\(\int (1-x^2)^3 (1+b x^4)^p \, dx\) [182]

Optimal result

Integrand size = 19, antiderivative size = 108 \[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=-\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )+\frac {(1-b (7+4 p)) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right )}{b (7+4 p)}+\frac {3}{5} x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-b x^4\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1221, 1907, 251, 371} \[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )+\frac {3}{5} x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-b x^4\right )-x^3 \left (1-\frac {1}{4 b p+7 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right )-\frac {x^3 \left (b x^4+1\right )^{p+1}}{b (4 p+7)} \]

[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

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 1221

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e^q*x^(2*q - 3)*((a + c*x^4)^(p +
1)/(c*(4*p + 2*q + 1))), x] + Dist[1/(c*(4*p + 2*q + 1)), Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d
+ e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, c, d, e, p},
x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]

Rule 1907

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+\frac {\int \left (1+b x^4\right )^p \left (b (7+4 p)+3 (1-b (7+4 p)) x^2+3 b (7+4 p) x^4\right ) \, dx}{b (7+4 p)} \\ & = -\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+\frac {\int \left (b (7+4 p) \left (1+b x^4\right )^p+3 (1-b (7+4 p)) x^2 \left (1+b x^4\right )^p+3 b (7+4 p) x^4 \left (1+b x^4\right )^p\right ) \, dx}{b (7+4 p)} \\ & = -\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+3 \int x^4 \left (1+b x^4\right )^p \, dx-\left (3 \left (1-\frac {1}{7 b+4 b p}\right )\right ) \int x^2 \left (1+b x^4\right )^p \, dx+\int \left (1+b x^4\right )^p \, dx \\ & = -\frac {x^3 \left (1+b x^4\right )^{1+p}}{b (7+4 p)}+x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right )-\left (1-\frac {1}{7 b+4 b p}\right ) x^3 \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-b x^4\right )+\frac {3}{5} x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-b x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-b x^4\right )-x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-b x^4\right )+\frac {3}{5} x^5 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-b x^4\right )-\frac {1}{7} x^7 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-b x^4\right ) \]

[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*Hypergeom
etric2F1[5/4, -p, 9/4, -(b*x^4)])/5 - (x^7*Hypergeometric2F1[7/4, -p, 11/4, -(b*x^4)])/7

Maple [A] (verified)

Time = 4.73 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69

method	result	size
meijerg	\(-\frac {x^{7} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {7}{4},-p ;\frac {11}{4};-b \,x^{4}\right )}{7}+\frac {3 x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {5}{4},-p ;\frac {9}{4};-b \,x^{4}\right )}{5}-x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},-p ;\frac {7}{4};-b \,x^{4}\right )+x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},-p ;\frac {5}{4};-b \,x^{4}\right )\)	\(75\)

[In]

int((-x^2+1)^3*(b*x^4+1)^p,x,method=_RETURNVERBOSE)

[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)

Fricas [F]

\[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )}^{3} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 50.67 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.19 \[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=- \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {3 x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {3 x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

-x**7*gamma(7/4)*hyper((7/4, -p), (11/4,), b*x**4*exp_polar(I*pi))/(4*gamma(11/4)) + 3*x**5*gamma(5/4)*hyper((
5/4, -p), (9/4,), b*x**4*exp_polar(I*pi))/(4*gamma(9/4)) - 3*x**3*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**4*e
xp_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]

\[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )}^{3} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[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)

Giac [F]

\[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=\int { -{\left (x^{2} - 1\right )}^{3} {\left (b x^{4} + 1\right )}^{p} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (1-x^2\right )^3 \left (1+b x^4\right )^p \, dx=-\int {\left (x^2-1\right )}^3\,{\left (b\,x^4+1\right )}^p \,d x \]

[In]

int(-(x^2 - 1)^3*(b*x^4 + 1)^p,x)

[Out]

-int((x^2 - 1)^3*(b*x^4 + 1)^p, x)