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)/Sqrt[-a^2 + 2*E*r^2 - 2*K*r^4]

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Rubi [A]  time = 0.0127465, antiderivative size = 25, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0000295, size = 25, normalized size = 1. \[ \frac{r x}{\sqrt{-a^2-2 K r^4+2 e r^2}} \]

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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Maple [A]  time = 0.002, size = 24, normalized size = 1. \begin{align*}{rx{\frac{1}{\sqrt{-2\,K{r}^{4}+2\,E{r}^{2}-{a}^{2}}}}} \end{align*}

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 = 0.930996, size = 31, normalized size = 1.24 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \end{align*}

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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Fricas [A]  time = 1.88055, size = 85, normalized size = 3.4 \begin{align*} -\frac{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}} r x}{2 \, K r^{4} - 2 \, E r^{2} + a^{2}} \end{align*}

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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Sympy [A]  time = 0.053411, size = 22, normalized size = 0.88 \begin{align*} \frac{r x}{\sqrt{- 2 K r^{4} - a^{2} + 2 e r^{2}}} \end{align*}

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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Giac [A]  time = 1.06028, size = 31, normalized size = 1.24 \begin{align*} \frac{r x}{\sqrt{-2 \, K r^{4} + 2 \, E r^{2} - a^{2}}} \end{align*}

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)