Optimal. Leaf size=108 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )+\frac{3}{5} x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-b x^4\right )+\frac{x^3 (1-b (4 p+7)) \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right )}{b (4 p+7)}-\frac{x^3 \left (b x^4+1\right )^{p+1}}{b (4 p+7)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.204596, antiderivative size = 103, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )+\frac{3}{5} x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-b x^4\right )-x^3 \left (1-\frac{1}{4 b p+7 b}\right ) \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right )-\frac{x^3 \left (b x^4+1\right )^{p+1}}{b (4 p+7)} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^2)^3*(1 + b*x^4)^p,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.2858, size = 70, normalized size = 0.65 \[ - \frac{x^{7}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{- b x^{4}} \right )}}{7} + \frac{3 x^{5}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- b x^{4}} \right )}}{5} - x^{3}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- b x^{4}} \right )} + x{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- b x^{4}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)**3*(b*x**4+1)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0333192, size = 86, normalized size = 0.8 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{7} x^7 \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-b x^4\right )+\frac{3}{5} x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-b x^4\right )-x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^2)^3*(1 + b*x^4)^p,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.424, size = 75, normalized size = 0.7 \[ -{\frac{{x}^{7}}{7}{\mbox{$_2$F$_1$}({\frac{7}{4}},-p;\,{\frac{11}{4}};\,-b{x}^{4})}}+{\frac{3\,{x}^{5}}{5}{\mbox{$_2$F$_1$}({\frac{5}{4}},-p;\,{\frac{9}{4}};\,-b{x}^{4})}}-{x}^{3}{\mbox{$_2$F$_1$}({\frac{3}{4}},-p;\,{\frac{7}{4}};\,-b{x}^{4})}+x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)^3*(b*x^4+1)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int{\left (x^{2} - 1\right )}^{3}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)^3*(b*x^4 + 1)^p,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (x^{6} - 3 \, x^{4} + 3 \, x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)^3*(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)**3*(b*x**4+1)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -{\left (x^{2} - 1\right )}^{3}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^2 - 1)^3*(b*x^4 + 1)^p,x, algorithm="giac")
[Out]