\(\int \frac {(1+x^3) (-1+2 x^3)^{2/3}}{x^6 (1+2 x^3)} \, dx\) [2071]

Optimal result

Integrand size = 29, antiderivative size = 150 \[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\frac {\left (-1+2 x^3\right )^{2/3} \left (-2+9 x^3\right )}{10 x^5}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{-1+2 x^3}+2^{2/3} \left (-1+2 x^3\right )^{2/3}\right ) \]

[Out]

1/10*(2*x^3-1)^(2/3)*(9*x^3-2)/x^5-2/3*2^(1/3)*arctan(3^(1/2)*x/(x+2^(1/3)*(2*x^3-1)^(1/3)))*3^(1/2)+2/3*2^(1/
3)*ln(-2*x+2^(1/3)*(2*x^3-1)^(1/3))-1/3*2^(1/3)*ln(4*x^2+2*2^(1/3)*x*(2*x^3-1)^(1/3)+2^(2/3)*(2*x^3-1)^(2/3))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x^3+1\right )+\sqrt [3]{2} \log \left (2^{2/3} x-\sqrt [3]{2 x^3-1}\right )-\frac {\left (2 x^3-1\right )^{2/3}}{5 x^5}+\frac {9 \left (2 x^3-1\right )^{2/3}}{10 x^2} \]

[In]

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

[Out]

-1/5*(-1 + 2*x^3)^(2/3)/x^5 + (9*(-1 + 2*x^3)^(2/3))/(10*x^2) - (2*2^(1/3)*ArcTan[(1 + (2*2^(2/3)*x)/(-1 + 2*x
^3)^(1/3))/Sqrt[3]])/Sqrt[3] - (2^(1/3)*Log[1 + 2*x^3])/3 + 2^(1/3)*Log[2^(2/3)*x - (-1 + 2*x^3)^(1/3)]

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*} \text {integral}& = -\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {9-2 x^3}{x^3 \sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {40}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-4 \int \frac {1}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {1+\frac {2\ 2^{2/3} x}{\sqrt [3]{-1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (1+2 x^3\right )+\sqrt [3]{2} \log \left (2^{2/3} x-\sqrt [3]{-1+2 x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\frac {1}{30} \left (-\frac {6 \left (-1+2 x^3\right )^{2/3}}{x^5}+\frac {27 \left (-1+2 x^3\right )^{2/3}}{x^2}-20 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{-2+4 x^3}}\right )+20 \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{-2+4 x^3}\right )-10 \sqrt [3]{2} \log \left (4 x^2+2 x \sqrt [3]{-2+4 x^3}+\left (-2+4 x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((-6*(-1 + 2*x^3)^(2/3))/x^5 + (27*(-1 + 2*x^3)^(2/3))/x^2 - 20*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + (-2 +
4*x^3)^(1/3))] + 20*2^(1/3)*Log[-2*x + (-2 + 4*x^3)^(1/3)] - 10*2^(1/3)*Log[4*x^2 + 2*x*(-2 + 4*x^3)^(1/3) + (
-2 + 4*x^3)^(2/3)])/30

Maple [A] (verified)

Time = 13.19 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {20 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {1}{3}} \left (2 x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+20 \,2^{\frac {1}{3}} \ln \left (\frac {-2^{\frac {2}{3}} x +\left (2 x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-10 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} \left (2 x^{3}-1\right )^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} x^{2}+\left (2 x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+27 \left (2 x^{3}-1\right )^{\frac {2}{3}} x^{3}-6 \left (2 x^{3}-1\right )^{\frac {2}{3}}}{30 x^{5}}\)	\(141\)
risch	\(\text {Expression too large to display}\)	\(923\)
trager	\(\text {Expression too large to display}\)	\(1147\)

[In]

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

[Out]

1/30*(20*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2^(1/3)*(2*x^3-1)^(1/3)))*x^5+20*2^(1/3)*ln((-2^(2/3)*x+(2*x^
3-1)^(1/3))/x)*x^5-10*2^(1/3)*ln((2^(2/3)*(2*x^3-1)^(1/3)*x+2*2^(1/3)*x^2+(2*x^3-1)^(2/3))/x^2)*x^5+27*(2*x^3-
1)^(2/3)*x^3-6*(2*x^3-1)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (116) = 232\).

Time = 1.82 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.93 \[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=-\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{5} + x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{6} + 66 \, x^{3} - 1\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + 1\right )}}{2 \, x^{3} + 1}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{3} + 1\right )} + 24 \, {\left (4 \, x^{5} - x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{3} + 1}\right ) - 9 \, {\left (9 \, x^{3} - 2\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]

[In]

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

[Out]

-1/90*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(6*sqrt(3)*2^(2/3)*(20*x^7 + 8*x^4 - x)*(2*x^3 - 1)^(2/3) - 12*sqrt(3
)*2^(1/3)*(76*x^8 - 32*x^5 + x^2)*(2*x^3 - 1)^(1/3) - sqrt(3)*(568*x^9 - 444*x^6 + 66*x^3 - 1))/(872*x^9 - 420
*x^6 + 6*x^3 + 1)) - 20*2^(1/3)*x^5*log(-(6*2^(2/3)*(2*x^3 - 1)^(1/3)*x^2 - 6*(2*x^3 - 1)^(2/3)*x - 2^(1/3)*(2
*x^3 + 1))/(2*x^3 + 1)) + 10*2^(1/3)*x^5*log((6*2^(1/3)*(10*x^4 - x)*(2*x^3 - 1)^(2/3) + 2^(2/3)*(76*x^6 - 32*
x^3 + 1) + 24*(4*x^5 - x^2)*(2*x^3 - 1)^(1/3))/(4*x^6 + 4*x^3 + 1)) - 9*(9*x^3 - 2)*(2*x^3 - 1)^(2/3))/x^5

Sympy [F]

\[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\int \frac {\left (x + 1\right ) \left (2 x^{3} - 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )}{x^{6} \cdot \left (2 x^{3} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}}{{\left (2 \, x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}}{{\left (2 \, x^{3} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx=\int \frac {\left (x^3+1\right )\,{\left (2\,x^3-1\right )}^{2/3}}{x^6\,\left (2\,x^3+1\right )} \,d x \]

[In]

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

[Out]

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