3.26.47 \(\int \frac {(-1+x^3)^{2/3} (8+2 x^3+x^6)}{x^6 (-2+x^3)} \, dx\)

Optimal. Leaf size=214 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {2}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3+8\right )}{10 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )-\frac {1}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right ) \]

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Rubi [C]  time = 0.39, antiderivative size = 123, normalized size of antiderivative = 0.57, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6725, 264, 277, 239, 430, 429} \begin {gather*} -\frac {2 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{\left (1-x^3\right )^{2/3}}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {4 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]

[Out]

(3*(-1 + x^3)^(2/3))/(2*x^2) - (4*(-1 + x^3)^(5/3))/(5*x^5) - (2*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3
, x^3, x^3/2])/(1 - x^3)^(2/3) - Sqrt[3]*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + (3*Log[-x + (-1 + x^3)
^(1/3)])/2

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 264

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

Rule 277

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

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx &=\int \left (-\frac {4 \left (-1+x^3\right )^{2/3}}{x^6}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^3}+\frac {4 \left (-1+x^3\right )^{2/3}}{-2+x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )-4 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+4 \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx\\ &=\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {4 \left (-1+x^3\right )^{5/3}}{5 x^5}-3 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (4 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{\left (1-x^3\right )^{2/3}}\\ &=\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {4 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{\left (1-x^3\right )^{2/3}}-\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {3}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.27, size = 181, normalized size = 0.85 \begin {gather*} -\frac {\sqrt [3]{1-x^3} x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,\frac {x^3}{2}\right )}{8 \sqrt [3]{x^3-1}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3+8\right )}{10 x^5}-\frac {-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )}{3\ 2^{2/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]

[Out]

((-1 + x^3)^(2/3)*(8 + 7*x^3))/(10*x^5) - (x^4*(1 - x^3)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, x^3, x^3/2])/(8*(-1
+ x^3)^(1/3)) - (2*Sqrt[3]*ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]] - 2*Log[2 - (2^(2/3)*x)/(1 - x^3)
^(1/3)] + Log[2 + (2^(1/3)*x^2)/(1 - x^3)^(2/3) + (2^(2/3)*x)/(1 - x^3)^(1/3)])/(3*2^(2/3))

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IntegrateAlgebraic [A]  time = 0.45, size = 214, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (8+7 x^3\right )}{10 x^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {2}{3} \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(8 + 2*x^3 + x^6))/(x^6*(-2 + x^3)),x]

[Out]

((-1 + x^3)^(2/3)*(8 + 7*x^3))/(10*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))]/Sqrt[3] - (2*2^(1/3)*Ar
cTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/3 + (2*2^(1/3)*Log[-x
+ 2^(1/3)*(-1 + x^3)^(1/3)])/3 + Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6 - (2^(1/3)*Log[x^2 + 2^(1
/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)])/3

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fricas [B]  time = 25.42, size = 360, normalized size = 1.68 \begin {gather*} \frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {12 \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )}}{3 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 30 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) + 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{3} - 2}\right ) - 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 15 \, x^{5} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 9 \, {\left (7 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="fricas")

[Out]

1/90*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(12*sqrt(3)*2^(2/3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 6*sqrt(3)*
2^(1/3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3) + sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8))/(53*x^9 - 48*x^6
- 12*x^3 + 8)) + 30*sqrt(3)*x^5*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 - 1)^(2/3)*x +
sqrt(3)*(5831*x^3 - 7200))/(58653*x^3 - 8000)) + 20*2^(1/3)*x^5*log(-(3*2^(2/3)*(x^3 - 1)^(1/3)*x^2 - 6*(x^3
- 1)^(2/3)*x + 2^(1/3)*(x^3 - 2))/(x^3 - 2)) - 10*2^(1/3)*x^5*log((12*2^(1/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 2^
(2/3)*(19*x^6 - 22*x^3 + 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 15*x^5*log(-3*(x^3 - 1)^
(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) + 9*(7*x^3 + 8)*(x^3 - 1)^(2/3))/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="giac")

[Out]

integrate((x^6 + 2*x^3 + 8)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^6), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+2 x^{3}+8\right )}{x^{6} \left (x^{3}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x)

[Out]

int((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3+8)/x^6/(x^3-2),x, algorithm="maxima")

[Out]

integrate((x^6 + 2*x^3 + 8)*(x^3 - 1)^(2/3)/((x^3 - 2)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3+8\right )}{x^6\,\left (x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 + 8))/(x^6*(x^3 - 2)),x)

[Out]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 + 8))/(x^6*(x^3 - 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} + 8\right )}{x^{6} \left (x^{3} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(x**6+2*x**3+8)/x**6/(x**3-2),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + 2*x**3 + 8)/(x**6*(x**3 - 2)), x)

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