Optimal. Leaf size=54 \[ \frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {857, 12, 266, 63, 208} \[ \frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 857
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\int \frac {d e^2}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{d e^2}\\ &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 (d+e x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.96 \[ \frac {\frac {\sqrt {d^2-e^2 x^2}}{d+e x}-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 62, normalized size = 1.15 \[ \frac {e x + {\left (e x + d\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + d + \sqrt {-e^{2} x^{2} + d^{2}}}{d^{2} e x + d^{3}} \]
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 = 88, normalized size = 1.63 \[ -\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}{\left (x +\frac {d}{e}\right ) d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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