Optimal. Leaf size=88 \[ \frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Rubi [A] time = 0.0754648, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {857, 823, 12, 266, 63, 208} \[ \frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]
Antiderivative was successfully verified.
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Rule 857
Rule 823
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-3 d e^2+2 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\int -\frac{3 d^3 e^4}{x \sqrt{d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^3}\\ &=\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^3}\\ &=\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\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^3 e^2}\\ &=\frac{3 d-2 e x}{3 d^4 \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4}\\ \end{align*}
Mathematica [A] time = 0.108529, size = 83, normalized size = 0.94 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (4 d^2+d e x-2 e^2 x^2\right )}{(d-e x) (d+e x)^2}-3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+3 \log (x)}{3 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 142, normalized size = 1.6 \begin{align*}{\frac{1}{{d}^{3}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{3}}\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}{3\,e{d}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{2\,ex}{3\,{d}^{4}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \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}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60024, size = 301, normalized size = 3.42 \begin{align*} \frac{4 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} - 4 \, d^{2} e x - 4 \, d^{3} + 3 \,{\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (2 \, e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} - d^{6} e x - d^{7}\right )}} \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 \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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