3.184 \(\int (1-x^2) (1+b x^4)^p \, dx\)

Optimal. Leaf size=42 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

[Out]

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

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Rubi [A]  time = 0.0205814, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1204, 245, 364} \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)*(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

Rule 1204

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)*(a + c*x^4)
^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx &=\int \left (\left (1+b x^4\right )^p-x^2 \left (1+b x^4\right )^p\right ) \, dx\\ &=\int \left (1+b x^4\right )^p \, dx-\int x^2 \left (1+b x^4\right )^p \, dx\\ &=x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.006765, size = 42, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )-\frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)*(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

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Maple [A]  time = 0.027, size = 37, normalized size = 0.9 \begin{align*} x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})}-{\frac{{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{4}},-p;\,{\frac{7}{4}};\,-b{x}^{4})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)*(b*x^4+1)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)*(b*x^4+1)^p,x, algorithm="fricas")

[Out]

integral(-(x^2 - 1)*(b*x^4 + 1)^p, x)

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Sympy [C]  time = 45.9657, size = 61, normalized size = 1.45 \begin{align*} - \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, - p \\ \frac{7}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**4*exp_polar(I*pi))/(4*gamma(7/4)) + x*gamma(1/4)*hyper((1/4, -p
), (5/4,), b*x**4*exp_polar(I*pi))/(4*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (x^{2} - 1\right )}{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)*(b*x^4+1)^p,x, algorithm="giac")

[Out]

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