3.2.8 \(\int \frac {1}{r \sqrt {-a^2-e^2-2 K r+2 H r^2}} \, dx\) [108]

Optimal. Leaf size=29 \[ \frac {x}{r \sqrt {-a^2-e^2-2 r (K-H r)}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {8} \begin {gather*} \frac {x}{r \sqrt {-a^2-e^2-2 r (K-H r)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(r*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 {1}{r \sqrt {-a^2-e^2-2 K r+2 H r^2}} \, dx &=\frac {x}{r \sqrt {-a^2-e^2-2 r (K-H r)}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.03 \begin {gather*} \frac {x}{r \sqrt {-a^2-e^2-2 K r+2 H r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 29, normalized size = 1.00

method	result	size
default	\(\frac {x}{r \sqrt {2 H \,r^{2}-2 K r -a^{2}-e^{2}}}\)	\(29\)
norman	\(\frac {x}{r \sqrt {2 H \,r^{2}-2 K r -a^{2}-e^{2}}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.30, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.86, size = 50, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} x}{2 \, H r^{3} - a^{2} r - 2 \, K r^{2} - r e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 24, normalized size = 0.83 \begin {gather*} \frac {x}{r \sqrt {2 H r^{2} - 2 K r - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.83, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.00, size = 28, normalized size = 0.97 \begin {gather*} \frac {x}{r\,\sqrt {-a^2-e^2+2\,H\,r^2-2\,K\,r}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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