Optimal. Leaf size=206 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac{e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{15}{16} d e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{3 e^6 (16 d-5 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]
[Out]
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Rubi [A] time = 0.621374, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^6}-\frac{e^2 (24 d+5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{40 x^5}-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{15}{16} d e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-\frac{3 e^6 (16 d-5 e x) \sqrt{d^2-e^2 x^2}}{16 x}+\frac{e^4 (16 d+5 e x) \left (d^2-e^2 x^2\right )^{3/2}}{16 x^3} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]
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Rubi in Sympy [A] time = 97.3131, size = 245, normalized size = 1.19 \[ - \frac{d^{7} \sqrt{d^{2} - e^{2} x^{2}}}{7 x^{7}} - \frac{d^{6} e \sqrt{d^{2} - e^{2} x^{2}}}{2 x^{6}} - \frac{6 d^{5} e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{35 x^{5}} + \frac{11 d^{4} e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{4}} + \frac{62 d^{3} e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{35 x^{3}} - \frac{15 d^{2} e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{16 x^{2}} - 3 d e^{7} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \frac{15 d e^{7} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16} - \frac{156 d e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{35 x} + e^{7} \sqrt{d^{2} - e^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**8,x)
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Mathematica [A] time = 0.273002, size = 161, normalized size = 0.78 \[ -\frac{15}{16} d e^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 d e^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{\sqrt{d^2-e^2 x^2} \left (80 d^7+280 d^6 e x+96 d^5 e^2 x^2-770 d^4 e^3 x^3-992 d^3 e^4 x^4+525 d^2 e^5 x^5+2496 d e^6 x^6-560 e^7 x^7\right )}{560 x^7}+\frac{15}{16} d e^7 \log (x) \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^8,x]
[Out]
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Maple [B] time = 0.062, size = 377, normalized size = 1.8 \[ -{\frac{d}{7\,{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{3}}{8\,{d}^{2}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{5}}{16\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{e}^{7}}{16\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{7}}{16\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{7}}{16}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{15\,{e}^{7}{d}^{2}}{16}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{3\,{e}^{2}}{5\,d{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{e}^{4}}{5\,{d}^{3}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{6}}{5\,{d}^{5}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{8}x}{5\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{e}^{8}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{3}}}-3\,{\frac{{e}^{8}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{d}}-3\,{\frac{d{e}^{8}}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }-{\frac{e}{2\,{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^8,x)
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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^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.302656, size = 988, normalized size = 4.8 \[ -\frac{3920 \, d e^{15} x^{15} - 19968 \, d^{2} e^{14} x^{14} - 35560 \, d^{3} e^{13} x^{13} + 227584 \, d^{4} e^{12} x^{12} + 115080 \, d^{5} e^{11} x^{11} - 766976 \, d^{6} e^{10} x^{10} - 248640 \, d^{7} e^{9} x^{9} + 1076352 \, d^{8} e^{8} x^{8} + 402080 \, d^{9} e^{7} x^{7} - 656000 \, d^{10} e^{6} x^{6} - 389760 \, d^{11} e^{5} x^{5} + 135936 \, d^{12} e^{4} x^{4} + 188160 \, d^{13} e^{3} x^{3} + 13312 \, d^{14} e^{2} x^{2} - 35840 \, d^{15} e x - 10240 \, d^{16} - 3360 \,{\left (d e^{15} x^{15} - 32 \, d^{3} e^{13} x^{13} + 160 \, d^{5} e^{11} x^{11} - 256 \, d^{7} e^{9} x^{9} + 128 \, d^{9} e^{7} x^{7} + 8 \,{\left (d^{2} e^{13} x^{13} - 10 \, d^{4} e^{11} x^{11} + 24 \, d^{6} e^{9} x^{9} - 16 \, d^{8} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 525 \,{\left (d e^{15} x^{15} - 32 \, d^{3} e^{13} x^{13} + 160 \, d^{5} e^{11} x^{11} - 256 \, d^{7} e^{9} x^{9} + 128 \, d^{9} e^{7} x^{7} + 8 \,{\left (d^{2} e^{13} x^{13} - 10 \, d^{4} e^{11} x^{11} + 24 \, d^{6} e^{9} x^{9} - 16 \, d^{8} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (560 \, e^{15} x^{15} - 2496 \, d e^{14} x^{14} - 13965 \, d^{2} e^{13} x^{13} + 80864 \, d^{3} e^{12} x^{12} + 62370 \, d^{4} e^{11} x^{11} - 431200 \, d^{5} e^{10} x^{10} - 144760 \, d^{6} e^{9} x^{9} + 800688 \, d^{7} e^{8} x^{8} + 266560 \, d^{8} e^{7} x^{7} - 586240 \, d^{9} e^{6} x^{6} - 309120 \, d^{10} e^{5} x^{5} + 138752 \, d^{11} e^{4} x^{4} + 170240 \, d^{12} e^{3} x^{3} + 8192 \, d^{13} e^{2} x^{2} - 35840 \, d^{14} e x - 10240 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{560 \,{\left (e^{8} x^{15} - 32 \, d^{2} e^{6} x^{13} + 160 \, d^{4} e^{4} x^{11} - 256 \, d^{6} e^{2} x^{9} + 128 \, d^{8} x^{7} + 8 \,{\left (d e^{6} x^{13} - 10 \, d^{3} e^{4} x^{11} + 24 \, d^{5} e^{2} x^{9} - 16 \, d^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="fricas")
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Sympy [A] time = 60.842, size = 1513, normalized size = 7.34 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**8,x)
[Out]
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GIAC/XCAS [A] time = 0.298732, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3/x^8,x, algorithm="giac")
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