3.278 \(\int \frac{1}{\frac{1}{a^2}+\sqrt{-a} x} \, dx\)

Optimal. Leaf size=20 \[ \frac{\log \left ((-a)^{5/2} x+1\right )}{\sqrt{-a}} \]

[Out]

Log[1 + (-a)^(5/2)*x]/Sqrt[-a]

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Rubi [A]  time = 0.0152459, antiderivative size = 20, 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 \[ \frac{\log \left ((-a)^{5/2} x+1\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]  Int[(a^(-2) + Sqrt[-a]*x)^(-1),x]

[Out]

Log[1 + (-a)^(5/2)*x]/Sqrt[-a]

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Rubi in Sympy [A]  time = 2.12812, size = 17, normalized size = 0.85 \[ \frac{\log{\left (x \left (- a\right )^{\frac{5}{2}} + 1 \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x*(-a)**(5/2) + 1)/sqrt(-a)

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Mathematica [A]  time = 0.0132575, size = 22, normalized size = 1.1 \[ \frac{\log \left (\frac{1}{a^2}+\sqrt{-a} x\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^(-2) + Sqrt[-a]*x)^(-1),x]

[Out]

Log[a^(-2) + Sqrt[-a]*x]/Sqrt[-a]

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \[{1\ln \left ({a}^{-2}+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/a^2+x*(-a)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.50498, size = 24, normalized size = 1.2 \[ \frac{\log \left (\sqrt{-a} x + \frac{1}{a^{2}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(-a)*x + 1/a^2)/sqrt(-a)

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Fricas [A]  time = 0.218365, size = 26, normalized size = 1.3 \[ \frac{\log \left (\sqrt{-a} a^{2} x + 1\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(-a)*a^2*x + 1)/sqrt(-a)

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Sympy [A]  time = 0.124067, size = 20, normalized size = 1. \[ \frac{\log{\left (a^{2} x \sqrt{- a} + 1 \right )}}{\sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a**2*x*sqrt(-a) + 1)/sqrt(-a)

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GIAC/XCAS [A]  time = 0.205269, size = 26, normalized size = 1.3 \[ \frac{{\rm ln}\left ({\left | \sqrt{-a} x + \frac{1}{a^{2}} \right |}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(sqrt(-a)*x + 1/a^2))/sqrt(-a)