3.54 \(\int \frac{\log (a+b \sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 \left (a+b \sqrt{x}\right ) \log \left (a+b \sqrt{x}\right )}{b}-2 \sqrt{x} \]

[Out]

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

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Rubi [A]  time = 0.0180764, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac{2 \left (a+b \sqrt{x}\right ) \log \left (a+b \sqrt{x}\right )}{b}-2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[Log[a + b*Sqrt[x]]/Sqrt[x],x]

[Out]

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

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

Mathematica [A]  time = 0.0093172, size = 33, normalized size = 1.03 \[ 2 \left (\frac{\left (a+b \sqrt{x}\right ) \log \left (a+b \sqrt{x}\right )}{b}-\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*Sqrt[x]]/Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.06, size = 40, normalized size = 1.3 \begin{align*} 2\,\ln \left ( a+b\sqrt{x} \right ) \sqrt{x}+2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) a}{b}}-2\,\sqrt{x}-2\,{\frac{a}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00635, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )} \log \left (b \sqrt{x} + a\right ) - b \sqrt{x} - a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08938, size = 73, normalized size = 2.28 \begin{align*} \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )} \log \left (b \sqrt{x} + a\right ) - b \sqrt{x}\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.28765, size = 133, normalized size = 4.16 \begin{align*} \begin{cases} \frac{2 a^{2} \log{\left (a + b \sqrt{x} \right )}}{a b + b^{2} \sqrt{x}} + \frac{2 a^{2}}{a b + b^{2} \sqrt{x}} + \frac{4 a b \sqrt{x} \log{\left (a + b \sqrt{x} \right )}}{a b + b^{2} \sqrt{x}} + \frac{2 b^{2} x \log{\left (a + b \sqrt{x} \right )}}{a b + b^{2} \sqrt{x}} - \frac{2 b^{2} x}{a b + b^{2} \sqrt{x}} & \text{for}\: b \neq 0 \\2 \sqrt{x} \log{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a**2*log(a + b*sqrt(x))/(a*b + b**2*sqrt(x)) + 2*a**2/(a*b + b**2*sqrt(x)) + 4*a*b*sqrt(x)*log(a
+ b*sqrt(x))/(a*b + b**2*sqrt(x)) + 2*b**2*x*log(a + b*sqrt(x))/(a*b + b**2*sqrt(x)) - 2*b**2*x/(a*b + b**2*sq
rt(x)), Ne(b, 0)), (2*sqrt(x)*log(a), True))

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Giac [A]  time = 1.24872, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )} \log \left (b \sqrt{x} + a\right ) - b \sqrt{x} - a\right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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