Optimal. Leaf size=207 \[ \frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{85}{16} d^6 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d^6 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.312916, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1807, 815, 844, 217, 203, 266, 63, 208} \[ \frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{85}{16} d^6 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d^6 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 815
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-6 d^4 e-d^3 e^2 x-2 d^2 e^3 x^2\right )}{x^2} \, dx}{2 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{\int \frac{\left (d^5 e^2-34 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{2 d^4}\\ &=\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac{\int \frac{\left (-6 d^7 e^4+170 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4 e^2}\\ &=\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{\int \frac{\left (24 d^9 e^6-510 d^8 e^7 x\right ) \sqrt{d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^4}\\ &=\frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac{\int \frac{-48 d^{11} e^8+510 d^{10} e^9 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{96 d^4 e^6}\\ &=\frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{1}{2} \left (d^7 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-\frac{1}{16} \left (85 d^6 e^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac{1}{4} \left (d^7 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-\frac{1}{16} \left (85 d^6 e^3\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{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac{85}{16} d^6 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d^7 \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{1}{16} d^4 e^2 (8 d-85 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{24} d^2 e^2 (4 d-85 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac{1}{30} e^2 (3 d-85 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac{85}{16} d^6 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d^6 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [C] time = 0.686788, size = 259, normalized size = 1.25 \[ -\frac{e \left (5040 d^9 \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right )+e x \left (240 \left (d^2-e^2 x^2\right )^4 \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )-7 d \left (-1632 d^5 e^2 x^2-295 d^4 e^3 x^3+672 d^3 e^4 x^4+170 d^2 e^5 x^5+75 d^7 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )-720 d^6 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )+165 d^6 e x+1104 d^7-144 d e^6 x^6-40 e^7 x^7\right )\right )\right )}{1680 d x \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 252, normalized size = 1.2 \begin{align*} -{\frac{17\,{e}^{3}x}{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{85\,{e}^{3}{d}^{2}x}{24} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{85\,{e}^{3}{d}^{4}x}{16}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{85\,{e}^{3}{d}^{6}}{16}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d{e}^{2}}{10} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}{e}^{2}}{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}{e}^{2}}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{{d}^{7}{e}^{2}}{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{d}{2\,{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-3\,{\frac{e \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{7/2}}{x}} \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 = 2.02622, size = 392, normalized size = 1.89 \begin{align*} \frac{2550 \, d^{6} e^{2} x^{2} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 120 \, d^{6} e^{2} x^{2} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 544 \, d^{6} e^{2} x^{2} +{\left (40 \, e^{7} x^{7} + 144 \, d e^{6} x^{6} + 50 \, d^{2} e^{5} x^{5} - 448 \, d^{3} e^{4} x^{4} - 645 \, d^{4} e^{3} x^{3} + 544 \, d^{5} e^{2} x^{2} - 720 \, d^{6} e x - 120 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 25.1735, size = 1073, normalized size = 5.18 \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 [A] time = 1.18552, size = 354, normalized size = 1.71 \begin{align*} -\frac{85}{16} \, d^{6} \arcsin \left (\frac{x e}{d}\right ) e^{2} \mathrm{sgn}\left (d\right ) - \frac{1}{2} \, d^{6} e^{2} \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 ) - \frac{1}{8} \,{\left (\frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{6}}{x^{2}}\right )} e^{\left (-8\right )} + \frac{1}{240} \,{\left (544 \, d^{5} e^{2} -{\left (645 \, d^{4} e^{3} + 2 \,{\left (224 \, d^{3} e^{4} -{\left (25 \, d^{2} e^{5} + 4 \,{\left (5 \, x e^{7} + 18 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} + \frac{{\left (d^{6} e^{6} + \frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{4}}{x}\right )} x^{2}}{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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