3.23.72 \(\int \frac {(-1+x^3)^{2/3} (1+x^3)}{x^6 (2+x^3)} \, dx\) [2272]

Optimal. Leaf size=173 \[ \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {\sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3} \sqrt [3]{3}} \]

[Out]

1/40*(-x^3-4)*(x^3-1)^(2/3)/x^5+1/8*2^(1/3)*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(1/3)*(x^3-1)^(1/3)))-1/24
*2^(1/3)*3^(2/3)*ln(-3*x+2^(1/3)*3^(2/3)*(x^3-1)^(1/3))+1/48*ln(3*x^2+2^(1/3)*3^(2/3)*x*(x^3-1)^(1/3)+2^(2/3)*
3^(1/3)*(x^3-1)^(2/3))*2^(1/3)*3^(2/3)

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Rubi [A]
time = 0.07, antiderivative size = 130, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {594, 597, 12, 384} \begin {gather*} \frac {\sqrt [6]{3} \text {ArcTan}\left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (x^3+2\right )}{8\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{40 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(-1 + x^3)^(2/3)/x^5 - (-1 + x^3)^(2/3)/(40*x^2) + (3^(1/6)*ArcTan[(1 + (2^(2/3)*3^(1/3)*x)/(-1 + x^3)^(
1/3))/Sqrt[3]])/(4*2^(2/3)) + Log[2 + x^3]/(8*2^(2/3)*3^(1/3)) - ((3/2)^(2/3)*Log[(3/2)^(1/3)*x - (-1 + x^3)^(
1/3)])/8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{10} \int \frac {-1+7 x^3}{x^3 \sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {1}{40} \int \frac {30}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{2-3 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {2 \sqrt [3]{2}+\sqrt [3]{3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {3 \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8 \sqrt [3]{2}}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 167, normalized size = 0.97 \begin {gather*} \frac {1}{240} \left (-\frac {6 \left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^5}+30 \sqrt [3]{2} \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*(-1 + x^3)^(2/3)*(4 + x^3))/x^5 + 30*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)
^(1/3))] - 10*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)] + 5*2^(1/3)*3^(2/3)*Log[3*x^2 + 2^(
1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)])/240

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 9.99, size = 804, normalized size = 4.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(804\)
trager	\(\text {Expression too large to display}\)	\(1113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(x^6+3*x^3-4)/x^5/(x^3-1)^(1/3)-1/24*ln((3*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*Root
Of(_Z^3+18)^3*x^3+108*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3-3*(x^3-
1)^(1/3)*RootOf(_Z^3+18)^2*x^2-54*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootO
f(_Z^3+18)*x^2-5*RootOf(_Z^3+18)*x^3-180*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3+18*x*(x^
3-1)^(2/3)+2*RootOf(_Z^3+18)+72*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(_Z^3
+18)-3/4*ln((3*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^3*x^3+108*RootOf(RootO
f(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3-3*(x^3-1)^(1/3)*RootOf(_Z^3+18)^2*x^2-54*
(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)*x^2-5*RootOf(_Z^3+18)*x
^3-180*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3+18*x*(x^3-1)^(2/3)+2*RootOf(_Z^3+18)+72*Ro
otOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18
)+324*_Z^2)+1/24*RootOf(_Z^3+18)*ln((-3*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+1
8)^3*x^3-135*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+21*(x^3-1)^(2/3)
*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)^2*x-4*(x^3-1)^(1/3)*RootOf(_Z^3+18)^
2*x^2-9*(x^3-1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*RootOf(_Z^3+18)*x^2-2*RootOf(_Z
^3+18)*x^3-90*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2)*x^3+3*x*(x^3-1)^(2/3)+2*RootOf(_Z^3+18)
+90*RootOf(RootOf(_Z^3+18)^2+18*_Z*RootOf(_Z^3+18)+324*_Z^2))/(x^3+2))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (123) = 246\).
time = 2.04, size = 289, normalized size = 1.67 \begin {gather*} \frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 36 \, {\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1440*(10*12^(2/3)*(-1)^(1/3)*x^5*log(-(18*12^(1/3)*(-1)^(2/3)*(x^3 - 1)^(1/3)*x^2 + 12^(2/3)*(-1)^(1/3)*(x^3
+ 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) - 5*12^(2/3)*(-1)^(1/3)*x^5*log(-(6*12^(2/3)*(-1)^(1/3)*(4*x^4 - x)*(
x^3 - 1)^(2/3) - 12^(1/3)*(-1)^(2/3)*(55*x^6 - 50*x^3 + 4) - 18*(7*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3
+ 4)) - 60*12^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*12^(1/6)*(12*12^(2/3)*(-1)^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)
^(2/3) + 36*(-1)^(1/3)*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1)^(1/3) - 12^(1/3)*(377*x^9 - 600*x^6 + 204*x^3 - 8))
/(487*x^9 - 480*x^6 + 12*x^3 + 8)) - 36*(x^3 + 4)*(x^3 - 1)^(2/3))/x^5

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

[Out]

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

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

[Out]

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

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

[Out]

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

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