3.1.54 \(\int \frac {\log (a+b \sqrt {x})}{\sqrt {x}} \, dx\) [54]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {2504, 2436, 2332} \begin {gather*} \frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-2 \sqrt {x} \end {gather*}

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 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]

Rule 2504

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])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.03 \begin {gather*} 2 \left (-\sqrt {x}+\frac {\left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}\right ) \end {gather*}

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 = 0.28, size = 32, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {2 \left (a +b \sqrt {x}\right ) \ln \left (a +b \sqrt {x}\right )-2 b \sqrt {x}-2 a}{b}\)	\(32\)
default	\(\frac {2 \left (a +b \sqrt {x}\right ) \ln \left (a +b \sqrt {x}\right )-2 b \sqrt {x}-2 a}{b}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 31, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \end {gather*}

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 = 0.38, size = 28, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x}\right )}}{b} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (27) = 54\).
time = 0.24, size = 133, normalized size = 4.16 \begin {gather*} \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 {gather*}

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 = 2.46, size = 31, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \end {gather*}

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

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Mupad [B]
time = 0.28, size = 33, normalized size = 1.03 \begin {gather*} 2\,\sqrt {x}\,\ln \left (a+b\,\sqrt {x}\right )-2\,\sqrt {x}+\frac {2\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x^(1/2))/x^(1/2),x)

[Out]

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

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