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

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log \left ((-a)^{5/2} x+1\right )}{\sqrt {-a}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.10 \begin {gather*} \frac {\log \left (\frac {1}{a^2}+\sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\frac {1}{a^2}+\sqrt {-a} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.01, size = 24, normalized size = 1.20 \begin {gather*} -\frac {\sqrt {-a} \log \left (a^{3} x - \sqrt {-a}\right )}{a} \end {gather*}

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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giac [A]  time = 1.08, size = 19, normalized size = 0.95 \begin {gather*} \frac {\log \left ({\left | \sqrt {-a} x + \frac {1}{a^{2}} \right |}\right )}{\sqrt {-a}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {\ln \left (\sqrt {-a}\, x +\frac {1}{a^{2}}\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 18, normalized size = 0.90 \begin {gather*} \frac {\log \left (\sqrt {-a} x + \frac {1}{a^{2}}\right )}{\sqrt {-a}} \end {gather*}

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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mupad [B]  time = 0.18, size = 14, normalized size = 0.70 \begin {gather*} \frac {\ln \left (x+\frac {1}{{\left (-a\right )}^{5/2}}\right )}{\sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/a^2 + (-a)^(1/2)*x),x)

[Out]

log(x + 1/(-a)^(5/2))/(-a)^(1/2)

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sympy [A]  time = 0.09, size = 20, normalized size = 1.00 \begin {gather*} \frac {\log {\left (a^{2} x \sqrt {- a} + 1 \right )}}{\sqrt {- a}} \end {gather*}

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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