3.107 \(\int \frac{1}{r \sqrt{-a^2-2 K r+2 H r^2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{x}{r \sqrt{-a^2-2 r (K-H r)}} \]

[Out]

x/(r*Sqrt[-a^2 - 2*r*(K - H*r)])

________________________________________________________________________________________

Rubi [A]  time = 0.0265996, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-2 r (K-H r)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-a^2 - 2*K*r + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 - 2*r*(K - H*r)])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{r \sqrt{-a^2-2 K r+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2-2 r (K-H r)}}\\ \end{align*}

Mathematica [A]  time = 0.0000408, size = 25, normalized size = 1.04 \[ \frac{x}{r \sqrt{-a^2+2 H r^2-2 K r}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-a^2 - 2*K*r + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 - 2*K*r + 2*H*r^2])

________________________________________________________________________________________

Maple [A]  time = 0., size = 24, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(2*H*r^2-2*K*r-a^2)^(1/2),x)

[Out]

1/r/(2*H*r^2-2*K*r-a^2)^(1/2)*x

________________________________________________________________________________________

Maxima [A]  time = 0.930996, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(2*H*r^2 - a^2 - 2*K*r)*r)

________________________________________________________________________________________

Fricas [A]  time = 1.87546, size = 80, normalized size = 3.33 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} x}{2 \, H r^{3} - a^{2} r - 2 \, K r^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2*H*r^2 - a^2 - 2*K*r)*x/(2*H*r^3 - a^2*r - 2*K*r^2)

________________________________________________________________________________________

Sympy [A]  time = 0.053192, size = 20, normalized size = 0.83 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r - a^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r**2-2*K*r-a**2)**(1/2),x)

[Out]

x/(r*sqrt(2*H*r**2 - 2*K*r - a**2))

________________________________________________________________________________________

Giac [A]  time = 1.24152, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(2*H*r^2 - a^2 - 2*K*r)*r)