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 = 0.0042515, 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, Rules used = {31} \[ \frac{\log \left ((-a)^{5/2} x+1\right )}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]

Int[(a^(-2) + Sqrt[-a]*x)^(-1),x]

[Out]

Log[1 + (-a)^(5/2)*x]/Sqrt[-a]

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*} \int \frac{1}{\frac{1}{a^2}+\sqrt{-a} x} \, dx &=\frac{\log \left (1+(-a)^{5/2} x\right )}{\sqrt{-a}}\\ \end{align*}

Mathematica [A]  time = 0.0183021, 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 = 0.001, size = 19, normalized size = 1. \begin{align*}{\ln \left ({a}^{-2}+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

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 = 1.03575, size = 24, normalized size = 1.2 \begin{align*} \frac{\log \left (\sqrt{-a} x + \frac{1}{a^{2}}\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(-a)*x + 1/a^2)/sqrt(-a)

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Fricas [A]  time = 1.45429, size = 47, normalized size = 2.35 \begin{align*} -\frac{\sqrt{-a} \log \left (a^{3} x - \sqrt{-a}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-a)*log(a^3*x - sqrt(-a))/a

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Sympy [A]  time = 0.11012, size = 20, normalized size = 1. \begin{align*} \frac{\log{\left (a^{2} x \sqrt{- a} + 1 \right )}}{\sqrt{- a}} \end{align*}

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 [A]  time = 1.2728, size = 26, normalized size = 1.3 \begin{align*} \frac{\log \left ({\left | \sqrt{-a} x + \frac{1}{a^{2}} \right |}\right )}{\sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(sqrt(-a)*x + 1/a^2))/sqrt(-a)