Optimal. Leaf size=169 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2} \]
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Rubi [A] time = 0.499527, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(x^6*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 42.136, size = 148, normalized size = 0.88 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 x^{5}} - \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{2 d x^{4}} - \frac{3 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} x^{3}} - \frac{3 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 d^{3} x^{2}} - \frac{3 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{4 d^{4}} - \frac{6 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/x**6/(-e**2*x**2+d**2)**(1/2),x)
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Mathematica [A] time = 0.171842, size = 106, normalized size = 0.63 \[ -\frac{15 e^5 x^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (4 d^4+10 d^3 e x+12 d^2 e^2 x^2+15 d e^3 x^3+24 e^4 x^4\right )-15 e^5 x^5 \log (x)}{20 d^4 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(x^6*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Maple [A] time = 0.022, size = 164, normalized size = 1. \[ -{\frac{1}{5\,{x}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{2}}{5\,{d}^{2}{x}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{6\,{e}^{4}}{5\,{d}^{4}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{e}{2\,d{x}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{3}}{4\,{d}^{3}{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{5}}{4\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/x^6/(-e^2*x^2+d^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(sqrt(-e^2*x^2 + d^2)*x^6),x, algorithm="maxima")
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Fricas [A] time = 0.295339, size = 533, normalized size = 3.15 \[ -\frac{24 \, e^{10} x^{10} + 15 \, d e^{9} x^{9} - 300 \, d^{2} e^{8} x^{8} - 185 \, d^{3} e^{7} x^{7} + 520 \, d^{4} e^{6} x^{6} + 290 \, d^{5} e^{5} x^{5} - 100 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 80 \, d^{8} e^{2} x^{2} - 160 \, d^{9} e x - 64 \, d^{10} - 15 \,{\left (5 \, d e^{9} x^{9} - 20 \, d^{3} e^{7} x^{7} + 16 \, d^{5} e^{5} x^{5} -{\left (e^{9} x^{9} - 12 \, d^{2} e^{7} x^{7} + 16 \, d^{4} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (120 \, d e^{8} x^{8} + 75 \, d^{2} e^{7} x^{7} - 420 \, d^{3} e^{6} x^{6} - 250 \, d^{4} e^{5} x^{5} + 164 \, d^{5} e^{4} x^{4} + 40 \, d^{6} e^{3} x^{3} + 112 \, d^{7} e^{2} x^{2} + 160 \, d^{8} e x + 64 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{20 \,{\left (5 \, d^{5} e^{4} x^{9} - 20 \, d^{7} e^{2} x^{7} + 16 \, d^{9} x^{5} -{\left (d^{4} e^{4} x^{9} - 12 \, d^{6} e^{2} x^{7} + 16 \, d^{8} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(sqrt(-e^2*x^2 + d^2)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 22.0968, size = 510, normalized size = 3.02 \[ d^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{5 d^{2} x^{4}} - \frac{4 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{4} x^{2}} - \frac{8 e^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{6}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{5 d^{2} x^{4}} - \frac{4 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{4} x^{2}} - \frac{8 i e^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{6}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{1}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e}{8 d^{2} x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e^{3}}{8 d^{4} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{3 e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e}{8 d^{2} x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e^{3}}{8 d^{4} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{3 i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac{2 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac{2 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/x**6/(-e**2*x**2+d**2)**(1/2),x)
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GIAC/XCAS [A] time = 0.298571, size = 493, normalized size = 2.92 \[ \frac{x^{5}{\left (\frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{10}}{x} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{8}}{x^{2}} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{6}}{x^{3}} + \frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{4}}{x^{4}} + e^{12}\right )} e^{3}}{160 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4}} - \frac{3 \, e^{5}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{4 \, d^{4}} - \frac{{\left (\frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{38}}{x} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{36}}{x^{2}} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{34}}{x^{3}} + \frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{16} e^{32}}{x^{4}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{16} e^{30}}{x^{5}}\right )} e^{\left (-35\right )}}{160 \, d^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(sqrt(-e^2*x^2 + d^2)*x^6),x, algorithm="giac")
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