Optimal. Leaf size=83 \[ d x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e (p+1)} \]
[Out]
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Rubi [A] time = 0.0567295, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ d x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(d^2 - e^2*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 25.1072, size = 66, normalized size = 0.8 \[ - \frac{2 d \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{e \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**p,x)
[Out]
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Mathematica [A] time = 0.0719559, size = 116, normalized size = 1.4 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p-d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )\right )}{2 e (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(d^2 - e^2*x^2)^p,x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(-e^2*x^2+d^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.31458, size = 82, normalized size = 0.99 \[ d d^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(-e**2*x**2+d**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="giac")
[Out]