Optimal. Leaf size=86 \[ -\frac{x (1-b (4 p+5)) \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )}{b (4 p+5)}-\frac{2}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right )+\frac{x \left (b x^4+1\right )^{p+1}}{b (4 p+5)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.148459, antiderivative size = 79, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ x \left (1-\frac{1}{4 b p+5 b}\right ) \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{2}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right )+\frac{x \left (b x^4+1\right )^{p+1}}{b (4 p+5)} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2)^2*(1 + b*x^4)^p,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.3513, size = 53, normalized size = 0.62 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- b x^{4}} \right )}}{5} - \frac{2 x^{3}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- b x^{4}} \right )}}{3} + x{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- b x^{4}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)**2*(b*x**4+1)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0214872, size = 65, normalized size = 0.76 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )+\frac{1}{5} x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-b x^4\right )-\frac{2}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)^2*(1 + b*x^4)^p,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.069, size = 56, normalized size = 0.7 \[{\frac{{x}^{5}}{5}{\mbox{$_2$F$_1$}({\frac{5}{4}},-p;\,{\frac{9}{4}};\,-b{x}^{4})}}-{\frac{2\,{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{4}},-p;\,{\frac{7}{4}};\,-b{x}^{4})}}+x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)^2*(b*x^4+1)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} - 1\right )}^{2}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^2*(b*x^4 + 1)^p,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{4} - 2 \, x^{2} + 1\right )}{\left (b x^{4} + 1\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^2*(b*x^4 + 1)^p,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)**2*(b*x**4+1)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2} - 1\right )}^{2}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^2*(b*x^4 + 1)^p,x, algorithm="giac")
[Out]