3.186 \(\int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx\)

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]

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

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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]

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

-x**7*hyper((-p, 7/4), (11/4,), -b*x**4)/7 + 3*x**5*hyper((-p, 5/4), (9/4,), -b*
x**4)/5 - x**3*hyper((-p, 3/4), (7/4,), -b*x**4) + x*hyper((-p, 1/4), (5/4,), -b
*x**4)

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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]

x*Hypergeometric2F1[1/4, -p, 5/4, -(b*x^4)] - x^3*Hypergeometric2F1[3/4, -p, 7/4
, -(b*x^4)] + (3*x^5*Hypergeometric2F1[5/4, -p, 9/4, -(b*x^4)])/5 - (x^7*Hyperge
ometric2F1[7/4, -p, 11/4, -(b*x^4)])/7

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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]

-1/7*x^7*hypergeom([7/4,-p],[11/4],-b*x^4)+3/5*x^5*hypergeom([5/4,-p],[9/4],-b*x
^4)-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. \[ -\int{\left (x^{2} - 1\right )}^{3}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]

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]

-integrate((x^2 - 1)^3*(b*x^4 + 1)^p, x)

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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]

integral(-(x^6 - 3*x^4 + 3*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)**3*(b*x**4+1)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -{\left (x^{2} - 1\right )}^{3}{\left (b x^{4} + 1\right )}^{p}\,{d x} \]

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]

integrate(-(x^2 - 1)^3*(b*x^4 + 1)^p, x)