Optimal. Leaf size=144 \[ -\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^4} \]
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Rubi [A] time = 0.186042, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1809, 833, 780, 217, 203} \[ -\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^4} \]
Antiderivative was successfully verified.
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Rule 1809
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{x^3 \left (-9 d^2 e^2-10 d e^3 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{5 e^2}\\ &=-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{x^2 \left (30 d^3 e^3+36 d^2 e^4 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{20 e^4}\\ &=-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{x \left (-72 d^4 e^4-90 d^3 e^5 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{60 e^6}\\ &=-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}+\frac{\left (3 d^5\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{4 e^3}\\ &=-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}+\frac{\left (3 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^3}\\ &=-\frac{3 d^2 x^2 \sqrt{d^2-e^2 x^2}}{5 e^2}-\frac{d x^3 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{5} x^4 \sqrt{d^2-e^2 x^2}-\frac{3 d^3 (8 d+5 e x) \sqrt{d^2-e^2 x^2}}{20 e^4}+\frac{3 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4 e^4}\\ \end{align*}
Mathematica [A] time = 0.0963739, size = 92, normalized size = 0.64 \[ \frac{15 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (12 d^2 e^2 x^2+15 d^3 e x+24 d^4+10 d e^3 x^3+4 e^4 x^4\right )}{20 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 149, normalized size = 1. \begin{align*} -{\frac{{x}^{4}}{5}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{d}^{2}{x}^{2}}{5\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{6\,{d}^{4}}{5\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{d{x}^{3}}{2\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{d}^{3}x}{4\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{d}^{5}}{4\,{e}^{3}}\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 = 1.4709, size = 190, normalized size = 1.32 \begin{align*} -\frac{1}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} x^{4} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} d x^{3}}{2 \, e} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x^{2}}{5 \, e^{2}} + \frac{3 \, d^{5} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{4 \, \sqrt{e^{2}} e^{3}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x}{4 \, e^{3}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7826, size = 204, normalized size = 1.42 \begin{align*} -\frac{30 \, d^{5} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (4 \, e^{4} x^{4} + 10 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x + 24 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{20 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.01119, size = 359, normalized size = 2.49 \begin{align*} d^{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 ) + 2 d e \left (\begin{cases} - \frac{3 i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{5}} + \frac{3 i d^{3} x}{8 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d x^{3}}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i x^{5}}{4 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{3 d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{5}} - \frac{3 d^{3} x}{8 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d x^{3}}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{8 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6}} - \frac{4 d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{6}}{6 \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 = 1.13264, size = 99, normalized size = 0.69 \begin{align*} \frac{3}{4} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{20} \,{\left (24 \, d^{4} e^{\left (-4\right )} +{\left (15 \, d^{3} e^{\left (-3\right )} + 2 \,{\left (6 \, d^{2} e^{\left (-2\right )} +{\left (5 \, d e^{\left (-1\right )} + 2 \, x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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