\(\int (1-x^2) (1+b x^4)^p \, dx\) [184]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )-\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1218, 251, 371} \[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )-\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right ) \]

[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 251

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1218

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )-\frac {1}{3} x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right ) \]

[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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88

method	result	size
meijerg	\(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},-p ;\frac {5}{4};-b \,x^{4}\right )-\frac {x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},-p ;\frac {7}{4};-b \,x^{4}\right )}{3}\)	\(37\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[In]

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

[Out]

integral(-(x^2 - 1)*(b*x^4 + 1)^p, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 13.87 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.45 \[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=- \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[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))

Maxima [F]

\[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (1-x^2\right ) \left (1+b x^4\right )^p \, dx=-\int \left (x^2-1\right )\,{\left (b\,x^4+1\right )}^p \,d x \]

[In]

int(-(x^2 - 1)*(b*x^4 + 1)^p,x)

[Out]

-int((x^2 - 1)*(b*x^4 + 1)^p, x)