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

Optimal. Leaf size=209 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{\sqrt [3]{3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-6 x^3-1\right )}{5 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{2 \sqrt [3]{3}} \]

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Rubi [C]  time = 0.39, antiderivative size = 99, normalized size of antiderivative = 0.47, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6725, 264, 277, 239, 429} \begin {gather*} 2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,2 x^3\right )-\log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {\left (x^3+1\right )^{2/3}}{x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.41, size = 164, normalized size = 0.78 \begin {gather*} \frac {1}{45} \left (-20\ 3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )+60 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {9 \left (x^3+1\right )^{2/3}}{x^5}-\frac {54 \left (x^3+1\right )^{2/3}}{x^2}+10\ 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )-\frac {1}{2} x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-x^3,2 x^3\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(x^4*AppellF1[4/3, 1/3, 1, 7/3, -x^3, 2*x^3]) + ((-9*(1 + x^3)^(2/3))/x^5 - (54*(1 + x^3)^(2/3))/x^2 + 60
*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(1 + x^3)^(1/3))] - 20*3^(2/3)*Log[1 - (3^(1/3)*x)/(1 + x^3)^(1/3)]
+ 10*3^(2/3)*Log[1 + (3^(2/3)*x^2)/(1 + x^3)^(2/3) + (3^(1/3)*x)/(1 + x^3)^(1/3)])/45

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IntegrateAlgebraic [A]  time = 0.41, size = 209, normalized size = 1.00 \begin {gather*} \frac {\left (-1-6 x^3\right ) \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {\log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 - 6*x^3)*(1 + x^3)^(2/3))/(5*x^5) + ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + 3^(1/6)*ArcTan[
(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - Log[-x + (1 + x^3)^(1/3)]/3 - Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3
)]/3^(1/3) + Log[x^2 + x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6 + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3
)*(1 + x^3)^(2/3)]/(2*3^(1/3))

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fricas [B]  time = 30.55, size = 382, normalized size = 1.83 \begin {gather*} \frac {10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 5 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 30 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\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 ) - 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 ) - 18 \, {\left (6 \, x^{3} + 1\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)*(2*x^6-1)/x^6/(2*x^3-1),x, algorithm="fricas")

[Out]

1/90*(10*3^(2/3)*(-1)^(1/3)*x^5*log((9*3^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 + 3^(2/3)*(-1)^(1/3)*(2*x^3 - 1)
- 9*(x^3 + 1)^(2/3)*x)/(2*x^3 - 1)) - 5*3^(2/3)*(-1)^(1/3)*x^5*log(-(3*3^(2/3)*(-1)^(1/3)*(7*x^4 + x)*(x^3 +
1)^(2/3) - 3^(1/3)*(-1)^(2/3)*(31*x^6 + 23*x^3 + 1) - 9*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1))
- 30*3^(1/6)*(-1)^(1/3)*x^5*arctan(1/3*3^(1/6)*(6*3^(2/3)*(-1)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 18
*(-1)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - 3^(1/3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 + 231
*x^6 + 6*x^3 - 1)) + 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)) - 15*x^5*log(3*(x^3 + 1)^(1/3)*x^2 - 3*(x^3 + 1)^(2/3)*x +
1) - 18*(6*x^3 + 1)*(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 (2 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^3 - 1)*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 (2\,x^6-1\right )}{x^6\,\left (2\,x^3-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^3 + 1)^(2/3)*(2*x^6 - 1))/(x^6*(2*x^3 - 1)), 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 (2 x^{6} - 1\right )}{x^{6} \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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