3.15.97 \(\int \frac {(1+x^3)^{2/3} (-1+3 x^6)}{x^9 (1+2 x^3)} \, dx\)

Optimal. Leaf size=104 \[ \frac {1}{3} \log \left (\sqrt [3]{x^3+1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\left (x^3+1\right )^{2/3} \left (x^6-14 x^3+5\right )}{40 x^8} \]

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Rubi [C]  time = 0.40, antiderivative size = 119, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6725, 271, 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 )+\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\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}}{8 x^8}-\frac {19 \left (x^3+1\right )^{5/3}}{40 x^5}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]

[Out]

(1 + x^3)^(2/3)/(2*x^2) + (1 + x^3)^(5/3)/(8*x^8) - (19*(1 + x^3)^(5/3))/(40*x^5) + 2*x*AppellF1[1/3, -2/3, 1,
4/3, -x^3, -2*x^3] - ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[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 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(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+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^9}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}-\frac {\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^6} \, 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^9} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {2 \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 {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {19 \left (1+x^3\right )^{5/3}}{40 x^5}+2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 104, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\frac {x}{\sqrt [3]{x^3+1}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right )+\frac {\left (x^3+1\right )^{2/3} \left (x^6-14 x^3+5\right )}{40 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]

[Out]

((1 + x^3)^(2/3)*(5 - 14*x^3 + x^6))/(40*x^8) + ArcTan[(-1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - Log[1 +
x^2/(1 + x^3)^(2/3) - x/(1 + x^3)^(1/3)]/6 + Log[1 + x/(1 + x^3)^(1/3)]/3

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IntegrateAlgebraic [A]  time = 0.20, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(-1 + 3*x^6))/(x^9*(1 + 2*x^3)),x]

[Out]

((1 + x^3)^(2/3)*(5 - 14*x^3 + x^6))/(40*x^8) + ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))]/Sqrt[3] + Log[x +
(1 + x^3)^(1/3)]/3 - Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)]/6

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fricas [A]  time = 1.01, size = 127, normalized size = 1.22 \begin {gather*} -\frac {40 \, \sqrt {3} x^{8} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 20 \, x^{8} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) - 3 \, {\left (x^{6} - 14 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{120 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="fricas")

[Out]

-1/120*(40*sqrt(3)*x^8*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(x^3 + 1)
)/(7*x^3 - 1)) - 20*x^8*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 + 1)^(2/3)*x + 1)/(2*x^3 + 1)) - 3*(x^6 -
14*x^3 + 5)*(x^3 + 1)^(2/3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="giac")

[Out]

integrate((3*x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^3 + 1)*x^9), x)

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maple [C]  time = 3.96, size = 425, normalized size = 4.09

method	result	size
risch	\(\frac {x^{9}-13 x^{6}-9 x^{3}+5}{40 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}+1}\right )}{3}-\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}+1}\right )\)	\(425\)
trager	\(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{6}-14 x^{3}+5\right )}{40 x^{8}}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4}{2 x^{3}+1}\right )-\frac {\ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\)	\(489\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/40*(x^9-13*x^6-9*x^3+5)/x^8/(x^3+1)^(1/3)-1/3*ln(-(9*RootOf(9*_Z^2+3*_Z+1)^2*x^3-3*RootOf(9*_Z^2+3*_Z+1)*(x^
3+1)^(2/3)*x+3*(x^3+1)^(1/3)*RootOf(9*_Z^2+3*_Z+1)*x^2+6*RootOf(9*_Z^2+3*_Z+1)*x^3-2*x*(x^3+1)^(2/3)+2*x^2*(x^
3+1)^(1/3)+x^3+3*RootOf(9*_Z^2+3*_Z+1)+1)/(2*x^3+1))-ln(-(9*RootOf(9*_Z^2+3*_Z+1)^2*x^3-3*RootOf(9*_Z^2+3*_Z+1
)*(x^3+1)^(2/3)*x+3*(x^3+1)^(1/3)*RootOf(9*_Z^2+3*_Z+1)*x^2+6*RootOf(9*_Z^2+3*_Z+1)*x^3-2*x*(x^3+1)^(2/3)+2*x^
2*(x^3+1)^(1/3)+x^3+3*RootOf(9*_Z^2+3*_Z+1)+1)/(2*x^3+1))*RootOf(9*_Z^2+3*_Z+1)+RootOf(9*_Z^2+3*_Z+1)*ln(-(9*R
ootOf(9*_Z^2+3*_Z+1)^2*x^3+3*RootOf(9*_Z^2+3*_Z+1)*(x^3+1)^(2/3)*x-3*(x^3+1)^(1/3)*RootOf(9*_Z^2+3*_Z+1)*x^2-x
*(x^3+1)^(2/3)+x^2*(x^3+1)^(1/3)-3*RootOf(9*_Z^2+3*_Z+1))/(2*x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(3*x^6-1)/x^9/(2*x^3+1),x, algorithm="maxima")

[Out]

integrate((3*x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^3 + 1)*x^9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (3\,x^6-1\right )}{x^9\,\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)*(3*x^6 - 1))/(x^9*(2*x^3 + 1)),x)

[Out]

int(((x^3 + 1)^(2/3)*(3*x^6 - 1))/(x^9*(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 (3 x^{6} - 1\right )}{x^{9} \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)*(3*x**6-1)/x**9/(2*x**3+1),x)

[Out]

Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(3*x**6 - 1)/(x**9*(2*x**3 + 1)), x)

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