Optimal. Leaf size=173 \[ \frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A] time = 0.0768588, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 \sqrt{d^2-e^2 x^2} (d+e x)}{2 e}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
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)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{5} \int \frac{(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{21}{5} \int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}-21 \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} (63 d) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} \left (63 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{1}{2} \left (63 d^2\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)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{42 (d+e x)^3}{5 e \sqrt{d^2-e^2 x^2}}+\frac{63 d \sqrt{d^2-e^2 x^2}}{2 e}+\frac{21 (d+e x) \sqrt{d^2-e^2 x^2}}{2 e}-\frac{63 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.282841, size = 131, normalized size = 0.76 \[ \frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (801 d^2 e^2 x^2-1163 d^3 e x+496 d^4-65 d e^3 x^3-5 e^4 x^4\right )-315 d (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{10 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 284, normalized size = 1.6 \begin{align*} -{\frac{76\,{d}^{6}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{e}^{6}{x}^{7}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{248\,{d}^{7}}{5\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{21\,{d}^{2}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+35\,{\frac{{d}^{4}{e}^{2}{x}^{3}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{63\,{e}^{4}{d}^{2}{x}^{5}}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-8\,{\frac{{e}^{5}d{x}^{6}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}+104\,{\frac{{e}^{3}{d}^{3}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-120\,{\frac{e{d}^{5}{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{63\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{27\,{d}^{4}x}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{423\,{d}^{2}x}{10}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.81844, size = 485, normalized size = 2.8 \begin{align*} -\frac{e^{6} x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{21}{10} \, d^{2} 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{8 \, d e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{21}{2} \, d^{2} 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{104 \, d^{3} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{35 \, d^{4} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{120 \, d^{5} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{76 \, d^{6} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{248 \, d^{7}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{69 \, d^{4} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{39 \, d^{2} x}{10 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{63 \, d^{2} \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 = 2.3162, size = 409, normalized size = 2.36 \begin{align*} \frac{496 \, d^{2} e^{3} x^{3} - 1488 \, d^{3} e^{2} x^{2} + 1488 \, d^{4} e x - 496 \, d^{5} + 630 \,{\left (d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (5 \, e^{4} x^{4} + 65 \, d e^{3} x^{3} - 801 \, d^{2} e^{2} x^{2} + 1163 \, d^{3} e x - 496 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\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 )^{8}}{\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 = 1.34107, size = 159, normalized size = 0.92 \begin{align*} -\frac{63}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (496 \, d^{7} e^{\left (-1\right )} +{\left (325 \, d^{6} -{\left (1200 \, d^{5} e +{\left (655 \, d^{4} e^{2} -{\left (1040 \, d^{3} e^{3} +{\left (591 \, d^{2} e^{4} - 5 \,{\left (x e^{6} + 16 \, d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{10 \,{\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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