Optimal. Leaf size=57 \[ -\frac{d^2 2^{p+1} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-1,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]
[Out]
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Rubi [A] time = 0.0554342, antiderivative size = 56, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ d x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{p+1}}{2 e (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(1 - (e^2*x^2)/d^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 31.2582, size = 80, normalized size = 1.4 \[ - \frac{2 d^{3} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{- p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{1}{2} - \frac{e x}{2 d}} \right )}}{e \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(1-e**2*x**2/d**2)**p,x)
[Out]
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Mathematica [A] time = 0.0581515, size = 85, normalized size = 1.49 \[ \frac{2 d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p-d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )}{2 e (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(1 - (e^2*x^2)/d^2)^p,x]
[Out]
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Maple [A] time = 0.059, size = 47, normalized size = 0.8 \[{\frac{e{x}^{2}}{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+dx{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(1-e^2*x^2/d^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2/d^2 + 1)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x + d\right )} \left (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2/d^2 + 1)^p,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.47406, size = 78, normalized size = 1.37 \[ d x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: e^{2} = 0 \\- \frac{d^{2} \left (\begin{cases} \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )} & \text{otherwise} \end{cases}\right )}{2 e^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(1-e**2*x**2/d**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2/d^2 + 1)^p,x, algorithm="giac")
[Out]