Optimal. Leaf size=177 \[ \frac{127 d^2 (d-e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}-\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.723625, antiderivative size = 177, 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 \[ \frac{127 d^2 (d-e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{3 d \sqrt{d^2-e^2 x^2}}{e^6}+\frac{13 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^6}-\frac{x \sqrt{d^2-e^2 x^2}}{2 e^5}+\frac{d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 55.2287, size = 156, normalized size = 0.88 \[ \frac{d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{6} \left (d + e x\right )^{3}} - \frac{23 d^{3} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6} \left (d + e x\right )^{2}} + \frac{13 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{6}} + \frac{127 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6} \left (d + e x\right )} + \frac{3 d \sqrt{d^{2} - e^{2} x^{2}}}{e^{6}} - \frac{x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.149257, size = 98, normalized size = 0.55 \[ \frac{195 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{\sqrt{d^2-e^2 x^2} \left (304 d^4+717 d^3 e x+479 d^2 e^2 x^2+45 d e^3 x^3-15 e^4 x^4\right )}{(d+e x)^3}}{30 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Maple [A] time = 0.025, size = 212, normalized size = 1.2 \[ -{\frac{x}{2\,{e}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{13\,{d}^{2}}{2\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+3\,{\frac{d\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{e}^{6}}}+{\frac{127\,{d}^{2}}{15\,{e}^{7}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}-{\frac{23\,{d}^{3}}{15\,{e}^{8}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{{d}^{4}}{5\,{e}^{9}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(e*x+d)^3/(-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(x^5/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.304596, size = 730, normalized size = 4.12 \[ -\frac{15 \, e^{9} x^{9} - 120 \, d e^{8} x^{8} - 678 \, d^{2} e^{7} x^{7} + 210 \, d^{3} e^{6} x^{6} + 5421 \, d^{4} e^{5} x^{5} + 6500 \, d^{5} e^{4} x^{4} - 2860 \, d^{6} e^{3} x^{3} - 7800 \, d^{7} e^{2} x^{2} - 3120 \, d^{8} e x + 390 \,{\left (d^{2} e^{7} x^{7} + 7 \, d^{3} e^{6} x^{6} + 3 \, d^{4} e^{5} x^{5} - 31 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} + 12 \, d^{7} e^{2} x^{2} + 40 \, d^{8} e x + 16 \, d^{9} -{\left (d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} - 19 \, d^{4} e^{4} x^{4} - 20 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} + 40 \, d^{7} e x + 16 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{8} x^{8} + 15 \, d e^{7} x^{7} - 535 \, d^{2} e^{6} x^{6} - 2821 \, d^{3} e^{5} x^{5} - 2600 \, d^{4} e^{4} x^{4} + 4420 \, d^{5} e^{3} x^{3} + 7800 \, d^{6} e^{2} x^{2} + 3120 \, d^{7} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 3 \, d^{2} e^{11} x^{5} - 31 \, d^{3} e^{10} x^{4} - 40 \, d^{4} e^{9} x^{3} + 12 \, d^{5} e^{8} x^{2} + 40 \, d^{6} e^{7} x + 16 \, d^{7} e^{6} -{\left (e^{12} x^{6} - 2 \, d e^{11} x^{5} - 19 \, d^{2} e^{10} x^{4} - 20 \, d^{3} e^{9} x^{3} + 20 \, d^{4} e^{8} x^{2} + 40 \, d^{5} e^{7} x + 16 \, d^{6} e^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="giac")
[Out]