3.17.40 \(\int \frac {(2+x^3) (1+2 x^3)^{2/3}}{x^6 (1+x^3)} \, dx\) [1640]

Optimal. Leaf size=111 \[ \frac {\left (-4-3 x^3\right ) \left (1+2 x^3\right )^{2/3}}{10 x^5}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+2 x^3}+\left (1+2 x^3\right )^{2/3}\right ) \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 0.87, 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*} -\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{2 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (x^3+1\right )+\frac {1}{2} \log \left (x-\sqrt [3]{2 x^3+1}\right )-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}-\frac {3 \left (2 x^3+1\right )^{2/3}}{10 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.22, size = 111, normalized size = 1.00 \begin {gather*} \frac {\left (-4-3 x^3\right ) \left (1+2 x^3\right )^{2/3}}{10 x^5}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+2 x^3}+\left (1+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.03, size = 358, normalized size = 3.23

method	result	size
trager	\(-\frac {\left (3 x^{3}+4\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}+\frac {\ln \left (\frac {12321 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+48096 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x -99369 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+56439 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-17091 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -16032 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+51230 x^{3}-12321 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-3048 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+15369}{\left (x^{2}-x +1\right ) \left (1+x \right )}\right )}{3}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+48096 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +51273 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-134214 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+33123 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -16032 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-12321 x^{3}-46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-65583 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-5476}{\left (x^{2}-x +1\right ) \left (1+x \right )}\right )\)	\(358\)
risch	\(-\frac {6 x^{6}+11 x^{3}+4}{10 x^{5} \left (2 x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (\frac {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x -15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (2 x^{3}+1\right )^{\frac {2}{3}} x -\left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 x^{3}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (x^{2}-x +1\right ) \left (1+x \right )}\right )}{3}-\ln \left (\frac {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x -15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-\left (2 x^{3}+1\right )^{\frac {2}{3}} x -\left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 x^{3}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (x^{2}-x +1\right ) \left (1+x \right )}\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 {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+4 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +4 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+9 x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3}{\left (x^{2}-x +1\right ) \left (1+x \right )}\right )\)	\(490\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(3*x^3+4)/x^5*(2*x^3+1)^(2/3)+1/3*ln((12321*RootOf(9*_Z^2+3*_Z+1)^2*x^3+48096*RootOf(9*_Z^2+3*_Z+1)*(2*x
^3+1)^(2/3)*x-99369*RootOf(9*_Z^2+3*_Z+1)*(2*x^3+1)^(1/3)*x^2+56439*RootOf(9*_Z^2+3*_Z+1)*x^3-17091*(2*x^3+1)^
(2/3)*x-16032*(2*x^3+1)^(1/3)*x^2+51230*x^3-12321*RootOf(9*_Z^2+3*_Z+1)^2-3048*RootOf(9*_Z^2+3*_Z+1)+15369)/(x
^2-x+1)/(1+x))+RootOf(9*_Z^2+3*_Z+1)*ln(-(46107*RootOf(9*_Z^2+3*_Z+1)^2*x^3+48096*RootOf(9*_Z^2+3*_Z+1)*(2*x^3
+1)^(2/3)*x+51273*RootOf(9*_Z^2+3*_Z+1)*(2*x^3+1)^(1/3)*x^2-134214*RootOf(9*_Z^2+3*_Z+1)*x^3+33123*(2*x^3+1)^(
2/3)*x-16032*(2*x^3+1)^(1/3)*x^2-12321*x^3-46107*RootOf(9*_Z^2+3*_Z+1)^2-65583*RootOf(9*_Z^2+3*_Z+1)-5476)/(x^
2-x+1)/(1+x))

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

[Out]

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

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Fricas [A]
time = 0.77, size = 133, normalized size = 1.20 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} + 1\right )}}{10 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{3} + 1}\right ) + 3 \, {\left (3 \, x^{3} + 4\right )} {\left (2 \, 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)*(2*x^3+1)^(2/3)/x^6/(x^3+1),x, algorithm="fricas")

[Out]

-1/30*(10*sqrt(3)*x^5*arctan(-(4*sqrt(3)*(2*x^3 + 1)^(1/3)*x^2 - 2*sqrt(3)*(2*x^3 + 1)^(2/3)*x + sqrt(3)*(2*x^
3 + 1))/(10*x^3 + 1)) - 5*x^5*log((x^3 + 3*(2*x^3 + 1)^(1/3)*x^2 - 3*(2*x^3 + 1)^(2/3)*x + 1)/(x^3 + 1)) + 3*(
3*x^3 + 4)*(2*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 (x^{3} + 2\right ) \left (2 x^{3} + 1\right )^{\frac {2}{3}}}{x^{6} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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