Optimal. Leaf size=104 \[ e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d (p+1)} \]
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Rubi [A] time = 0.126423, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ e x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{e^2 x^2}{d^2}\right )}{2 d (p+1)} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(d^2 - e^2*x^2)^p)/x,x]
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Rubi in Sympy [A] time = 30.4703, size = 82, normalized size = 0.79 \[ e x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 d \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**p/x,x)
[Out]
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Mathematica [A] time = 0.0755646, size = 104, normalized size = 1. \[ \frac{1}{2} \left (d^2-e^2 x^2\right )^p \left (\frac{d \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+2 e x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(d^2 - e^2*x^2)^p)/x,x]
[Out]
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Maple [F] time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(-e^2*x^2+d^2)^p/x,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p/x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.95561, size = 78, normalized size = 0.75 \[ - \frac{d e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (- p + 1\right )} + d^{2 p} e x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(-e**2*x**2+d**2)**p/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p/x,x, algorithm="giac")
[Out]