Integrand size = 22, antiderivative size = 101 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {\left (x^2+x^3\right )^{2/3} \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )}{52360 x^7 (1+x)}-\frac {1}{3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {9 \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )-52360 x^{17/3} \sqrt [3]{1+x} \text {RootSum}\left [1-3 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right )+3 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]}{471240 x^5 \sqrt [3]{x^2 (1+x)}} \]
(9*(-9240 + 660*x - 900*x^2 + 20985*x^3 - 6357*x^4 + 19071*x^5 + 109573*x^ 6) - 52360*x^(17/3)*(1 + x)^(1/3)*RootSum[1 - 3*#1^3 + 3*#1^6 & , (-2*Log[ (x/(1 + x))^(1/3) - #1] + 3*Log[(x/(1 + x))^(1/3) - #1]*#1^3)/(-#1^2 + 2*# 1^5) & ])/(471240*x^5*(x^2*(1 + x))^(1/3))
Result contains complex when optimal does not.
Time = 1.01 (sec) , antiderivative size = 552, normalized size of antiderivative = 5.47, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2467, 2035, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^6 \left (x^3+1\right ) \sqrt [3]{x^3+x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^{20/3} \sqrt [3]{x+1} \left (x^3+1\right )}dx}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \int \frac {1}{x^6 \sqrt [3]{x+1} \left (x^3+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \int \left (\frac {2-\sqrt [3]{x}}{9 \left (x^{2/3}-\sqrt [3]{x}+1\right ) \sqrt [3]{x+1}}+\frac {1}{9 \left (\sqrt [3]{x}+1\right ) \sqrt [3]{x+1}}-\frac {1}{x^3 \sqrt [3]{x+1}}+\frac {1}{x^6 \sqrt [3]{x+1}}+\frac {2-x}{3 \sqrt [3]{x+1} \left (x^2-x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3+x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x+1} \left (\frac {\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {19071 (x+1)^{2/3}}{52360 x^{2/3}}-\frac {6357 (x+1)^{2/3}}{26180 x^{5/3}}+\frac {2119 (x+1)^{2/3}}{10472 x^{8/3}}-\frac {90 (x+1)^{2/3}}{1309 x^{11/3}}+\frac {15 (x+1)^{2/3}}{238 x^{14/3}}-\frac {(x+1)^{2/3}}{17 x^{17/3}}+\frac {\sqrt [3]{x}}{3 \sqrt [3]{x+1}}+\frac {1}{18} \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \log \left (2 x-i \sqrt {3}-1\right )+\frac {1}{18} \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \log \left (2 x+i \sqrt {3}-1\right )-\frac {1}{6} \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}} \log \left (-\sqrt [3]{x+1}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+3 i}}}\right )-\frac {1}{6} \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}} \log \left (-\sqrt [3]{x+1}+\frac {\sqrt [3]{x}}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+3 i}}}\right )\right )}{\sqrt [3]{x^3+x^2}}\) |
(3*x^(2/3)*(1 + x)^(1/3)*(x^(1/3)/(3*(1 + x)^(1/3)) - (1 + x)^(2/3)/(17*x^ (17/3)) + (15*(1 + x)^(2/3))/(238*x^(14/3)) - (90*(1 + x)^(2/3))/(1309*x^( 11/3)) + (2119*(1 + x)^(2/3))/(10472*x^(8/3)) - (6357*(1 + x)^(2/3))/(2618 0*x^(5/3)) + (19071*(1 + x)^(2/3))/(52360*x^(2/3)) + (((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)*ArcTan[(1 + (2*x^(1/3))/(((I - Sqrt[3])/(3*I - Sqrt[3])) ^(1/3)*(1 + x)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]) + (((I + Sqrt[3])/(3*I + Sqrt [3]))^(1/3)*ArcTan[(1 + (2*x^(1/3))/(((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3) *(1 + x)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]) + (((I - Sqrt[3])/(3*I - Sqrt[3]))^ (1/3)*Log[-1 - I*Sqrt[3] + 2*x])/18 + (((I + Sqrt[3])/(3*I + Sqrt[3]))^(1/ 3)*Log[-1 + I*Sqrt[3] + 2*x])/18 - (((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3)* Log[x^(1/3)/((I - Sqrt[3])/(3*I - Sqrt[3]))^(1/3) - (1 + x)^(1/3)])/6 - (( (I + Sqrt[3])/(3*I + Sqrt[3]))^(1/3)*Log[x^(1/3)/((I + Sqrt[3])/(3*I + Sqr t[3]))^(1/3) - (1 + x)^(1/3)])/6))/(x^2 + x^3)^(1/3)
