Optimal. Leaf size=57 \[ -\frac{2^{p-2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d e (p+1)} \]
[Out]
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Rubi [A] time = 0.110468, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{2^{p-2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d e (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 30.5739, size = 75, normalized size = 1.32 \[ - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{- p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{1}{2} - \frac{e x}{2 d}} \right )}}{4 e \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-e**2*x**2/d**2)**p/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.106841, size = 76, normalized size = 1.33 \[ -\frac{2^{p-2} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \left (1-\frac{e^2 x^2}{d^2}\right )^p \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{d^2 e (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - (e^2*x^2)/d^2)^p/(d + e*x)^2,x]
[Out]
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Maple [F] time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( 1-{\frac{{e}^{2}{x}^{2}}{{d}^{2}}} \right ) ^{p}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-e^2*x^2/d^2)^p/(e*x+d)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (-1 + \frac{e x}{d}\right ) \left (1 + \frac{e x}{d}\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-e**2*x**2/d**2)**p/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2/d^2 + 1)^p/(e*x + d)^2,x, algorithm="giac")
[Out]