Optimal. Leaf size=83 \[ -\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.084863, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1809, 780, 217, 203} \[ -\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 1809
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{x \left (-5 d^2 e^2-6 d e^3 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{3 e^2}\\ &=-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}-\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{d^3 \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e}\\ &=-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}-\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ &=-\frac{1}{3} x^2 \sqrt{d^2-e^2 x^2}-\frac{d (5 d+3 e x) \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0621549, size = 69, normalized size = 0.83 \[ \frac{3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (5 d^2+3 d e x+e^2 x^2\right )}{3 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 98, normalized size = 1.2 \begin{align*} -{\frac{{x}^{2}}{3}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{5\,{d}^{2}}{3\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{dx}{e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{3}}{e}\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 [A] time = 1.47, size = 122, normalized size = 1.47 \begin{align*} -\frac{1}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} x^{2} + \frac{d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} d x}{e} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81565, size = 150, normalized size = 1.81 \begin{align*} -\frac{6 \, d^{3} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{2} x^{2} + 3 \, d e x + 5 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.57588, size = 219, normalized size = 2.64 \begin{align*} d^{2} \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{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^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{d^{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 = 1.1695, size = 66, normalized size = 0.8 \begin{align*} d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-2\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{3} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (5 \, d^{2} e^{\left (-2\right )} +{\left (3 \, d e^{\left (-1\right )} + x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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