Optimal. Leaf size=81 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]
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Rubi [A] time = 0.063554, antiderivative size = 81, 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 = {857, 807, 266, 63, 208} \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]
Antiderivative was successfully verified.
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Rule 857
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{\int \frac{-2 d e^2+e^3 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{e \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (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^2 e}\\ &=-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0556433, size = 62, normalized size = 0.77 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{(d+2 e x) \sqrt{d^2-e^2 x^2}}{x (d+e x)}}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 108, normalized size = 1.3 \begin{align*}{\frac{e}{{d}^{2}}\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{1}{{d}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{1}{{d}^{3}x}\sqrt{-{x}^{2}{e}^{2}+{d}^{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{1}{\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 = 1.56651, size = 176, normalized size = 2.17 \begin{align*} -\frac{e^{2} x^{2} + d e x +{\left (e^{2} x^{2} + d e x\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}}{\left (2 \, e x + d\right )}}{d^{3} e x^{2} + d^{4} 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{1}{x^{2} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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