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

Optimal. Leaf size=24 \[ \frac{r x}{\sqrt{-a^2-e^2+2 e r^2}} \]

[Out]

(r*x)/Sqrt[-a^2 - e^2 + 2*E*r^2]

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Rubi [A]  time = 0.0111275, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {8} \[ \frac{r x}{\sqrt{-a^2-e^2+2 e r^2}} \]

Antiderivative was successfully verified.

[In]

Int[r/Sqrt[-a^2 - e^2 + 2*E*r^2],x]

[Out]

(r*x)/Sqrt[-a^2 - e^2 + 2*E*r^2]

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 e r^2}} \, dx &=\frac{r x}{\sqrt{-a^2-e^2+2 e r^2}}\\ \end{align*}

Mathematica [A]  time = 0.0000289, size = 24, normalized size = 1. \[ \frac{r x}{\sqrt{-a^2-e^2+2 e r^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(r*x)/Sqrt[-a^2 - e^2 + 2*E*r^2]

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Maple [A]  time = 0.002, size = 23, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{2\,E{r}^{2}-{a}^{2}-{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.937238, size = 30, normalized size = 1.25 \begin{align*} \frac{r x}{\sqrt{2 \, E r^{2} - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2 - e^2)

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Fricas [A]  time = 1.81658, size = 42, normalized size = 1.75 \begin{align*} \frac{r x}{\sqrt{2 \, E r^{2} - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2 - e^2)

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Sympy [A]  time = 0.05323, size = 19, normalized size = 0.79 \begin{align*} \frac{r x}{\sqrt{- a^{2} - e^{2} + 2 e r^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(-a**2 - e**2 + 2*E*r**2)

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Giac [A]  time = 1.0504, size = 28, normalized size = 1.17 \begin{align*} \frac{r x}{\sqrt{2 \, E r^{2} - a^{2} - e^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2 - e^2)