3.2.4 \(\int \frac {1}{r \sqrt {-a^2-e^2+2 H r^2}} \, dx\) [104]

Optimal. Leaf size=26 \[ \frac {x}{r \sqrt {-a^2-e^2+2 H r^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {8} \begin {gather*} \frac {x}{r \sqrt {-a^2-e^2+2 H r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(r*Sqrt[-a^2 - e^2 + 2*H*r^2])

Rule 8

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {x}{r \sqrt {-a^2-e^2+2 H r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(r*Sqrt[-a^2 - e^2 + 2*H*r^2])

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Maple [A]
time = 0.02, size = 25, normalized size = 0.96

method	result	size
default	\(\frac {x}{r \sqrt {2 H \,r^{2}-a^{2}-e^{2}}}\)	\(25\)
norman	\(\frac {x}{r \sqrt {2 H \,r^{2}-a^{2}-e^{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.42, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - e^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(2*H*r^2 - a^2 - e^2)*r)

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Fricas [A]
time = 0.68, size = 40, normalized size = 1.54 \begin {gather*} \frac {\sqrt {2 \, H r^{2} - a^{2} - e^{2}} x}{2 \, H r^{3} - a^{2} r - r e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 19, normalized size = 0.73 \begin {gather*} \frac {x}{r \sqrt {2 H r^{2} - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(r*sqrt(2*H*r**2 - a**2 - e**2))

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Giac [A]
time = 1.09, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - e^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(2*H*r^2 - a^2 - e^2)*r)

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Mupad [B]
time = 0.00, size = 24, normalized size = 0.92 \begin {gather*} \frac {x}{r\,\sqrt {-a^2-e^2+2\,H\,r^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(r*(2*H*r^2 - a^2 - e^2)^(1/2))

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