Optimal. Leaf size=186 \[ -\frac {\left (\sqrt [3]{x-1}+1\right ) \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )^2 (x-1)^{2/3} \sqrt [3]{(x-1)^2 \left (x^2-x-1\right )^3} \left (\frac {(x-1)^{2/3} \left (6 x^2-21 x+10\right )}{10 x}-\frac {1}{3} \log \left (\sqrt [3]{x-1}+1\right )+\frac {1}{6} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{\left (-x+(x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) x \left (x^3-2 x^2+1\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 299, normalized size of antiderivative = 1.61, number of steps used = 12, number of rules used = 11, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.239, Rules used = {6688, 6719, 897, 1482, 1489, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {3 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} (1-x)}{5 \left (-x^2+x+1\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}{x \left (-x^2+x+1\right )}+\frac {3 \sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3}}{2 \left (-x^2+x+1\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \log \left (\sqrt [3]{x-1}+1\right )}{3 (x-1)^{2/3} \left (-x^2+x+1\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )}{6 (x-1)^{2/3} \left (-x^2+x+1\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (-x^2+x+1\right )^3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3} (x-1)^{2/3} \left (-x^2+x+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 292
Rule 618
Rule 628
Rule 634
Rule 897
Rule 1482
Rule 1489
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}}{x^2} \, dx &=\int \frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}}{x^2} \, dx\\ &=\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \int \frac {(-1+x)^{2/3} \left (-1-x+x^2\right )}{x^2} \, dx}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {\left (3 \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-1+x^3+x^6\right )}{\left (1+x^3\right )^2} \, dx,x,\sqrt [3]{-1+x}\right )}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}-\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {x \left (2-3 x^6\right )}{1+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}-\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \left (3 x-3 x^4-\frac {x}{1+x^3}\right ) \, dx,x,\sqrt [3]{-1+x}\right )}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{2 \left (1+x-x^2\right )}+\frac {3 (1-x) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{5 \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}+\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\sqrt [3]{-1+x}\right )}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{2 \left (1+x-x^2\right )}+\frac {3 (1-x) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{5 \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}-\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x}\right )}{3 (-1+x)^{2/3} \left (-1-x+x^2\right )}+\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{3 (-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{2 \left (1+x-x^2\right )}+\frac {3 (1-x) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{5 \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \log \left (1+\sqrt [3]{-1+x}\right )}{3 (-1+x)^{2/3} \left (1+x-x^2\right )}+\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{6 (-1+x)^{2/3} \left (-1-x+x^2\right )}+\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )}{2 (-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{2 \left (1+x-x^2\right )}+\frac {3 (1-x) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{5 \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \log \left (1+\sqrt [3]{-1+x}\right )}{3 (-1+x)^{2/3} \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 (-1+x)^{2/3} \left (1+x-x^2\right )}-\frac {\sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )}{(-1+x)^{2/3} \left (-1-x+x^2\right )}\\ &=\frac {3 \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{2 \left (1+x-x^2\right )}+\frac {3 (1-x) \sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{5 \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3}}{x \left (1+x-x^2\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{\sqrt {3} (-1+x)^{2/3} \left (1+x-x^2\right )}+\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \log \left (1+\sqrt [3]{-1+x}\right )}{3 (-1+x)^{2/3} \left (1+x-x^2\right )}-\frac {\sqrt [3]{-(1-x)^2 \left (1+x-x^2\right )^3} \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )}{6 (-1+x)^{2/3} \left (1+x-x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 63, normalized size = 0.34 \begin {gather*} \frac {\sqrt [3]{(x-1)^2 \left (x^2-x-1\right )^3} \left (5 x \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x\right )+6 x^2-21 x+10\right )}{10 x \left (x^2-x-1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 23.38, size = 190, normalized size = 1.02 \begin {gather*} -\frac {\left (\sqrt [3]{x-1}+1\right ) \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )^2 (x-1)^{2/3} \sqrt [3]{(x-1)^2 \left (x^2-x-1\right )^3} \left (\frac {(x-1)^{2/3} \left (6 (x-1)^2-9 (x-1)-5\right )}{10 x}-\frac {1}{3} \log \left (\sqrt [3]{x-1}+1\right )+\frac {1}{6} \log \left ((x-1)^{2/3}-\sqrt [3]{x-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{\sqrt {3}}\right )}{\left (-x+(x-1)^{2/3}-\sqrt [3]{x-1}+1\right ) x \left (x^3-2 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 399, normalized size = 2.15 \begin {gather*} -\frac {10 \, \sqrt {3} {\left (x^{3} - x^{2} - x\right )} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} - 2 \, x^{2} + 1\right )} - 2 \, \sqrt {3} {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x^{3} - 2 \, x^{2} + 1\right )}}\right ) - 5 \, {\left (x^{3} - x^{2} - x\right )} \log \left (\frac {x^{6} - 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} - {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + 1\right )} + {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {2}{3}} + 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} + 1}\right ) + 10 \, {\left (x^{3} - x^{2} - x\right )} \log \left (\frac {x^{3} - 2 \, x^{2} + {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1}{x^{3} - 2 \, x^{2} + 1}\right ) - 3 \, {\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}} {\left (6 \, x^{2} - 21 \, x + 10\right )}}{30 \, {\left (x^{3} - x^{2} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 137, normalized size = 0.74 \begin {gather*} \frac {\left (6 x^{3}-27 x^{2}+31 x -10\right ) \left (\left (-1+x \right )^{2} \left (x^{2}-x -1\right )^{3}\right )^{\frac {1}{3}}}{10 x \left (-1+x \right ) \left (x^{2}-x -1\right )}+\frac {\left (-\frac {\ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{3}+\frac {\ln \left (1-\left (-1+x \right )^{\frac {1}{3}}+\left (-1+x \right )^{\frac {2}{3}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 \left (-1+x \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )}{3}\right ) \left (\left (-1+x \right )^{2} \left (x^{2}-x -1\right )^{3}\right )^{\frac {1}{3}}}{\left (-1+x \right )^{\frac {2}{3}} \left (x^{2}-x -1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 5 \, x^{7} + 7 \, x^{6} + 2 \, x^{5} - 10 \, x^{4} + 2 \, x^{3} + 5 \, x^{2} - x - 1\right )}^{\frac {1}{3}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^8-5\,x^7+7\,x^6+2\,x^5-10\,x^4+2\,x^3+5\,x^2-x-1\right )}^{1/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\left (x - 1\right )^{2} \left (x^{2} - x - 1\right )^{3}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________