3.2.12 \(\int \frac {r}{\sqrt {-a^2-e^2+2 e r^2-2 K r^4}} \, dx\) [112]

Optimal. Leaf size=30 \[ \frac {r x}{\sqrt {-a^2-e^2+2 e r^2-2 K r^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 30, 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 {r x}{\sqrt {-a^2-e^2-2 K r^4+2 e r^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {r x}{\sqrt {-a^2-e^2+2 e r^2-2 K r^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(r*x)/Sqrt[-a^2 - e^2 + 2*E*r^2 - 2*K*r^4]

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 30, normalized size = 1.00

method	result	size
default	\(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}-2 K \,r^{4}}}\)	\(30\)
norman	\(\frac {r x}{\sqrt {-a^{2}-e^{2}+2 \,{\mathrm e} r^{2}-2 K \,r^{4}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 2.10, size = 28, normalized size = 0.93 \begin {gather*} \frac {r x}{\sqrt {-2 \, K r^{4} + 2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.67, size = 50, normalized size = 1.67 \begin {gather*} -\frac {\sqrt {-2 \, K r^{4} + 2 \, r^{2} e - a^{2} - e^{2}} r x}{2 \, K r^{4} - 2 \, r^{2} e + a^{2} + e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 27, normalized size = 0.90 \begin {gather*} \frac {r x}{\sqrt {- 2 K r^{4} - a^{2} - e^{2} + 2 e r^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.76, size = 28, normalized size = 0.93 \begin {gather*} \frac {r x}{\sqrt {-2 \, K r^{4} + 2 \, r^{2} e - a^{2} - e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.00, size = 29, normalized size = 0.97 \begin {gather*} \frac {r\,x}{\sqrt {-a^2-e^2-2\,K\,r^4+2\,\mathrm {e}\,r^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________