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

Optimal. Leaf size=27 \[ \frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}} \]

[Out]

(r*x)/Sqrt[-a^2 - e^2 - 2*r*(K - H*r)]

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Rubi [A]  time = 0.0056563, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {8} \[ \frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(r*x)/Sqrt[-a^2 - e^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{r}{\sqrt{-a^2-e^2-2 K r+2 H r^2}} \, dx &=\frac{r x}{\sqrt{-a^2-e^2-2 r (K-H r)}}\\ \end{align*}

Mathematica [A]  time = 0.0000306, size = 28, normalized size = 1.04 \[ \frac{r x}{\sqrt{-a^2-e^2+2 H r^2-2 K r}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 27, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}-{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.925849, size = 35, normalized size = 1.3 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - e^2 - 2*K*r)

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Fricas [A]  time = 1.81226, size = 53, normalized size = 1.96 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - e^2 - 2*K*r)

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Sympy [A]  time = 0.05365, size = 24, normalized size = 0.89 \begin{align*} \frac{r x}{\sqrt{2 H r^{2} - 2 K r - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05683, size = 34, normalized size = 1.26 \begin{align*} \frac{r x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - 2*K*r - e^2)