Optimal. Leaf size=134 \[ -\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+e^4 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{13}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.215096, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1807, 811, 844, 217, 203, 266, 63, 208} \[ -\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+e^4 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{13}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 811
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{x^5} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{\int \frac{\sqrt{d^2-e^2 x^2} \left (-12 d^4 e-13 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}+\frac{\int \frac{\left (39 d^5 e^2+12 d^4 e^3 x\right ) \sqrt{d^2-e^2 x^2}}{x^3} \, dx}{12 d^4}\\ &=-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac{\int \frac{78 d^7 e^4+48 d^6 e^5 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{48 d^6}\\ &=-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac{1}{8} \left (13 d e^4\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-e^5 \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-\frac{1}{16} \left (13 d e^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-e^5 \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{8} \left (13 d e^2\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 )\\ &=-\frac{e^2 (13 d+8 e x) \sqrt{d^2-e^2 x^2}}{8 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{x^3}-e^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{13}{8} e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [C] time = 0.261596, size = 196, normalized size = 1.46 \[ -\frac{e \sqrt{d^2-e^2 x^2} \left (2 e^3 x^3 \left (d^2-e^2 x^2\right ) \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};1-\frac{e^2 x^2}{d^2}\right )+6 d^5 \sqrt{1-\frac{e^2 x^2}{d^2}}+9 d^4 e x \sqrt{1-\frac{e^2 x^2}{d^2}}+6 d^2 e^3 x^3 \sin ^{-1}\left (\frac{e x}{d}\right )-9 d^2 e^3 x^3 \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )\right )}{6 d^3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 212, normalized size = 1.6 \begin{align*} -{\frac{e}{{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{d}{4\,{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{e}^{2}}{8\,d{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{e}^{4}}{8\,d}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{13\,d{e}^{4}}{8}\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}^{3}}{{d}^{2}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{5}x}{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{{e}^{5}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{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 = 1.76287, size = 230, normalized size = 1.72 \begin{align*} \frac{16 \, e^{4} x^{4} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13 \, e^{4} x^{4} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (11 \, d e^{2} x^{2} + 8 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.60595, size = 556, normalized size = 4.15 \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 = 1.17322, size = 398, normalized size = 2.97 \begin{align*} -\arcsin \left (\frac{x e}{d}\right ) e^{4} \mathrm{sgn}\left (d\right ) + \frac{x^{4}{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{8}}{x} + \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{6}}{x^{2}} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{4}}{x^{3}} + e^{10}\right )} e^{2}}{64 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4}} - \frac{1}{64} \,{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{26}}{x} + \frac{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{24}}{x^{2}} + \frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{22}}{x^{3}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{20}}{x^{4}}\right )} e^{\left (-24\right )} + \frac{13}{8} \, e^{4} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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