Optimal. Leaf size=194 \[ \frac{e x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}-\frac{3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p-2}}{e^4 (2-p)}-\frac{2 e (3 p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 (2 p+1)}-\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^4 (1-p)} \]
[Out]
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Rubi [A] time = 0.492174, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ \frac{e x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}-\frac{3 d \left (d^2-e^2 x^2\right )^p}{2 e^4 p}+\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p-2}}{e^4 (2-p)}-\frac{2 e (3 p+4) x^5 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{5}{2},3-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 (2 p+1)}-\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p-1}}{2 e^4 (1-p)} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 109.922, size = 182, normalized size = 0.94 \[ \frac{2 d^{5} \left (d^{2} - e^{2} x^{2}\right )^{p - 2}}{e^{4} \left (- p + 2\right )} - \frac{7 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p - 1}}{2 e^{4} \left (- p + 1\right )} - \frac{3 d \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p} - \frac{3 e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5 d^{4}} - \frac{e^{3} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 3, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7 d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d)**3,x)
[Out]
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Mathematica [C] time = 0.444837, size = 140, normalized size = 0.72 \[ -\frac{5 d x^4 (d-e x)^p (d+e x)^{p-3} F_1\left (4;-p,3-p;5;\frac{e x}{d},-\frac{e x}{d}\right )}{4 \left (e x \left (p F_1\left (5;1-p,3-p;6;\frac{e x}{d},-\frac{e x}{d}\right )-(p-3) F_1\left (5;-p,4-p;6;\frac{e x}{d},-\frac{e x}{d}\right )\right )-5 d F_1\left (4;-p,3-p;5;\frac{e x}{d},-\frac{e x}{d}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x^3*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]
[Out]
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Maple [F] time = 0.118, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(-e^2*x^2+d^2)^p/(e*x+d)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(-e**2*x**2+d**2)**p/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^p*x^3/(e*x + d)^3,x, algorithm="giac")
[Out]