Optimal. Leaf size=113 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac{5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac{15 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
[Out]
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Rubi [A] time = 0.115866, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac{5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac{15 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 23.0227, size = 95, normalized size = 0.84 \[ - \frac{15 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{15 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} - \frac{5 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.0997608, size = 75, normalized size = 0.66 \[ \sqrt{d^2-e^2 x^2} \left (-\frac{8 d^2}{e (d+e x)}-\frac{4 d}{e}+\frac{x}{2}\right )-\frac{15 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.008, size = 284, normalized size = 2.5 \[ -{\frac{1}{{e}^{5}d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}-3\,{\frac{1}{{e}^{4}{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-4\,{\frac{1}{{e}^{3}{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{7/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}-4\,{\frac{1}{e{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{5/2}}-5\,{\frac{x}{{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2}}-{\frac{15\,x}{2}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{15\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(5/2)/(e*x+d)^4,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^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289165, size = 366, normalized size = 3.24 \[ \frac{e^{5} x^{5} - 10 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 18 \, d^{3} e^{2} x^{2} + 68 \, d^{4} e x + 30 \,{\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} - 2 \, d^{4} e x - 4 \, d^{5} -{\left (d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{4} x^{4} - 5 \, d e^{3} x^{3} - 18 \, d^{2} e^{2} x^{2} - 68 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} - 2 \, d^{2} e^{2} x - 4 \, d^{3} e -{\left (e^{3} x^{2} - 2 \, d e^{2} x - 4 \, d^{2} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(5/2)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.301862, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/(e*x + d)^4,x, algorithm="giac")
[Out]