x\right )-x+\frac {a\,\ln \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x),x)

[Out]

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

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