Optimal. Leaf size=51 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d}-\frac{\sqrt{d^2-e^2 x^2}}{d x} \]
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Rubi [A] time = 0.0573296, antiderivative size = 51, normalized size of antiderivative = 1., 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 = {850, 807, 266, 63, 208} \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d}-\frac{\sqrt{d^2-e^2 x^2}}{d x} \]
Antiderivative was successfully verified.
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Rule 850
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d^2-e^2 x^2}}{x^2 (d+e x)} \, dx &=\int \frac{d-e x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{d x}-e \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{d x}-\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{d 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 )}{e}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{d x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0691569, size = 53, normalized size = 1.04 \[ -\frac{\sqrt{d^2-e^2 x^2}-e x \log \left (\sqrt{d^2-e^2 x^2}+d\right )+e x \log (x)}{d x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 222, normalized size = 4.4 \begin{align*} -{\frac{e}{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{e\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{e}{{d}^{2}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{{e}^{2}}{d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{{d}^{3}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}x}{{d}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{{e}^{2}}{d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.593, size = 97, normalized size = 1.9 \begin{align*} -\frac{e x \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}}}{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{2} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23455, size = 138, normalized size = 2.71 \begin{align*} \frac{e \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d} + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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