3.2.5 \(\int \frac {1}{r \sqrt {-a^2+2 H r^2-2 K r^4}} \, dx\) [105]

Optimal. Leaf size=27 \[ \frac {x}{r \sqrt {-a^2+2 H r^2-2 K r^4}} \]

[Out]

x/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {8} \begin {gather*} \frac {x}{r \sqrt {-a^2+2 H r^2-2 K r^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4])

Rule 8

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

Rubi steps

\begin {align*} \int \frac {1}{r \sqrt {-a^2+2 H r^2-2 K r^4}} \, dx &=\frac {x}{r \sqrt {-a^2+2 H r^2-2 K r^4}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x}{r \sqrt {-a^2+2 H r^2-2 K r^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2 - 2*K*r^4])

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Maple [A]
time = 0.02, size = 26, normalized size = 0.96

method	result	size
default	\(\frac {x}{r \sqrt {-2 K \,r^{4}+2 H \,r^{2}-a^{2}}}\)	\(26\)
norman	\(\frac {x}{r \sqrt {-2 K \,r^{4}+2 H \,r^{2}-a^{2}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

x/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2)

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Maxima [A]
time = 1.93, size = 25, normalized size = 0.93 \begin {gather*} \frac {x}{\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="maxima")

[Out]

x/(sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*r)

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Fricas [A]
time = 0.54, size = 43, normalized size = 1.59 \begin {gather*} -\frac {\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} x}{2 \, K r^{5} - 2 \, H r^{3} + a^{2} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*x/(2*K*r^5 - 2*H*r^3 + a^2*r)

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Sympy [A]
time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} \frac {x}{r \sqrt {2 H r^{2} - 2 K r^{4} - a^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r**4+2*H*r**2-a**2)**(1/2),x)

[Out]

x/(r*sqrt(2*H*r**2 - 2*K*r**4 - a**2))

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Giac [A]
time = 1.18, size = 25, normalized size = 0.93 \begin {gather*} \frac {x}{\sqrt {-2 \, K r^{4} + 2 \, H r^{2} - a^{2}} r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(-2*K*r^4+2*H*r^2-a^2)^(1/2),x, algorithm="giac")

[Out]

x/(sqrt(-2*K*r^4 + 2*H*r^2 - a^2)*r)

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Mupad [B]
time = 0.00, size = 25, normalized size = 0.93 \begin {gather*} \frac {x}{r\,\sqrt {-a^2-2\,K\,r^4+2\,H\,r^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(r*(2*H*r^2 - 2*K*r^4 - a^2)^(1/2)),x)

[Out]

x/(r*(2*H*r^2 - 2*K*r^4 - a^2)^(1/2))

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