Optimal. Leaf size=133 \[ -\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.168746, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 18.8941, size = 110, normalized size = 0.83 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{7 d e \left (d + e x\right )^{4}} - \frac{3 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{2} e \left (d + e x\right )^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{3} e \left (d + e x\right )^{2}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{35 d^{4} e \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0447698, size = 63, normalized size = 0.47 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (12 d^3+13 d^2 e x+8 d e^2 x^2+2 e^3 x^3\right )}{35 d^4 e (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^4*Sqrt[d^2 - e^2*x^2]),x]
[Out]
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Maple [A] time = 0.011, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{3}{x}^{3}+8\,{e}^{2}{x}^{2}d+13\,x{d}^{2}e+12\,{d}^{3} \right ) }{35\,e{d}^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^4/(-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)^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228687, size = 377, normalized size = 2.83 \[ -\frac{10 \, e^{6} x^{7} - 14 \, d e^{5} x^{6} - 189 \, d^{2} e^{4} x^{5} - 315 \, d^{3} e^{3} x^{4} + 420 \, d^{5} e x^{2} + 280 \, d^{6} x + 7 \,{\left (2 \, e^{5} x^{6} + 12 \, d e^{4} x^{5} + 15 \, d^{2} e^{3} x^{4} - 20 \, d^{3} e^{2} x^{3} - 60 \, d^{4} e x^{2} - 40 \, d^{5} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (d^{4} e^{7} x^{7} - 14 \, d^{6} e^{5} x^{5} - 28 \, d^{7} e^{4} x^{4} - 7 \, d^{8} e^{3} x^{3} + 28 \, d^{9} e^{2} x^{2} + 28 \, d^{10} e x + 8 \, d^{11} +{\left (d^{4} e^{6} x^{6} + 7 \, d^{5} e^{5} x^{5} + 11 \, d^{6} e^{4} x^{4} - 7 \, d^{7} e^{3} x^{3} - 32 \, d^{8} e^{2} x^{2} - 28 \, d^{9} e x - 8 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239057, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)^4),x, algorithm="giac")
[Out]