\(\int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr\) [209]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {-\alpha ^2+2 e r^2}}{2 e} \]

[Out]

1/2*(2*e*r^2-alpha^2)^(1/2)/e

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {267} \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {2 e r^2-\alpha ^2}}{2 e} \]

[In]

Int[r/Sqrt[-alpha^2 + 2*e*r^2],r]

[Out]

Sqrt[-alpha^2 + 2*e*r^2]/(2*e)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-\alpha ^2+2 e r^2}}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {-\alpha ^2+2 e r^2}}{2 e} \]

[In]

Integrate[r/Sqrt[-alpha^2 + 2*e*r^2],r]

[Out]

Sqrt[-alpha^2 + 2*e*r^2]/(2*e)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
gosper	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\)	\(20\)
derivativedivides	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\)	\(20\)
default	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\)	\(20\)
trager	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\)	\(20\)
pseudoelliptic	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}}}{2 e}\)	\(20\)
risch	\(-\frac {-2 e \,r^{2}+\alpha ^{2}}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}}}\)	\(30\)

[In]

int(r/(2*e*r^2-alpha^2)^(1/2),r,method=_RETURNVERBOSE)

[Out]

1/2*(2*e*r^2-alpha^2)^(1/2)/e

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="fricas")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\begin {cases} \frac {\sqrt {- \alpha ^{2} + 2 e r^{2}}}{2 e} & \text {for}\: e \neq 0 \\\frac {r^{2}}{2 \sqrt {- \alpha ^{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(r/(2*e*r**2-alpha**2)**(1/2),r)

[Out]

Piecewise((sqrt(-alpha**2 + 2*e*r**2)/(2*e), Ne(e, 0)), (r**2/(2*sqrt(-alpha**2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="maxima")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {2 \, e r^{2} - \alpha ^{2}}}{2 \, e} \]

[In]

integrate(r/(2*e*r^2-alpha^2)^(1/2),r, algorithm="giac")

[Out]

1/2*sqrt(2*e*r^2 - alpha^2)/e

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2}} \, dr=\frac {\sqrt {2\,e\,r^2-\alpha ^2}}{2\,e} \]

[In]

int(r/(2*e*r^2 - alpha^2)^(1/2),r)

[Out]

(2*e*r^2 - alpha^2)^(1/2)/(2*e)