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 = {651} \[ -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 651
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 = 32, normalized size = 1.03 \[ -\frac {\sqrt {d^2-e^2 x^2}}{d^2 e+d e^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 35, normalized size = 1.13 \[ -\frac {e x + d + \sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.94 \[ -\frac {-e x +d}{\sqrt {-e^{2} x^{2}+d^{2}}\, d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 30, normalized size = 0.97 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]
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 \[ -\frac {\sqrt {d^2-e^2\,x^2}}{d\,e\,\left (d+e\,x\right )} \]
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 \[ \int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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