\(\int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx\) [277]

Optimal result

Integrand size = 15, antiderivative size = 21 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log \left (1-(-a)^{3/2} x\right )}{\sqrt {-a}} \]

[Out]

ln(1-(-a)^(3/2)*x)/(-a)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {31} \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log \left (1-(-a)^{3/2} x\right )}{\sqrt {-a}} \]

[In]

Int[(a^(-1) + Sqrt[-a]*x)^(-1),x]

[Out]

Log[1 - (-a)^(3/2)*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 (1-(-a)^{3/2} x\right )}{\sqrt {-a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log \left (1+\sqrt {-a} a x\right )}{\sqrt {-a}} \]

[In]

Integrate[(a^(-1) + Sqrt[-a]*x)^(-1),x]

[Out]

Log[1 + Sqrt[-a]*a*x]/Sqrt[-a]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\ln \left (\frac {1}{a}+x \sqrt {-a}\right )}{\sqrt {-a}}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=-\frac {\sqrt {-a} \log \left (a^{2} x - \sqrt {-a}\right )}{a} \]

[In]

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

[Out]

-sqrt(-a)*log(a^2*x - sqrt(-a))/a

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log {\left (a x \sqrt {- a} + 1 \right )}}{\sqrt {- a}} \]

[In]

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

[Out]

log(a*x*sqrt(-a) + 1)/sqrt(-a)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log \left (\sqrt {-a} x + \frac {1}{a}\right )}{\sqrt {-a}} \]

[In]

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

[Out]

log(sqrt(-a)*x + 1/a)/sqrt(-a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\log \left ({\left | \sqrt {-a} x + \frac {1}{a} \right |}\right )}{\sqrt {-a}} \]

[In]

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

[Out]

log(abs(sqrt(-a)*x + 1/a))/sqrt(-a)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\frac {1}{a}+\sqrt {-a} x} \, dx=\frac {\ln \left (x-\frac {1}{{\left (-a\right )}^{3/2}}\right )}{\sqrt {-a}} \]

[In]

int(1/(1/a + (-a)^(1/2)*x),x)

[Out]

log(x - 1/(-a)^(3/2))/(-a)^(1/2)