3.15.14.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 4.80 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {-52360 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{5} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+328719 x^{6}+57213 x^{5}-19071 x^{4}+62955 x^{3}-2700 x^{2}+1980 x -27720}{157080 x^{5} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\) | \(96\) |
trager | \(\text {Expression too large to display}\) | \(2118\) |
risch | \(\text {Expression too large to display}\) | \(2477\) |
1/157080*(-52360*sum(ln((-_R*x+(x^2*(1+x))^(1/3))/x)/_R,_R=RootOf(_Z^6-3*_ Z^3+3))*x^5*(x^2*(1+x))^(1/3)+328719*x^6+57213*x^5-19071*x^4+62955*x^3-270 0*x^2+1980*x-27720)/x^5/(x^2*(1+x))^(1/3)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 521, normalized size of antiderivative = 5.16 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\frac {13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} - \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x + i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} + \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 26180 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7}\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (-i \, \sqrt {3} x - 3 \, x\right )} {\left (i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} + \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (i \, \sqrt {-3} x - i \, x\right )} - 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7} - \sqrt {-3} {\left (x^{8} + x^{7}\right )}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (\sqrt {3} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 3 \, \sqrt {-3} x + 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 24 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 26180 \cdot 6^{\frac {2}{3}} {\left (x^{8} + x^{7}\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {1}{3}} \log \left (\frac {6^{\frac {1}{3}} {\left (i \, \sqrt {3} x - 3 \, x\right )} {\left (-i \, \sqrt {3} - 3\right )}^{\frac {2}{3}} + 12 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 9 \, {\left (109573 \, x^{6} + 19071 \, x^{5} - 6357 \, x^{4} + 20985 \, x^{3} - 900 \, x^{2} + 660 \, x - 9240\right )} {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{471240 \, {\left (x^{8} + x^{7}\right )}} \]
-1/471240*(13090*6^(2/3)*(x^8 + x^7 - sqrt(-3)*(x^8 + x^7))*(I*sqrt(3) - 3 )^(1/3)*log((6^(1/3)*(sqrt(3)*(I*sqrt(-3)*x + I*x) + 3*sqrt(-3)*x + 3*x)*( I*sqrt(3) - 3)^(2/3) + 24*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x^7 + sqrt(-3)*(x^8 + x^7))*(I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(-I*s qrt(-3)*x + I*x) - 3*sqrt(-3)*x + 3*x)*(I*sqrt(3) - 3)^(2/3) + 24*(x^3 + x ^2)^(1/3))/x) - 26180*6^(2/3)*(x^8 + x^7)*(I*sqrt(3) - 3)^(1/3)*log((6^(1/ 3)*(-I*sqrt(3)*x - 3*x)*(I*sqrt(3) - 3)^(2/3) + 12*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x^7 + sqrt(-3)*(x^8 + x^7))*(-I*sqrt(3) - 3)^(1/3)*l og((6^(1/3)*(sqrt(3)*(I*sqrt(-3)*x - I*x) - 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3 ) - 3)^(2/3) + 24*(x^3 + x^2)^(1/3))/x) + 13090*6^(2/3)*(x^8 + x^7 - sqrt( -3)*(x^8 + x^7))*(-I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(sqrt(3)*(-I*sqrt(-3) *x - I*x) + 3*sqrt(-3)*x + 3*x)*(-I*sqrt(3) - 3)^(2/3) + 24*(x^3 + x^2)^(1 /3))/x) - 26180*6^(2/3)*(x^8 + x^7)*(-I*sqrt(3) - 3)^(1/3)*log((6^(1/3)*(I *sqrt(3)*x - 3*x)*(-I*sqrt(3) - 3)^(2/3) + 12*(x^3 + x^2)^(1/3))/x) - 9*(1 09573*x^6 + 19071*x^5 - 6357*x^4 + 20985*x^3 - 900*x^2 + 660*x - 9240)*(x^ 3 + x^2)^(2/3))/(x^8 + x^7)
Not integrable
Time = 0.99 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{6}} \,d x } \]
Exception generated. \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Not integrable
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \left (1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^6\,{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+1\right )} \,d x \]