3.2.10 \(\int \frac {r}{\sqrt {-a^2-e^2+2 e r^2}} \, dx\) [110]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 e r^2}} \end {gather*}

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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Mathics [A]
time = 1.81, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 E r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[r/Sqrt[2*E*r^2 - a^2 - e^2],x]')

[Out]

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

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Maple [A]
time = 0.01, size = 24, normalized size = 1.00

method	result	size
default	\(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}}}\)	\(24\)
norman	\(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(-a^2-e^2+2*exp(1)*r^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*r^2*e - a^2 - e^2)

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Fricas [A]
time = 0.41, size = 22, normalized size = 0.92 \begin {gather*} \frac {r x}{\sqrt {2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*r^2*e - a^2 - e^2)

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Sympy [A]
time = 0.03, size = 20, normalized size = 0.83 \begin {gather*} \frac {r x}{\sqrt {- a^{2} - e^{2} + 2 e r^{2}}} \end {gather*}

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 = 0.00, size = 23, normalized size = 0.96 \begin {gather*} \frac {r x}{\sqrt {-a^{2}+2 r^{2} \mathrm {e}-\mathrm {e}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(-a^2-e^2+2*exp(1)*r^2)^(1/2),x)

[Out]

r*x/sqrt(2*e*r^2 - a^2 - e^2)

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Mupad [B]
time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} \frac {r\,x}{\sqrt {-a^2-e^2+2\,\mathrm {e}\,r^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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