Optimal. Leaf size=209 \[ -\frac{14 e^2 \sqrt{d^2-e^2 x^2}}{3 d^8 x}+\frac{2 e^3 (45 d+53 e x)}{15 d^8 \sqrt{d^2-e^2 x^2}}-\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^8}-\frac{e \sqrt{d^2-e^2 x^2}}{d^7 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^6 x^3}+\frac{e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
[Out]
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Rubi [A] time = 0.722024, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{14 e^2 \sqrt{d^2-e^2 x^2}}{3 d^8 x}+\frac{2 e^3 (45 d+53 e x)}{15 d^8 \sqrt{d^2-e^2 x^2}}-\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^8}-\frac{e \sqrt{d^2-e^2 x^2}}{d^7 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^6 x^3}+\frac{e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
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Rubi in Sympy [A] time = 84.2737, size = 226, normalized size = 1.08 \[ - \frac{1}{3 d^{4} x^{3} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{e^{3}}{5 d^{5} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 e}{d^{5} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{23 e^{3}}{15 d^{6} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{13 e^{2}}{3 d^{6} x \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 e^{3}}{d^{7} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{3 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{7} x^{2}} + \frac{176 e^{4} x}{15 d^{8} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{7 e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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Mathematica [A] time = 0.126912, size = 138, normalized size = 0.66 \[ \frac{-105 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (5 d^6+5 d^5 e x+40 d^4 e^2 x^2-246 d^3 e^3 x^3+122 d^2 e^4 x^4+247 d e^5 x^5-176 e^6 x^6\right )}{x^3 (e x-d)^3 (d+e x)}+105 e^3 \log (x)}{15 d^8} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
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Maple [A] time = 0.025, size = 249, normalized size = 1.2 \[ -{\frac{1}{3\,{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{11\,{e}^{2}}{3\,{d}^{2}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{22\,{e}^{4}x}{5\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{88\,{e}^{4}x}{15\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{176\,{e}^{4}x}{15\,{d}^{8}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{e}{d{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{3}}{5\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{3}}{3\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+7\,{\frac{{e}^{3}}{{d}^{7}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-7\,{\frac{{e}^{3}}{{d}^{7}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/x^4/(-e^2*x^2+d^2)^(7/2),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/((-e^2*x^2 + d^2)^(7/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284712, size = 953, normalized size = 4.56 \[ -\frac{176 \, e^{12} x^{12} - 943 \, d e^{11} x^{11} - 1898 \, d^{2} e^{10} x^{10} + 8404 \, d^{3} e^{9} x^{9} + 1788 \, d^{4} e^{8} x^{8} - 19305 \, d^{5} e^{7} x^{7} + 4075 \, d^{6} e^{6} x^{6} + 16090 \, d^{7} e^{5} x^{5} - 5350 \, d^{8} e^{4} x^{4} - 4400 \, d^{9} e^{3} x^{3} + 1040 \, d^{10} e^{2} x^{2} + 160 \, d^{11} e x + 160 \, d^{12} - 105 \,{\left (6 \, d e^{11} x^{11} - 12 \, d^{2} e^{10} x^{10} - 32 \, d^{3} e^{9} x^{9} + 76 \, d^{4} e^{8} x^{8} + 26 \, d^{5} e^{7} x^{7} - 128 \, d^{6} e^{6} x^{6} + 32 \, d^{7} e^{5} x^{5} + 64 \, d^{8} e^{4} x^{4} - 32 \, d^{9} e^{3} x^{3} -{\left (e^{11} x^{11} - 2 \, d e^{10} x^{10} - 17 \, d^{2} e^{9} x^{9} + 36 \, d^{3} e^{8} x^{8} + 30 \, d^{4} e^{7} x^{7} - 96 \, d^{5} e^{6} x^{6} + 16 \, d^{6} e^{5} x^{5} + 64 \, d^{7} e^{4} x^{4} - 32 \, d^{8} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 2 \,{\left (58 \, e^{11} x^{11} + 412 \, d e^{10} x^{10} - 1727 \, d^{2} e^{9} x^{9} - 1094 \, d^{3} e^{8} x^{8} + 6430 \, d^{4} e^{7} x^{7} - 920 \, d^{5} e^{6} x^{6} - 6975 \, d^{6} e^{5} x^{5} + 2385 \, d^{7} e^{4} x^{4} + 2160 \, d^{8} e^{3} x^{3} - 560 \, d^{9} e^{2} x^{2} - 80 \, d^{10} e x - 80 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (6 \, d^{9} e^{8} x^{11} - 12 \, d^{10} e^{7} x^{10} - 32 \, d^{11} e^{6} x^{9} + 76 \, d^{12} e^{5} x^{8} + 26 \, d^{13} e^{4} x^{7} - 128 \, d^{14} e^{3} x^{6} + 32 \, d^{15} e^{2} x^{5} + 64 \, d^{16} e x^{4} - 32 \, d^{17} x^{3} -{\left (d^{8} e^{8} x^{11} - 2 \, d^{9} e^{7} x^{10} - 17 \, d^{10} e^{6} x^{9} + 36 \, d^{11} e^{5} x^{8} + 30 \, d^{12} e^{4} x^{7} - 96 \, d^{13} e^{3} x^{6} + 16 \, d^{14} e^{2} x^{5} + 64 \, d^{15} e x^{4} - 32 \, d^{16} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/((-e^2*x^2 + d^2)^(7/2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/x**4/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.303236, size = 439, normalized size = 2.1 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (2 \, x{\left (\frac{53 \, x e^{8}}{d^{8}} + \frac{45 \, e^{7}}{d^{7}}\right )} - \frac{235 \, e^{6}}{d^{6}}\right )} x - \frac{200 \, e^{5}}{d^{5}}\right )} x + \frac{135 \, e^{4}}{d^{4}}\right )} x + \frac{116 \, e^{3}}{d^{3}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} + \frac{x^{3}{\left (\frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{x} + \frac{57 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} + e^{8}\right )} e}{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8}} - \frac{7 \, e^{3}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{8}} - \frac{{\left (\frac{57 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{16}}{x} + \frac{6 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{14}}{x^{2}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{12}}{x^{3}}\right )} e^{\left (-15\right )}}{24 \, d^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/((-e^2*x^2 + d^2)^(7/2)*x^4),x, algorithm="giac")
[Out]