Optimal. Leaf size=58 \[ \frac{x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}} \]
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Rubi [A] time = 0.0262578, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {796, 12, 261} \[ \frac{x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 796
Rule 12
Rule 261
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{2 d^2 e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \int \frac{x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e}\\ &=\frac{x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{3 e^3 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0246986, size = 52, normalized size = 0.9 \[ \frac{-2 d^2+2 d e x+e^2 x^2}{3 d e^3 (d-e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 55, normalized size = 1. \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -{x}^{2}{e}^{2}-2\,dex+2\,{d}^{2} \right ) }{3\,d{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0003, size = 119, normalized size = 2.05 \begin{align*} \frac{x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{d x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{3}} - \frac{x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14648, size = 204, normalized size = 3.52 \begin{align*} -\frac{2 \, e^{3} x^{3} - 2 \, d e^{2} x^{2} - 2 \, d^{2} e x + 2 \, d^{3} -{\left (e^{2} x^{2} + 2 \, d e x - 2 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d e^{6} x^{3} - d^{2} e^{5} x^{2} - d^{3} e^{4} x + d^{4} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.63715, size = 233, normalized size = 4.02 \begin{align*} d \left (\begin{cases} \frac{i x^{3}}{- 3 d^{5} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\- \frac{x^{3}}{- 3 d^{5} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} \frac{2 d^{2}}{- 3 d^{2} e^{4} \sqrt{d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{3 e^{2} x^{2}}{- 3 d^{2} e^{4} \sqrt{d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17522, size = 69, normalized size = 1.19 \begin{align*} \frac{{\left (x^{2}{\left (\frac{x}{d} + 3 \, e^{\left (-1\right )}\right )} - 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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