Optimal. Leaf size=100 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]
[Out]
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Rubi [A] time = 0.120878, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d+e x)^2}-\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 13.2549, size = 82, normalized size = 0.82 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d e \left (d + e x\right )^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{2} e \left (d + e x\right )^{2}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} e \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0393493, size = 52, normalized size = 0.52 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (7 d^2+6 d e x+2 e^2 x^2\right )}{15 d^3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^3*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Maple [A] time = 0.01, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{2}{x}^{2}+6\,dex+7\,{d}^{2} \right ) }{15\,e{d}^{3} \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(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(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225133, size = 273, normalized size = 2.73 \[ -\frac{9 \, e^{4} x^{5} + 35 \, d e^{3} x^{4} + 20 \, d^{2} e^{2} x^{3} - 60 \, d^{3} e x^{2} - 60 \, d^{4} x - 5 \,{\left (e^{3} x^{4} - 2 \, d e^{2} x^{3} - 12 \, d^{2} e x^{2} - 12 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{5} + 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} - 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x - 4 \, d^{8} -{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^3),x, algorithm="giac")
[Out]