3.111 \(\int \frac {r}{\sqrt {-a^2+2 e r^2-2 K r^4}} \, dx\)

Optimal. Leaf size=25 \[ \frac {r x}{\sqrt {-a^2-2 K r^4+2 e r^2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {8} \[ \frac {r x}{\sqrt {-a^2-2 K r^4+2 e r^2}} \]

Antiderivative was successfully verified.

[In]

Int[r/Sqrt[-a^2 + 2*E*r^2 - 2*K*r^4],x]

[Out]

(r*x)/Sqrt[-a^2 + 2*E*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 {r}{\sqrt {-a^2+2 e r^2-2 K r^4}} \, dx &=\frac {r x}{\sqrt {-a^2+2 e r^2-2 K r^4}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ \frac {r x}{\sqrt {-a^2-2 K r^4+2 e r^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[r/Sqrt[-a^2 + 2*E*r^2 - 2*K*r^4],x]

[Out]

(r*x)/Sqrt[-a^2 + 2*E*r^2 - 2*K*r^4]

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fricas [A]  time = 0.40, size = 42, normalized size = 1.68 \[ -\frac {\sqrt {-2 \, K r^{4} + 2 \, E r^{2} - a^{2}} r x}{2 \, K r^{4} - 2 \, E r^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-2*K*r^4 + 2*E*r^2 - a^2)*r*x/(2*K*r^4 - 2*E*r^2 + a^2)

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giac [A]  time = 0.92, size = 23, normalized size = 0.92 \[ \frac {r x}{\sqrt {-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(-2*K*r^4 + 2*E*r^2 - a^2)

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maple [A]  time = 0.00, size = 24, normalized size = 0.96 \[ \frac {r x}{\sqrt {-2 K \,r^{4}+2 E \,r^{2}-a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 23, normalized size = 0.92 \[ \frac {r x}{\sqrt {-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(-2*K*r^4 + 2*E*r^2 - a^2)

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mupad [B]  time = 0.00, size = 24, normalized size = 0.96 \[ \frac {r\,x}{\sqrt {-a^2-2\,K\,r^4+2\,\mathrm {e}\,r^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.06, size = 24, normalized size = 0.96 \[ \frac {r x}{\sqrt {- 2 K r^{4} - a^{2} + 2 e r^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

r*x/sqrt(-2*K*r**4 - a**2 + 2*E*r**2)

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