3.277 \(\int \frac{1}{\frac{1}{a}+\sqrt{-a} x} \, dx\)

Optimal. Leaf size=21 \[ \frac{\log \left (1-(-a)^{3/2} x\right )}{\sqrt{-a}} \]

[Out]

Log[1 - (-a)^(3/2)*x]/Sqrt[-a]

_______________________________________________________________________________________

Rubi [A]  time = 0.0125565, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\log \left (1-(-a)^{3/2} x\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

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

[Out]

Log[1 - (-a)^(3/2)*x]/Sqrt[-a]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.86478, size = 17, normalized size = 0.81 \[ \frac{\log{\left (- x \left (- a\right )^{\frac{3}{2}} + 1 \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1/a+x*(-a)**(1/2)),x)

[Out]

log(-x*(-a)**(3/2) + 1)/sqrt(-a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00974668, size = 21, normalized size = 1. \[ \frac{\log \left (\sqrt{-a} a x+1\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + Sqrt[-a]*a*x]/Sqrt[-a]

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 19, normalized size = 0.9 \[{1\ln \left ({a}^{-1}+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/a+x*(-a)^(1/2)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34056, size = 24, normalized size = 1.14 \[ \frac{\log \left (\sqrt{-a} x + \frac{1}{a}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a)*x + 1/a),x, algorithm="maxima")

[Out]

log(sqrt(-a)*x + 1/a)/sqrt(-a)

_______________________________________________________________________________________

Fricas [A]  time = 0.217417, size = 23, normalized size = 1.1 \[ \frac{\log \left (\sqrt{-a} a x + 1\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a)*x + 1/a),x, algorithm="fricas")

[Out]

log(sqrt(-a)*a*x + 1)/sqrt(-a)

_______________________________________________________________________________________

Sympy [A]  time = 0.113746, size = 19, normalized size = 0.9 \[ \frac{\log{\left (a x \sqrt{- a} + 1 \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x*sqrt(-a) + 1)/sqrt(-a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206434, size = 26, normalized size = 1.24 \[ \frac{{\rm ln}\left ({\left | \sqrt{-a} x + \frac{1}{a} \right |}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-a)*x + 1/a),x, algorithm="giac")

[Out]

ln(abs(sqrt(-a)*x + 1/a))/sqrt(-a)