3.2.13 \(\int \frac {r}{\sqrt {-a^2-e^2-2 K r+2 H r^2}} \, dx\) [113]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {8} \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2-2 r (K-H r)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.04 \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2-2 K r+2 H r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [A]
time = 1.92, size = 26, normalized size = 0.96 \begin {gather*} \frac {r x}{\sqrt {2 H r^2-2 K r-a^2-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

r x / Sqrt[2 H r ^ 2 - 2 K r - a ^ 2 - e ^ 2]

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Maple [A]
time = 0.01, size = 27, normalized size = 1.00

method	result	size
default	\(\frac {r x}{\sqrt {2 H \,r^{2}-2 K r -a^{2}-e^{2}}}\)	\(27\)
norman	\(\frac {r x}{\sqrt {2 H \,r^{2}-2 K r -a^{2}-e^{2}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 25, normalized size = 0.93 \begin {gather*} \frac {r x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - 2*K*r - e^2)

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Fricas [A]
time = 0.33, size = 25, normalized size = 0.93 \begin {gather*} \frac {r x}{\sqrt {2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - 2*K*r - e^2)

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Sympy [A]
time = 0.03, size = 24, normalized size = 0.89 \begin {gather*} \frac {r x}{\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(r/(2*H*r**2-2*K*r-a**2-e**2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(2*H*r^2 - a^2 - e^2 - 2*K*r)

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Mupad [B]
time = 0.00, size = 26, normalized size = 0.96 \begin {gather*} \frac {r\,x}{\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(r/(2*H*r^2 - 2*K*r - a^2 - e^2)^(1/2),x)

[Out]

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

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