3.3.10 \(\int \frac {r}{\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}} \, dr\) [210]

Optimal. Leaf size=28 \[ \frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {267} \begin {gather*} \frac {\sqrt {-\alpha ^2+2 e r^2-\epsilon ^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-alpha^2 - epsilon^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*} \int \frac {r}{\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}} \, dr &=\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\alpha ^2-\epsilon ^2+2 e r^2}}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-alpha^2 - epsilon^2 + 2*e*r^2]/(2*e)

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.13, size = 48, normalized size = 1.71 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {-\text {alpha}^2+2 e r^2-\text {epsilon}^2}}{2 e},e\text {!=}0\right \}\right \},\frac {r^2}{2 \sqrt {-\text {alpha}^2-\text {epsilon}^2}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[r/Sqrt[2*e*r^2-alpha^2-epsilon^2],r]')

[Out]

Piecewise[{{Sqrt[-alpha ^ 2 + 2 e r ^ 2 - epsilon ^ 2] / (2 e), e != 0}}, r ^ 2 / (2 Sqrt[-alpha ^ 2 - epsilon
^ 2])]

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 25, normalized size = 0.89

method	result	size
gosper	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
derivativedivides	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
default	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
trager	\(\frac {\sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{2 e}\)	\(25\)
risch	\(-\frac {-2 e \,r^{2}+\alpha ^{2}+\epsilon ^{2}}{2 e \sqrt {2 e \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - \epsilon ^{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.31, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, r^{2} e - \alpha ^{2} - \epsilon ^{2}} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.30, size = 36, normalized size = 1.29 \begin {gather*} \begin {cases} \frac {\sqrt {- \alpha ^{2} + 2 e r^{2} - \epsilon ^{2}}}{2 e} & \text {for}\: e \neq 0 \\\frac {r^{2}}{2 \sqrt {- \alpha ^{2} - \epsilon ^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(-alpha**2 + 2*e*r**2 - epsilon**2)/(2*e), Ne(e, 0)), (r**2/(2*sqrt(-alpha**2 - epsilon**2)), T
rue))

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 18, normalized size = 0.64 \begin {gather*} \left (1.83939720586\times 10^{-13}\right ) \sqrt {-\left (1.0\times 10^{24}\right ) \alpha ^{2}+\left (2.0\times 10^{24}\right ) r^{2}\cdot 2.71828182846-1.0} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1.83939720586000e-13)*sqrt(-1.00000000000000e24*alpha^2 + 5.43656365692000e24*r^2 - 1.00000000000000)

________________________________________________________________________________________

Mupad [B]
time = 0.27, size = 24, normalized size = 0.86 \begin {gather*} \frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,e\,r^2}}{2\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*e*r^2 - alpha^2 - epsilon^2)^(1/2),r)

[Out]

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

________________________________________________________________________________________