Optimal. Leaf size=147 \[ \frac{1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^5 (p+3)}-\frac{d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)} \]
[Out]
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Rubi [A] time = 0.214114, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{1}{5} d x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^5 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^5 (p+3)}-\frac{d^4 \left (d^2-e^2 x^2\right )^{p+1}}{2 e^5 (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^4*(d + e*x)*(d^2 - e^2*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 38.6495, size = 119, normalized size = 0.81 \[ - \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{5} \left (p + 1\right )} + \frac{d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{e^{5} \left (p + 2\right )} + \frac{d x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 3}}{2 e^{5} \left (p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)*(-e**2*x**2+d**2)**p,x)
[Out]
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Mathematica [A] time = 0.210056, size = 208, normalized size = 1.41 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 d e^5 \left (p^3+6 p^2+11 p+6\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-5 \left (-e^6 \left (p^2+3 p+2\right ) x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^p+d^2 e^4 p (p+1) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 d^6 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+2 d^4 e^2 p x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )\right )}{10 e^5 (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(d + e*x)*(d^2 - e^2*x^2)^p,x]
[Out]
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Maple [F] time = 0.073, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(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. \[ \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{5} + d x^{4}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.9238, size = 972, normalized size = 6.61 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(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. \[ \int{\left (e x + d\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="giac")
[Out]