Integrand size = 20, antiderivative size = 111 \[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
3/2*(x^3-x^2)^(2/3)/x^2+3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^3-x^2)^(1/3)))-ln (-x+(x^3-x^2)^(1/3))+1/2*ln(x^2+x*(x^3-x^2)^(1/3)+(x^3-x^2)^(2/3))
Time = 0.18 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24 \[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {-3+3 x+2 \sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\sqrt [3]{-1+x} x^{2/3} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )}{2 \sqrt [3]{(-1+x) x^2}} \]
(-3 + 3*x + 2*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[(Sqrt[3]*x^(1/3))/(2*( -1 + x)^(1/3) + x^(1/3))] - 2*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3) - x^(1/3)] + (-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(2/3) + (-1 + x)^(1/3)*x^(1 /3) + x^(2/3)])/(2*((-1 + x)*x^2)^(1/3))
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1944, 1917, 71}
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.
| \(\displaystyle \int \frac {x+1}{x \sqrt [3]{x^3-x^2}} \, dx\) |
| \(\Big \downarrow \) 1944 |
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3-x^2}}dx+\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \int \frac {1}{\sqrt [3]{x-1} x^{2/3}}dx}{\sqrt [3]{x^3-x^2}}+\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{x-1} x^{2/3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )-\frac {\log (x)}{2}\right )}{\sqrt [3]{x^3-x^2}}+\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}\) |
(3*(-x^2 + x^3)^(2/3))/(2*x^2) + ((-1 + x)^(1/3)*x^(2/3)*(-(Sqrt[3]*ArcTan [1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))]) - (3*Log[-1 + (-1 + x) ^(1/3)/x^(1/3)])/2 - Log[x]/2))/(-x^2 + x^3)^(1/3)
3.17.36.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]) Int[ x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b *c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)) Int[(e*x)^(m + n)*(a*x^ j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 ] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {-\frac {3}{2}+\frac {3 x}{2}}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(42\) |
| meijerg | \(-\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{2 \operatorname {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}}}+\frac {3 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(54\) |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) x^{2}+\ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) x^{2}-2 \ln \left (\frac {\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right ) x^{2}+3 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(104\) |
| trager | \(\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}+6 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {180 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-18 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right )-6 \ln \left (-\frac {-180 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +360 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +174 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -20 x^{2}+12 x}{x}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+\ln \left (-\frac {-180 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +360 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +174 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-138 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -20 x^{2}+12 x}{x}\right )\) | \(506\) |
3/2*(-1+x)/((-1+x)*x^2)^(1/3)+3/signum(-1+x)^(1/3)*(-signum(-1+x))^(1/3)*x ^(1/3)*hypergeom([1/3,1/3],[4/3],x)
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x^{2} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x^{2} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
-1/2*(2*sqrt(3)*x^2*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 - x^2)^(1/3))/x ) + 2*x^2*log(-(x - (x^3 - x^2)^(1/3))/x) - x^2*log((x^2 + (x^3 - x^2)^(1/ 3)*x + (x^3 - x^2)^(2/3))/x^2) - 3*(x^3 - x^2)^(2/3))/x^2
\[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {x + 1}{x \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \]
\[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {x + 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x + 1)^(1/3) + 1)) + 3/2*(-1/x + 1)^(2/ 3) + 1/2*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) - log(abs((-1/x + 1) ^(1/3) - 1))
Time = 5.54 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.40 \[ \int \frac {1+x}{x \sqrt [3]{-x^2+x^3}} \, dx=\frac {3\,{\left (x^3-x^2\right )}^{2/3}}{2\,x^2}+\frac {3\,x\,{\left (1-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x\right )}{{\left (x^3-x^2\right )}^{1/3}} \]