Optimal. Leaf size=143 \[ \frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]
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Rubi [A] time = 0.27113, antiderivative size = 143, 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 = {1635, 1814, 641, 217, 203} \[ \frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]
Antiderivative was successfully verified.
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Rule 1635
Rule 1814
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{(d+e x) \left (\frac{2 d^5}{e^5}+\frac{5 d^4 x}{e^4}+\frac{5 d^3 x^2}{e^3}+\frac{5 d^2 x^3}{e^2}+\frac{5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{\frac{16 d^5}{e^5}+\frac{45 d^4 x}{e^4}+\frac{30 d^3 x^2}{e^3}+\frac{15 d^2 x^3}{e^2}}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{\frac{30 d^5}{e^5}+\frac{15 d^4 x}{e^4}}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^4}\\ &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{(2 d) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac{d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 d (30 d+23 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\sqrt{d^2-e^2 x^2}}{e^6}-\frac{2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}\\ \end{align*}
Mathematica [A] time = 0.233577, size = 111, normalized size = 0.78 \[ \frac{-32 d^2 e^2 x^2-\frac{30 (d-e x)^3 (d+e x) \sin ^{-1}\left (\frac{e x}{d}\right )}{\sqrt{1-\frac{e^2 x^2}{d^2}}}-82 d^3 e x+56 d^4+76 d e^3 x^3-15 e^4 x^4}{15 e^6 (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 193, normalized size = 1.4 \begin{align*} -{{x}^{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+7\,{\frac{{d}^{2}{x}^{4}}{{e}^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{28\,{d}^{4}{x}^{2}}{3\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{56\,{d}^{6}}{15\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,d{x}^{5}}{5\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{2\,d{x}^{3}}{3\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{dx}{{e}^{5}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}-2\,{\frac{d}{{e}^{5}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55754, size = 390, normalized size = 2.73 \begin{align*} \frac{2}{15} \, d e 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{x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{2 \, d 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 )}}{3 \, e} + \frac{7 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{28 \, d^{4} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{56 \, d^{6}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{8 \, d^{3} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{14 \, d x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{2 \, d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01491, size = 397, normalized size = 2.78 \begin{align*} \frac{56 \, d e^{4} x^{4} - 112 \, d^{2} e^{3} x^{3} + 112 \, d^{4} e x - 56 \, d^{5} + 60 \,{\left (d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} + 2 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{4} x^{4} - 76 \, d e^{3} x^{3} + 32 \, d^{2} e^{2} x^{2} + 82 \, d^{3} e x - 56 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{10} x^{4} - 2 \, d e^{9} x^{3} + 2 \, d^{3} e^{7} x - d^{4} e^{6}\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{x^{5} \left (d + e x\right )^{2}}{\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 = 1.16453, size = 143, normalized size = 1. \begin{align*} -2 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (56 \, d^{6} e^{\left (-6\right )} +{\left (30 \, d^{5} e^{\left (-5\right )} -{\left (140 \, d^{4} e^{\left (-4\right )} +{\left (70 \, d^{3} e^{\left (-3\right )} -{\left (105 \, d^{2} e^{\left (-2\right )} +{\left (46 \, d e^{\left (-1\right )} - 15 \, x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\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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