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

Optimal. Leaf size=19 \[ \frac {r x}{\sqrt {2 e r^2-a^2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {8} \[ \frac {r x}{\sqrt {2 e r^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Int[r/Sqrt[-a^2 + 2*E*r^2],x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {r x}{\sqrt {2 e r^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[r/Sqrt[-a^2 + 2*E*r^2],x]

[Out]

(r*x)/Sqrt[-a^2 + 2*E*r^2]

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fricas [A]  time = 0.39, size = 17, normalized size = 0.89 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2)

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giac [A]  time = 0.98, size = 17, normalized size = 0.89 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2)

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maple [A]  time = 0.00, size = 18, normalized size = 0.95 \[ \frac {r x}{\sqrt {2 E \,r^{2}-a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 17, normalized size = 0.89 \[ \frac {r x}{\sqrt {2 \, E r^{2} - a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*E*r^2 - a^2)

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mupad [B]  time = 0.00, size = 18, normalized size = 0.95 \[ \frac {r\,x}{\sqrt {2\,r^2\,\mathrm {e}-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*r^2*exp(1) - a^2)^(1/2),x)

[Out]

(r*x)/(2*r^2*exp(1) - a^2)^(1/2)

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sympy [A]  time = 0.05, size = 17, normalized size = 0.89 \[ \frac {r x}{\sqrt {- a^{2} + 2 e r^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(-a**2 + 2*E*r**2)

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