Optimal. Leaf size=94 \[ \frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.045912, antiderivative size = 94, 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, 778, 191} \[ \frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 796
Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x \left (2 d^2 e-2 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e^2}\\ &=\frac{x^2 (d+e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{15 d^3 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0284838, size = 82, normalized size = 0.87 \[ \frac{3 d^2 e^2 x^2+2 d^3 e x-2 d^4+2 d e^3 x^3-2 e^4 x^4}{15 d^3 e^3 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 77, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 2\,{e}^{4}{x}^{4}-2\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}-2\,x{d}^{3}e+2\,{d}^{4} \right ) }{15\,{d}^{3}{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992948, size = 151, normalized size = 1.61 \begin{align*} \frac{x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \, d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d e^{2}} - \frac{2 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9073, size = 340, normalized size = 3.62 \begin{align*} -\frac{2 \, e^{5} x^{5} - 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} + 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x - 2 \, d^{5} -{\left (2 \, e^{4} x^{4} - 2 \, d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x + 2 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{8} x^{5} - d^{4} e^{7} x^{4} - 2 \, d^{5} e^{6} x^{3} + 2 \, d^{6} e^{5} x^{2} + d^{7} e^{4} x - d^{8} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.1192, size = 515, normalized size = 5.48 \begin{align*} d \left (\begin{cases} - \frac{5 i d^{2} x^{3}}{15 d^{9} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{2 i e^{2} x^{5}}{15 d^{9} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{5 d^{2} x^{3}}{15 d^{9} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{2 e^{2} x^{5}}{15 d^{9} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 30 d^{7} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 15 d^{5} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{2 d^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{5 e^{2} x^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{7}{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 = 1.18525, size = 86, normalized size = 0.91 \begin{align*} \frac{{\left ({\left (x{\left (\frac{2 \, x^{2} e^{2}}{d^{3}} - \frac{5}{d}\right )} - 5 \, e^{\left (-1\right )}\right )} x^{2} + 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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