Optimal. Leaf size=31 \[ -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {665}
\begin {gather*} -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 46, normalized size = 1.48
method | result | size |
gosper | \(-\frac {-e x +d}{d e \sqrt {-e^{2} x^{2}+d^{2}}}\) | \(29\) |
trager | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d e \left (e x +d \right )}\) | \(30\) |
default | \(-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{2} d \left (x +\frac {d}{e}\right )}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}}}{d x e^{2} + d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x e + d + \sqrt {-x^{2} e^{2} + d^{2}}}{d x e^{2} + d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.45, size = 38, normalized size = 1.23 \begin {gather*} \frac {2 \, e^{\left (-1\right )}}{d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.64, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}}{d\,e\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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