Optimal. Leaf size=206 \[ \frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A] time = 0.101482, antiderivative size = 206, 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 = {669, 671, 641, 217, 203} \[ \frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 \sqrt{d^2-e^2 x^2} (d+e x)^2}{5 e}+\frac{77 d \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 669
Rule 671
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+e x)^9}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{11}{5} \int \frac{(d+e x)^7}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{33}{5} \int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}-\frac{231}{5} \int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-(77 d) \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^2\right ) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{2} \left (231 d^3\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{2 (d+e x)^8}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 (d+e x)^6}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{66 (d+e x)^4}{5 e \sqrt{d^2-e^2 x^2}}+\frac{231 d^2 \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 d (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{77 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{231 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.342552, size = 144, normalized size = 0.7 \[ \frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (8711 d^3 e^2 x^2-815 d^2 e^3 x^3-12843 d^4 e x+5446 d^5-105 d e^4 x^4-10 e^5 x^5\right )-3465 d^2 (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{30 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.209, size = 309, normalized size = 1.5 \begin{align*} -{\frac{152\,{d}^{7}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{157\,{d}^{5}x}{15} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+63\,{\frac{{d}^{5}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{231\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{4093\,{d}^{3}x}{30}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{9\,d{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{231\,{d}^{3}{e}^{4}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{77\,{d}^{3}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{116\,{e}^{5}{d}^{2}{x}^{6}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+358\,{\frac{{e}^{3}{d}^{4}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{1348\,e{d}^{6}{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{e}^{7}{x}^{8}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2723\,{d}^{8}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.87807, size = 518, normalized size = 2.51 \begin{align*} -\frac{e^{7} x^{8}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{9 \, d e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{77}{10} \, d^{3} e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{116 \, d^{2} e^{5} x^{6}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{77}{2} \, d^{3} e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{358 \, d^{4} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{63 \, d^{5} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{1348 \, d^{6} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{152 \, d^{7} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2723 \, d^{8}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{619 \, d^{5} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{989 \, d^{3} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{231 \, d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54416, size = 447, normalized size = 2.17 \begin{align*} \frac{5446 \, d^{3} e^{3} x^{3} - 16338 \, d^{4} e^{2} x^{2} + 16338 \, d^{5} e x - 5446 \, d^{6} + 6930 \,{\left (d^{3} e^{3} x^{3} - 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x - d^{6}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (10 \, e^{5} x^{5} + 105 \, d e^{4} x^{4} + 815 \, d^{2} e^{3} x^{3} - 8711 \, d^{3} e^{2} x^{2} + 12843 \, d^{4} e x - 5446 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{9}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32293, size = 174, normalized size = 0.84 \begin{align*} -\frac{231}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (5446 \, d^{8} e^{\left (-1\right )} +{\left (3495 \, d^{7} -{\left (13480 \, d^{6} e +{\left (7765 \, d^{5} e^{2} -{\left (10740 \, d^{4} e^{3} +{\left (5941 \, d^{3} e^{4} - 5 \,{\left (232 \, d^{2} e^{5} +{\left (2 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{30 \,{\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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