Optimal. Leaf size=97 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.144726, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 d^2 e^2 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e^2 (d+e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[x/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 13.6547, size = 78, normalized size = 0.8 \[ \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 e^{2} \left (d + e x\right )^{3}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d e^{2} \left (d + e x\right )^{2}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} e^{2} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0390686, size = 49, normalized size = 0.51 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (d^2+3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Maple [A] time = 0.011, size = 52, normalized size = 0.5 \[ -{\frac{ \left ( -ex+d \right ) \left ({e}^{2}{x}^{2}+3\,dex+{d}^{2} \right ) }{5\,{e}^{2}{d}^{2} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(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/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281395, size = 238, normalized size = 2.45 \[ -\frac{2 \, e^{3} x^{5} + 5 \, d e^{2} x^{4} - 5 \, d^{2} e x^{3} - 10 \, d^{3} x^{2} + 5 \,{\left (d e x^{3} + 2 \, d^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{2} e^{5} x^{5} + 5 \, d^{3} e^{4} x^{4} + 5 \, d^{4} e^{3} x^{3} - 5 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7} -{\left (d^{2} e^{4} x^{4} - 7 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x - 4 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(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. \[ \int \frac{x}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(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/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="giac")
[Out]