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

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*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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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {1218, 251, 371} \begin {gather*} 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 {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.55, size = 42, normalized size = 1.00 \begin {gather*} 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 {gather*}

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 = 0.11, size = 37, normalized size = 0.88

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{4}, -p \right ], \left [\frac {5}{4}\right ], -b \,x^{4}\right )-\frac {x^{3} \hypergeom \left (\left [\frac {3}{4}, -p \right ], \left [\frac {7}{4}\right ], -b \,x^{4}\right )}{3}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 15.34, size = 61, normalized size = 1.45 \begin {gather*} - \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \left (x^2-1\right )\,{\left (b\,x^4+1\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 - 1)*(b*x^4 + 1)^p,x)

[Out]

-int((x^2 - 1)*(b*x^4 + 1)^p, x)

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