3.20.25 \(\int \frac {(-2+x^3) (1+x^3)^{2/3}}{x^6 (-1+2 x^3)} \, dx\) [1925]

Optimal. Leaf size=134 \[ \frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \]

[Out]

1/10*(-19*x^3-4)*(x^3+1)^(2/3)/x^5+3*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*(x^3+1)^(1/3)))-ln(-3*x+3^(2/3)*(x^
3+1)^(1/3))*3^(2/3)+1/2*ln(3*x^2+3^(2/3)*x*(x^3+1)^(1/3)+3^(1/3)*(x^3+1)^(2/3))*3^(2/3)

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Rubi [A]
time = 0.06, antiderivative size = 112, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 597, 12, 384} \begin {gather*} 3 \sqrt [6]{3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {1}{2} 3^{2/3} \log \left (2 x^3-1\right )-\frac {3}{2} 3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}-\frac {19 \left (x^3+1\right )^{2/3}}{10 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 + x^3)^(2/3))/(5*x^5) - (19*(1 + x^3)^(2/3))/(10*x^2) + 3*3^(1/6)*ArcTan[(1 + (2*3^(1/3)*x)/(1 + x^3)^(
1/3))/Sqrt[3]] + (3^(2/3)*Log[-1 + 2*x^3])/2 - (3*3^(2/3)*Log[3^(1/3)*x - (1 + x^3)^(1/3)])/2

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 (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {19+7 x^3}{x^3 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int \frac {90}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \text {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3 \text {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-3 \text {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {9}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )-\left (3\ 3^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 134, normalized size = 1.00 \begin {gather*} \frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4 - 19*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - 3^(
2/3)*Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)] + (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^
(2/3)])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.45, size = 551, normalized size = 4.11 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(19*x^6+23*x^3+4)/x^5/(x^3+1)^(1/3)+RootOf(_Z^3+9)*ln(-(-6*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+8
1*_Z^2)*RootOf(_Z^3+9)^3*x^3-27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x^3+27
*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2+8*RootOf(_Z^3+9)*x^3+36
*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^3+9*x*(x^3+1)^(2/3)+2*RootOf(_Z^3+9)+9*RootOf(RootOf(_
Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3-1))+9*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*ln(-(
12*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*RootOf(_Z^3+9)^3*x^3+81*RootOf(RootOf(_Z^3+9)^2+9*_Z*R
ootOf(_Z^3+9)+81*_Z^2)^2*RootOf(_Z^3+9)^2*x^3+15*(x^3+1)^(2/3)*RootOf(_Z^3+9)^2*RootOf(RootOf(_Z^3+9)^2+9*_Z*R
ootOf(_Z^3+9)+81*_Z^2)*x+5*(x^3+1)^(1/3)*RootOf(_Z^3+9)^2*x^2+63*(x^3+1)^(1/3)*RootOf(_Z^3+9)*RootOf(RootOf(_Z
^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2)*x^2+4*RootOf(_Z^3+9)*x^3+27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+8
1*_Z^2)*x^3+6*x*(x^3+1)^(2/3)+4*RootOf(_Z^3+9)+27*RootOf(RootOf(_Z^3+9)^2+9*_Z*RootOf(_Z^3+9)+81*_Z^2))/(2*x^3
-1))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (104) = 208\).
time = 1.67, size = 274, normalized size = 2.04 \begin {gather*} -\frac {10 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2 \, \sqrt {3} \left (-9\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 10 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-9\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-9\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{2 \, x^{3} - 1}\right ) + 5 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \, \left (-9\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-9\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 27 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) + 3 \, {\left (19 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(10*sqrt(3)*(-9)^(1/3)*x^5*arctan(1/3*(2*sqrt(3)*(-9)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 6*sqr
t(3)*(-9)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - sqrt(3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 +
231*x^6 + 6*x^3 - 1)) - 10*(-9)^(1/3)*x^5*log((3*(-9)^(2/3)*(x^3 + 1)^(1/3)*x^2 - 9*(x^3 + 1)^(2/3)*x + (-9)^
(1/3)*(2*x^3 - 1))/(2*x^3 - 1)) + 5*(-9)^(1/3)*x^5*log(-(9*(-9)^(1/3)*(7*x^4 + x)*(x^3 + 1)^(2/3) - (-9)^(2/3)
*(31*x^6 + 23*x^3 + 1) - 27*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1)) + 3*(19*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^{3} - 2\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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