Optimal. Leaf size=211 \[ \frac{e^2 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (2-p)}+\frac{e^2 (11-p) \left (d^2-e^2 x^2\right )^{p-3}}{2 (3-p)}+\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{x}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{2 x^2}-\frac{8 e^3 (4-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^7} \]
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Rubi [A] time = 0.634138, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{e^2 (10-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{2 d^2 (2-p)}+\frac{e^2 (11-p) \left (d^2-e^2 x^2\right )^{p-3}}{2 (3-p)}+\frac{4 d e \left (d^2-e^2 x^2\right )^{p-3}}{x}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{2 x^2}-\frac{8 e^3 (4-p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^7} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 88.1931, size = 204, normalized size = 0.97 \[ \frac{3 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 1, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{- p + 3} + \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}{{}_{2}F_{1}\left (\begin{matrix} 2, p - 3 \\ p - 2 \end{matrix}\middle |{1 - \frac{e^{2} x^{2}}{d^{2}}} \right )}}{2 \left (- p + 3\right )} + \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{p - 3}}{2 \left (- p + 3\right )} + \frac{4 e \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{5} x} - \frac{4 e^{3} x \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 4, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**p/x**3/(e*x+d)**4,x)
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Mathematica [A] time = 2.03137, size = 399, normalized size = 1.89 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{80 d e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{64 d^2 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{8 d^3 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{5 e^2 2^{p+4} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^2 2^{p+3} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^2 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{e^2 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{16 d^7} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^4),x]
[Out]
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Maple [F] time = 0.127, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{{x}^{3} \left ( ex+d \right ) ^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^p/x^3/(e*x+d)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{4} x^{7} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{5} + 4 \, d^{3} e x^{4} + d^{4} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^3),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{3} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**p/x**3/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p/((e*x + d)^4*x^3),x, algorithm="giac")
[Out]