Optimal. Leaf size=201 \[ \frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4} \]
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Rubi [A] time = 0.156395, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4} \]
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^9} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{\int \frac{\left (-8 d^2 e-3 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx}{8 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}+\frac{\int \frac{\left (21 d^3 e^2+16 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{\int \frac{\left (-96 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{336 d^6}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{e^4 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{16 d^3}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{e^4 \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{32 d^3}\\ &=-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{\left (3 e^6\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{128 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac{\left (3 e^8\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{256 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{\left (3 e^6\right ) \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 )}{128 d^3}\\ &=\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}\\ \end{align*}
Mathematica [C] time = 0.0296739, size = 73, normalized size = 0.36 \[ -\frac{e \left (d^2-e^2 x^2\right )^{5/2} \left (7 e^7 x^7 \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};1-\frac{e^2 x^2}{d^2}\right )+2 d^5 e^2 x^2+5 d^7\right )}{35 d^9 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 236, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,d{x}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{16\,{d}^{3}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{4}}{64\,{d}^{5}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{128\,{d}^{7}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{8}}{128\,{d}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{8}}{128\,{d}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{e}^{8}}{128\,{d}^{3}}\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{e}{7\,{d}^{2}{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{3}}{35\,{d}^{4}{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 = 2.09431, size = 292, normalized size = 1.45 \begin{align*} \frac{105 \, e^{8} x^{8} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (256 \, e^{7} x^{7} + 105 \, d e^{6} x^{6} + 128 \, d^{2} e^{5} x^{5} + 70 \, d^{3} e^{4} x^{4} - 1024 \, d^{4} e^{3} x^{3} - 840 \, d^{5} e^{2} x^{2} + 640 \, d^{6} e x + 560 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 30.6822, size = 1171, normalized size = 5.83 \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 = 1.23435, size = 582, normalized size = 2.9 \begin{align*} \frac{x^{8}{\left (\frac{80 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{16}}{x} - \frac{112 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{12}}{x^{3}} - \frac{280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{10}}{x^{4}} - \frac{560 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{8}}{x^{5}} + \frac{1680 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{4}}{x^{7}} + 35 \, e^{18}\right )} e^{6}}{71680 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{4}} - \frac{3 \, e^{8} \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 )}{128 \, d^{4}} - \frac{{\left (\frac{1680 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{28} e^{86}}{x} - \frac{560 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{28} e^{82}}{x^{3}} - \frac{280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{28} e^{80}}{x^{4}} - \frac{112 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{28} e^{78}}{x^{5}} + \frac{80 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{28} e^{74}}{x^{7}} + \frac{35 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{28} e^{72}}{x^{8}}\right )} e^{\left (-80\right )}}{71680 \, d^{32}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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