3.187 \(\int \left (1-x^2\right )^2 \left (1+b x^4\right )^p \, dx\)

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]

(x*(1 + b*x^4)^(1 + p))/(b*(5 + 4*p)) - ((1 - b*(5 + 4*p))*x*Hypergeometric2F1[1
/4, -p, 5/4, -(b*x^4)])/(b*(5 + 4*p)) - (2*x^3*Hypergeometric2F1[3/4, -p, 7/4, -
(b*x^4)])/3

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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]

(x*(1 + b*x^4)^(1 + p))/(b*(5 + 4*p)) + (1 - (5*b + 4*b*p)^(-1))*x*Hypergeometri
c2F1[1/4, -p, 5/4, -(b*x^4)] - (2*x^3*Hypergeometric2F1[3/4, -p, 7/4, -(b*x^4)])
/3

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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]

x**5*hyper((-p, 5/4), (9/4,), -b*x**4)/5 - 2*x**3*hyper((-p, 3/4), (7/4,), -b*x*
*4)/3 + x*hyper((-p, 1/4), (5/4,), -b*x**4)

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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]

x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - (2*x^3*Hypergeometric2F1[3/4, -p,
7/4, -(b*x^4)])/3 + (x^5*Hypergeometric2F1[5/4, -p, 9/4, -(b*x^4)])/5

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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]

1/5*x^5*hypergeom([5/4,-p],[9/4],-b*x^4)-2/3*x^3*hypergeom([3/4,-p],[7/4],-b*x^4
)+x*hypergeom([1/4,-p],[5/4],-b*x^4)

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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]

integrate((x^2 - 1)^2*(b*x^4 + 1)^p, x)

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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]

integral((x^4 - 2*x^2 + 1)*(b*x^4 + 1)^p, x)

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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]

Timed out

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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]

integrate((x^2 - 1)^2*(b*x^4 + 1)^p, x)