Optimal. Leaf size=172 \[ \frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
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Rubi [A] time = 0.125398, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {835, 807, 266, 47, 63, 208} \[ \frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
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Rule 835
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{\int \frac{\left (-7 d^2 e-2 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{7 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{\int \frac{\left (12 d^3 e^2+7 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{42 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^3 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^3 \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d^2}\\ &=-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^5 \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^5 \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{16 d^2}\\ &=\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end{align*}
Mathematica [C] time = 0.0219998, size = 72, normalized size = 0.42 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2} \left (7 e^7 x^7 \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right )+2 d^5 e^2 x^2+5 d^7\right )}{35 d^8 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 211, normalized size = 1.2 \begin{align*} -{\frac{e}{6\,{d}^{2}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}}{24\,{d}^{4}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{5}}{48\,{d}^{6}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{7}}{48\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{7}}{16\,{d}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{{e}^{7}}{16\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{7\,d{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{2}}{35\,{d}^{3}{x}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01573, size = 265, normalized size = 1.54 \begin{align*} \frac{105 \, e^{7} x^{7} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (96 \, e^{6} x^{6} + 105 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} - 384 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x + 240 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.0072, size = 1049, normalized size = 6.1 \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 [B] time = 1.28193, size = 667, normalized size = 3.88 \begin{align*} \frac{x^{7}{\left (\frac{35 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{14}}{x} - \frac{21 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{12}}{x^{2}} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{10}}{x^{3}} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{8}}{x^{4}} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{6}}{x^{5}} + \frac{315 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{4}}{x^{6}} + 15 \, e^{16}\right )} e^{5}}{13440 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{3}} - \frac{e^{7} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{16 \, d^{3}} - \frac{{\left (\frac{315 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{68}}{x} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{66}}{x^{2}} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e^{64}}{x^{3}} - \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{62}}{x^{4}} - \frac{21 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18} e^{60}}{x^{5}} + \frac{35 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{58}}{x^{6}} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{56}}{x^{7}}\right )} e^{\left (-63\right )}}{13440 \, d^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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