Optimal. Leaf size=121 \[ \frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.112722, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {850, 813, 844, 217, 203, 266, 63, 208} \[ \frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 850
Rule 813
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac{3}{8} \int \frac{\left (4 d^2 e+4 d e^2 x\right ) \sqrt{d^2-e^2 x^2}}{x^2} \, dx\\ &=\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{16} \int \frac{-8 d^3 e^2+8 d^2 e^3 x}{x \sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac{1}{2} \left (3 d^3 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx+\frac{1}{2} \left (3 d^2 e^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac{1}{4} \left (3 d^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )+\frac{1}{2} \left (3 d^2 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{2} \left (3 d^3\right ) \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 )\\ &=\frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.168974, size = 119, normalized size = 0.98 \[ \frac{1}{2} \left (\frac{\sqrt{d^2-e^2 x^2} \left (2 d^2 e x-d^3-2 d e^2 x^2+e^3 x^3\right )}{x^2}+3 d^2 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+3 d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-3 d^2 e^2 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 411, normalized size = 3.4 \begin{align*} -{\frac{3\,{e}^{2}}{10\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{2\,d} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,d{e}^{2}}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{e}^{2}{d}^{3}}{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{{e}^{2}}{5\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}x}{4\,{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{3}x}{8}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{3\,{e}^{3}{d}^{2}}{8}\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}{2\,{d}^{3}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{e}{{d}^{4}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{3}x}{{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{3}x}{4\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{3}x}{8}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{15\,{e}^{3}{d}^{2}}{8}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64981, size = 270, normalized size = 2.23 \begin{align*} -\frac{6 \, d^{2} e^{2} x^{2} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \, d^{2} e^{2} x^{2} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 2 \, d^{2} e^{2} x^{2} -{\left (e^{3} x^{3} - 2 \, d e^{2} x^{2} + 2 \, d^{2} e x - d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 11.3905, size = 471, normalized size = 3.89 \begin{align*} d^{3} \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) \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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