Optimal. Leaf size=121 \[ -\frac{17 d^2 (d+e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (15 d+13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}+\frac{d^3 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]
[Out]
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Rubi [A] time = 0.316319, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{17 d^2 (d+e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (15 d+13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}+\frac{d^3 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 58.0747, size = 121, normalized size = 1. \[ \frac{d^{3}}{5 e^{5} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{17 d^{2}}{15 e^{5} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 d}{e^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{26 x}{15 e^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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Mathematica [A] time = 0.118366, size = 106, normalized size = 0.88 \[ \frac{15 (d+e x) (d-e x)^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-16 d^3+17 d^2 e x+22 d e^2 x^2-26 e^3 x^3\right )}{15 e^5 (e x-d)^3 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Maple [B] time = 0.023, size = 236, normalized size = 2. \[{\frac{{d}^{2}{x}^{3}}{2\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{3\,{d}^{4}x}{10\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{{d}^{2}x}{10\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{6\,x}{5\,{e}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{x}^{5}}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{3}}{3\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+2\,{\frac{d{x}^{4}}{e \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{8\,{d}^{3}{x}^{2}}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{16\,{d}^{5}}{15\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x)
[Out]
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Maxima [A] time = 0.80027, size = 420, normalized size = 3.47 \[ \frac{1}{15} \, e^{2} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{2 \, d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{8 \, d^{3} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{3 \, d^{4} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{16 \, d^{5}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{5}} + \frac{11 \, d^{2} x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}} - \frac{4 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{4}} - \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{4}} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280343, size = 548, normalized size = 4.53 \[ \frac{16 \, e^{6} x^{6} + 46 \, d e^{5} x^{5} - 130 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} + 120 \, d^{4} e^{2} x^{2} - 60 \, d^{5} e x + 30 \,{\left (e^{6} x^{6} - 2 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} + 10 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} - 8 \, d^{5} e x + 4 \, d^{6} +{\left (3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (26 \, e^{5} x^{5} - 70 \, d e^{4} x^{4} - 25 \, d^{2} e^{3} x^{3} + 120 \, d^{3} e^{2} x^{2} - 60 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{11} x^{6} - 2 \, d e^{10} x^{5} - 4 \, d^{2} e^{9} x^{4} + 10 \, d^{3} e^{8} x^{3} - d^{4} e^{7} x^{2} - 8 \, d^{5} e^{6} x + 4 \, d^{6} e^{5} +{\left (3 \, d e^{9} x^{4} - 6 \, d^{2} e^{8} x^{3} - d^{3} e^{7} x^{2} + 8 \, d^{4} e^{6} x - 4 \, d^{5} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.292406, size = 128, normalized size = 1.06 \[ -\arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )}{\rm sign}\left (d\right ) - \frac{{\left (16 \, d^{5} e^{\left (-5\right )} +{\left (15 \, d^{4} e^{\left (-4\right )} -{\left (40 \, d^{3} e^{\left (-3\right )} +{\left (35 \, d^{2} e^{\left (-2\right )} - 2 \,{\left (15 \, d e^{\left (-1\right )} + 13 \, x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
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, algorithm="giac")
[Out]