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

Optimal result

Integrand size = 16, antiderivative size = 32 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=-2 \sqrt {x}+\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2436, 2332} \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}-2 \sqrt {x} \]

[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*} \text {integral}& = 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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=2 \left (-\sqrt {x}+\frac {\left (a+b \sqrt {x}\right ) \log \left (a+b \sqrt {x}\right )}{b}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 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	\(\frac {2 \left (a +b \sqrt {x}\right ) \ln \left (a +b \sqrt {x}\right )-2 b \sqrt {x}-2 a}{b}\)	\(32\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x}\right )}}{b} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (27) = 54\).

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.88 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: \left (a = 0 \vee a = - b \sqrt {x}\right ) \wedge \left (a = - b \sqrt {x} \vee b = 0\right ) \\2 \sqrt {x} \log {\left (a \right )} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 a^{2}}{a b + b^{2} \sqrt {x}} + \frac {4 a b \sqrt {x} \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} + \frac {2 b^{2} x \log {\left (a + b \sqrt {x} \right )}}{a b + b^{2} \sqrt {x}} - \frac {2 b^{2} x}{a b + b^{2} \sqrt {x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*sqrt(x), (Eq(a, 0) | Eq(a, -b*sqrt(x))) & (Eq(b, 0) | Eq(a, -b*sqrt(x)))), (2*sqrt(x)*log(a), E
q(b, 0)), (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)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left ({\left (b \sqrt {x} + a\right )} \log \left (b \sqrt {x} + a\right ) - b \sqrt {x} - a\right )}}{b} \]

[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

Mupad [B] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (a+b \sqrt {x}\right )}{\sqrt {x}} \, dx=2\,\sqrt {x}\,\ln \left (a+b\,\sqrt {x}\right )-2\,\sqrt {x}+\frac {2\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b} \]

[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