### 3.227 $$\int \log (\sqrt{a+b x}) \, dx$$

Optimal. Leaf size=25 $\frac{(a+b x) \log \left (\sqrt{a+b x}\right )}{b}-\frac{x}{2}$

[Out]

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

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Rubi [A]  time = 0.009634, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {2389, 2295} $\frac{(a+b x) \log \left (\sqrt{a+b x}\right )}{b}-\frac{x}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[Sqrt[a + b*x]],x]

[Out]

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

Rule 2389

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]

Rule 2295

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

Rubi steps

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

Mathematica [A]  time = 0.0044784, size = 23, normalized size = 0.92 $\frac{1}{2} \left (\frac{(a+b x) \log (a+b x)}{b}-x\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[Sqrt[a + b*x]],x]

[Out]

(-x + ((a + b*x)*Log[a + b*x])/b)/2

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Maple [A]  time = 0.001, size = 32, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( bx+a \right ) x}{2}}+{\frac{\ln \left ( bx+a \right ) a}{2\,b}}-{\frac{x}{2}}-{\frac{a}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*ln(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.02115, size = 31, normalized size = 1.24 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*log(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.74772, size = 53, normalized size = 2.12 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right )}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*log(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(b*x - (b*x + a)*log(b*x + a))/b

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Sympy [A]  time = 0.318819, size = 29, normalized size = 1.16 \begin{align*} - b \left (- \frac{a \log{\left (a + b x \right )}}{2 b^{2}} + \frac{x}{2 b}\right ) + \frac{x \log{\left (a + b x \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*ln(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.3112, size = 31, normalized size = 1.24 \begin{align*} -\frac{b x -{\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*log(b*x+a),x, algorithm="giac")

[Out]

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