Optimal. Leaf size=179 \[ \frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e} \]
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Rubi [A] time = 0.0669763, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \[ \frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e} \]
Antiderivative was successfully verified.
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Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{10} (11 d) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{10} \left (11 d^2\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{80} \left (77 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{96} \left (77 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{128} \left (77 d^8\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{256} \left (77 d^{10}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{1}{256} \left (77 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{77}{256} d^8 x \sqrt{d^2-e^2 x^2}+\frac{77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac{11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac{11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac{77 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}\\ \end{align*}
Mathematica [A] time = 0.355226, size = 118, normalized size = 0.66 \[ \frac{\left (d^2-e^2 x^2\right )^{9/2} \left (\frac{33 d^2 \left (-326 d^4 e^2 x^3+200 d^2 e^4 x^5+\frac{105 d^7 \sin ^{-1}\left (\frac{e x}{d}\right )}{e \sqrt{1-\frac{e^2 x^2}{d^2}}}+279 d^6 x-48 e^6 x^7\right )}{\left (d^2-e^2 x^2\right )^4}-\frac{2560 d}{e}-1152 x\right )}{11520} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 151, normalized size = 0.8 \begin{align*} -{\frac{x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}}+{\frac{11\,{d}^{2}x}{80} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{77\,{d}^{4}x}{480} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{77\,{d}^{6}x}{384} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{8}x}{256}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{77\,{d}^{10}}{256}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,d}{9\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71957, size = 193, normalized size = 1.08 \begin{align*} \frac{77 \, d^{10} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}}} + \frac{77}{256} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{8} x + \frac{77}{384} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{6} x + \frac{77}{480} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4} x + \frac{11}{80} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x - \frac{1}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} x - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} d}{9 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1594, size = 355, normalized size = 1.98 \begin{align*} -\frac{6930 \, d^{10} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (1152 \, e^{9} x^{9} + 2560 \, d e^{8} x^{8} - 3024 \, d^{2} e^{7} x^{7} - 10240 \, d^{3} e^{6} x^{6} + 312 \, d^{4} e^{5} x^{5} + 15360 \, d^{5} e^{4} x^{4} + 6150 \, d^{6} e^{3} x^{3} - 10240 \, d^{7} e^{2} x^{2} - 8055 \, d^{8} e x + 2560 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{11520 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 31.3627, size = 1420, normalized size = 7.93 \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 = 1.26565, size = 173, normalized size = 0.97 \begin{align*} \frac{77}{256} \, d^{10} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{11520} \,{\left (2560 \, d^{9} e^{\left (-1\right )} -{\left (8055 \, d^{8} + 2 \,{\left (5120 \, d^{7} e -{\left (3075 \, d^{6} e^{2} + 4 \,{\left (1920 \, d^{5} e^{3} +{\left (39 \, d^{4} e^{4} - 2 \,{\left (640 \, d^{3} e^{5} +{\left (189 \, d^{2} e^{6} - 8 \,{\left (9 \, x e^{8} + 20 \, d e^{7}\right )} x\right )} x\right )} x\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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