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]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2295

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

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]

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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.44, size = 22, normalized size = 0.88 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right )}{2 \, b} \]

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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giac [A]  time = 0.16, size = 23, normalized size = 0.92 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \]

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

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maple [A]  time = 0.07, size = 32, normalized size = 1.28 \[ \frac {x \ln \left (b x +a \right )}{2}+\frac {a \ln \left (b x +a \right )}{2 b}-\frac {x}{2}-\frac {a}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 23, normalized size = 0.92 \[ -\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{2 \, b} \]

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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mupad [B]  time = 0.06, size = 25, normalized size = 1.00 \[ \frac {x\,\ln \left (a+b\,x\right )}{2}-\frac {x}{2}+\frac {a\,\ln \left (a+b\,x\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x)/2,x)

[Out]

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

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sympy [A]  time = 0.16, size = 29, normalized size = 1.16 \[ - b \left (- \frac {a \log {\left (a + b x \right )}}{2 b^{2}} + \frac {x}{2 b}\right ) + \frac {x \log {\left (a + b x \right )}}{2} \]

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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