\(\int \frac {1}{a+\sqrt {a} x} \, dx\) [273]

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left (\sqrt {a}+x\right )}{\sqrt {a}} \]

[Out]

ln(x+a^(1/2))/a^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {31} \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left (\sqrt {a}+x\right )}{\sqrt {a}} \]

[In]

Int[(a + Sqrt[a]*x)^(-1),x]

[Out]

Log[Sqrt[a] + x]/Sqrt[a]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (\sqrt {a}+x\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left (a+\sqrt {a} x\right )}{\sqrt {a}} \]

[In]

Integrate[(a + Sqrt[a]*x)^(-1),x]

[Out]

Log[a + Sqrt[a]*x]/Sqrt[a]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\ln \left (a +x \sqrt {a}\right )}{\sqrt {a}}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left (x + \sqrt {a}\right )}{\sqrt {a}} \]

[In]

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

[Out]

log(x + sqrt(a))/sqrt(a)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log {\left (\sqrt {a} x + a \right )}}{\sqrt {a}} \]

[In]

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

[Out]

log(sqrt(a)*x + a)/sqrt(a)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left (\sqrt {a} x + a\right )}{\sqrt {a}} \]

[In]

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

[Out]

log(sqrt(a)*x + a)/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\log \left ({\left | \sqrt {a} x + a \right |}\right )}{\sqrt {a}} \]

[In]

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

[Out]

log(abs(sqrt(a)*x + a))/sqrt(a)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{a+\sqrt {a} x} \, dx=\frac {\ln \left (x+\sqrt {a}\right )}{\sqrt {a}} \]

[In]

int(1/(a + a^(1/2)*x),x)

[Out]

log(x + a^(1/2))/a^(1/2)