Optimal. Leaf size=115 \[ -\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.117931, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {850, 813, 815, 844, 217, 203, 266, 63, 208} \[ -\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 850
Rule 813
Rule 815
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^2 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{1}{2} \int \frac{\left (2 d^2 e+6 d e^2 x\right ) \sqrt{d^2-e^2 x^2}}{x} \, dx\\ &=-\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac{\int \frac{-4 d^4 e^3-6 d^3 e^4 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\left (d^4 e\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-\frac{1}{2} \left (3 d^3 e^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{1}{2} \left (d^4 e\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^3 e^2\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{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{d^4 \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{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.135068, size = 114, normalized size = 0.99 \[ \sqrt{d^2-e^2 x^2} \left (-\frac{4 d^2 e}{3}-\frac{d^3}{x}-\frac{1}{2} d e^2 x+\frac{e^3 x^2}{3}\right )+d^3 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 e \log (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 380, normalized size = 3.3 \begin{align*} -{\frac{e}{5\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{e}{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-e{d}^{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}+{e{d}^{4}\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}{5\,{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{2}x}{4\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,d{e}^{2}x}{8}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{3\,{e}^{2}{d}^{3}}{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}{{d}^{3}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{2}x}{{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{2}x}{4\,d} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{15\,d{e}^{2}x}{8}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{15\,{e}^{2}{d}^{3}}{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 = 1.60561, size = 257, normalized size = 2.23 \begin{align*} \frac{18 \, d^{3} e x \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 6 \, d^{3} e x \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 8 \, d^{3} e x +{\left (2 \, e^{3} x^{3} - 3 \, d e^{2} x^{2} - 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.2584, size = 393, normalized size = 3.42 \begin{align*} d^{3} \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^{2} e \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 ) - d e^{2} \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 ) + e^{3} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{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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