3.5.94 \(\int \frac {\sqrt [3]{-1+x^6} (1+x^6) (-1+x^3+x^6)}{x^8} \, dx\)

Optimal. Leaf size=38 \[ \frac {\sqrt [3]{x^6-1} \left (4 x^{12}+7 x^9-8 x^6-7 x^3+4\right )}{28 x^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 49, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1835, 1586, 1584, 449} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{7 x}-\frac {\left (x^6-1\right )^{4/3}}{7 x^7}+\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^6)^(1/3)*(1 + x^6)*(-1 + x^3 + x^6))/x^8,x]

[Out]

-1/7*(-1 + x^6)^(4/3)/x^7 + (-1 + x^6)^(4/3)/(4*x^4) + (-1 + x^6)^(4/3)/(7*x)

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1835

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq
0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[(2*a*(m + 1)*(Pq - Pq0))/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x^2+2 x^5+14 x^8+14 x^{11}\right )}{x^7} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x+2 x^4+14 x^7+14 x^{10}\right )}{x^6} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14+2 x^3+14 x^6+14 x^9\right )}{x^5} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16 x^2+112 x^8\right )}{x^4} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16+112 x^6\right )}{x^2} \, dx\\ &=-\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {\left (-1+x^6\right )^{4/3}}{7 x}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 100, normalized size = 2.63 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (20 \, _2F_1\left (-\frac {7}{6},-\frac {1}{3};-\frac {1}{6};x^6\right )+7 x^3 \left (10 x^6 \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {4}{3};x^6\right )-5 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {1}{3};x^6\right )+4 x^9 \, _2F_1\left (-\frac {1}{3},\frac {5}{6};\frac {11}{6};x^6\right )\right )\right )}{140 x^7 \sqrt [3]{1-x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6)*(-1 + x^3 + x^6))/x^8,x]

[Out]

((-1 + x^6)^(1/3)*(20*Hypergeometric2F1[-7/6, -1/3, -1/6, x^6] + 7*x^3*(-5*Hypergeometric2F1[-2/3, -1/3, 1/3,
x^6] + 10*x^6*Hypergeometric2F1[-1/3, 1/3, 4/3, x^6] + 4*x^9*Hypergeometric2F1[-1/3, 5/6, 11/6, x^6])))/(140*x
^7*(1 - x^6)^(1/3))

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IntegrateAlgebraic [A]  time = 0.70, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (-1+x^6\right )^{4/3} \left (-4+7 x^3+4 x^6\right )}{28 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^6)^(1/3)*(1 + x^6)*(-1 + x^3 + x^6))/x^8,x]

[Out]

((-1 + x^6)^(4/3)*(-4 + 7*x^3 + 4*x^6))/(28*x^7)

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fricas [A]  time = 0.46, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)*(x^6+x^3-1)/x^8,x, algorithm="fricas")

[Out]

1/28*(4*x^12 + 7*x^9 - 8*x^6 - 7*x^3 + 4)*(x^6 - 1)^(1/3)/x^7

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)*(x^6+x^3-1)/x^8,x, algorithm="giac")

[Out]

integrate((x^6 + x^3 - 1)*(x^6 + 1)*(x^6 - 1)^(1/3)/x^8, x)

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maple [A]  time = 0.16, size = 35, normalized size = 0.92

method	result	size
trager	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{12}+7 x^{9}-8 x^{6}-7 x^{3}+4\right )}{28 x^{7}}\)	\(35\)
gosper	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{6}+7 x^{3}-4\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{28 x^{7}}\)	\(45\)
risch	\(\frac {4 x^{18}+7 x^{15}-12 x^{12}-14 x^{9}+12 x^{6}+7 x^{3}-4}{28 \left (x^{6}-1\right )^{\frac {2}{3}} x^{7}}\)	\(45\)
meijerg	\(\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right ) x^{5}}{5 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}+\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right ) x^{2}}{2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{4}}+\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {7}{6}, -\frac {1}{3}\right ], \left [-\frac {1}{6}\right ], x^{6}\right )}{7 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{7}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/3)*(x^6+1)*(x^6+x^3-1)/x^8,x,method=_RETURNVERBOSE)

[Out]

1/28*(x^6-1)^(1/3)*(4*x^12+7*x^9-8*x^6-7*x^3+4)/x^7

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maxima [A]  time = 0.82, size = 55, normalized size = 1.45 \begin {gather*} \frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)*(x^6+x^3-1)/x^8,x, algorithm="maxima")

[Out]

1/28*(4*x^12 + 7*x^9 - 8*x^6 - 7*x^3 + 4)*(x^2 + x + 1)^(1/3)*(x^2 - x + 1)^(1/3)*(x + 1)^(1/3)*(x - 1)^(1/3)/
x^7

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mupad [B]  time = 0.46, size = 56, normalized size = 1.47 \begin {gather*} {\left (x^6-1\right )}^{1/3}\,\left (\frac {x^5}{7}+\frac {x^2}{4}\right )-\frac {2\,{\left (x^6-1\right )}^{1/3}}{7\,x}-\frac {{\left (x^6-1\right )}^{1/3}}{4\,x^4}+\frac {{\left (x^6-1\right )}^{1/3}}{7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(1/3)*(x^6 + 1)*(x^3 + x^6 - 1))/x^8,x)

[Out]

(x^6 - 1)^(1/3)*(x^2/4 + x^5/7) - (2*(x^6 - 1)^(1/3))/(7*x) - (x^6 - 1)^(1/3)/(4*x^4) + (x^6 - 1)^(1/3)/(7*x^7
)

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sympy [C]  time = 4.08, size = 148, normalized size = 3.89 \begin {gather*} \frac {x^{5} e^{\frac {i \pi }{3}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{3} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{7} \Gamma \left (- \frac {1}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(1/3)*(x**6+1)*(x**6+x**3-1)/x**8,x)

[Out]

x**5*exp(I*pi/3)*gamma(5/6)*hyper((-1/3, 5/6), (11/6,), x**6)/(6*gamma(11/6)) + x**2*exp(I*pi/3)*gamma(1/3)*hy
per((-1/3, 1/3), (4/3,), x**6)/(6*gamma(4/3)) - exp(-2*I*pi/3)*gamma(-2/3)*hyper((-2/3, -1/3), (1/3,), x**6)/(
6*x**4*gamma(1/3)) + exp(-2*I*pi/3)*gamma(-7/6)*hyper((-7/6, -1/3), (-1/6,), x**6)/(6*x**7*gamma(-1/6))

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