Optimal. Leaf size=159 \[ \frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3} \]
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Rubi [A] time = 0.10801, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {797, 641, 195, 217, 203} \[ \frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3} \]
Antiderivative was successfully verified.
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Rule 797
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac{\int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac{d^2 \int (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac{d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2}+\frac{\left (3 d^5\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac{3 d^5 x \sqrt{d^2-e^2 x^2}}{8 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^5\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{8 e^2}+\frac{\left (3 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^2}+\frac{\left (3 d^7\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^2}\\ &=\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{3 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{\left (5 d^7\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^2}\\ &=\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3}\\ \end{align*}
Mathematica [A] time = 0.191435, size = 135, normalized size = 0.85 \[ \frac{\sqrt{d^2-e^2 x^2} \left (105 d^6 \sin ^{-1}\left (\frac{e x}{d}\right )-\sqrt{1-\frac{e^2 x^2}{d^2}} \left (48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+105 d^5 e x+96 d^6+280 d e^5 x^5+240 e^6 x^6\right )\right )}{1680 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 148, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{7\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{d}^{2}}{35\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{dx}{6\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{24\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}x}{16\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{7}}{16\,{e}^{2}}\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 = 1.57686, size = 189, normalized size = 1.19 \begin{align*} \frac{d^{7} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{2}} + \frac{\sqrt{-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} x}{24 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x^{2}}{7 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x}{6 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{2}}{35 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91371, size = 262, normalized size = 1.65 \begin{align*} -\frac{210 \, d^{7} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (240 \, e^{6} x^{6} + 280 \, d e^{5} x^{5} - 384 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} + 48 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x + 96 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 14.9523, size = 656, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30189, size = 130, normalized size = 0.82 \begin{align*} \frac{1}{16} \, d^{7} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{1680} \,{\left (96 \, d^{6} e^{\left (-3\right )} +{\left (105 \, d^{5} e^{\left (-2\right )} + 2 \,{\left (24 \, d^{4} e^{\left (-1\right )} -{\left (245 \, d^{3} + 4 \,{\left (48 \, d^{2} e - 5 \,{\left (6 \, x e^{3} + 7 \, d e^{2}\right )} x\right )} 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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