Optimal. Leaf size=148 \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.454123, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 77.2055, size = 182, normalized size = 1.23 \[ - \frac{d^{5}}{5 e^{7} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{d^{4}}{e^{7} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{d^{3} x}{15 e^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{3 d^{2}}{e^{7} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 d x^{3}}{3 e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{13 d x}{15 e^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{7}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.161397, size = 115, normalized size = 0.78 \[ -\frac{15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (48 d^5+33 d^4 e x-87 d^3 e^2 x^2-52 d^2 e^3 x^3+38 d e^4 x^4+15 e^5 x^5\right )}{(d-e x)^2 (d+e x)^3}}{15 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]
[Out]
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Maple [B] time = 0.045, size = 288, normalized size = 2. \[ -{\frac{{x}^{4}}{{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{{d}^{2}{x}^{2}}{{e}^{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-3\,{\frac{{d}^{4}}{{e}^{7} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{{d}^{5}}{5\,{e}^{8}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}x}{15\,{e}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,dx}{15\,{e}^{6}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{2\,{d}^{3}x}{3\,{e}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,dx}{3\,{e}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{d{x}^{3}}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{d}{{e}^{6}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(e*x+d)/(-e^2*x^2+d^2)^(5/2),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(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.306824, size = 838, normalized size = 5.66 \[ \frac{27 \, d e^{9} x^{9} + 142 \, d^{2} e^{8} x^{8} + 112 \, d^{3} e^{7} x^{7} - 523 \, d^{4} e^{6} x^{6} - 523 \, d^{5} e^{5} x^{5} + 620 \, d^{6} e^{4} x^{4} + 620 \, d^{7} e^{3} x^{3} - 240 \, d^{8} e^{2} x^{2} - 240 \, d^{9} e x + 30 \,{\left (d e^{9} x^{9} + d^{2} e^{8} x^{8} - 14 \, d^{3} e^{7} x^{7} - 14 \, d^{4} e^{6} x^{6} + 41 \, d^{5} e^{5} x^{5} + 41 \, d^{6} e^{4} x^{4} - 44 \, d^{7} e^{3} x^{3} - 44 \, d^{8} e^{2} x^{2} + 16 \, d^{9} e x + 16 \, d^{10} +{\left (5 \, d^{2} e^{7} x^{7} + 5 \, d^{3} e^{6} x^{6} - 25 \, d^{4} e^{5} x^{5} - 25 \, d^{5} e^{4} x^{4} + 36 \, d^{6} e^{3} x^{3} + 36 \, d^{7} e^{2} x^{2} - 16 \, d^{8} e x - 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (15 \, e^{9} x^{9} + 38 \, d e^{8} x^{8} + 8 \, d^{2} e^{7} x^{7} - 303 \, d^{3} e^{6} x^{6} - 303 \, d^{4} e^{5} x^{5} + 500 \, d^{5} e^{4} x^{4} + 500 \, d^{6} e^{3} x^{3} - 240 \, d^{7} e^{2} x^{2} - 240 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{16} x^{9} + d e^{15} x^{8} - 14 \, d^{2} e^{14} x^{7} - 14 \, d^{3} e^{13} x^{6} + 41 \, d^{4} e^{12} x^{5} + 41 \, d^{5} e^{11} x^{4} - 44 \, d^{6} e^{10} x^{3} - 44 \, d^{7} e^{9} x^{2} + 16 \, d^{8} e^{8} x + 16 \, d^{9} e^{7} +{\left (5 \, d e^{14} x^{7} + 5 \, d^{2} e^{13} x^{6} - 25 \, d^{3} e^{12} x^{5} - 25 \, d^{4} e^{11} x^{4} + 36 \, d^{5} e^{10} x^{3} + 36 \, d^{6} e^{9} x^{2} - 16 \, d^{7} e^{8} x - 16 \, d^{8} e^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="giac")
[Out]