Optimal. Leaf size=112 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3+x}-x\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )}{81 \sqrt {3}}-\frac {5}{486} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )+\frac {1}{648} \sqrt [3]{x^3+x} \left (81 x^7+9 x^5-12 x^3+20 x\right ) \]
________________________________________________________________________________________
Rubi [B] time = 0.24, antiderivative size = 226, normalized size of antiderivative = 2.02, number of steps used = 14, number of rules used = 12, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {2021, 2024, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {1}{54} \sqrt [3]{x^3+x} x^3+\frac {5}{162} \sqrt [3]{x^3+x} x+\frac {1}{8} \sqrt [3]{x^3+x} x^7+\frac {1}{72} \sqrt [3]{x^3+x} x^5+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{243 \left (x^3+x\right )^{2/3}}-\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{486 \left (x^3+x\right )^{2/3}}+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3+x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x^6 \sqrt [3]{x+x^3} \, dx &=\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {1}{12} \int \frac {x^7}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {2}{27} \int \frac {x^5}{\left (x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5}{81} \int \frac {x^3}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {10}{243} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{243 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{162 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{81 \left (x+x^3\right )^{2/3}}\\ &=\frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{243 \left (x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{486 \left (x+x^3\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 68, normalized size = 0.61 \begin {gather*} \frac {x \sqrt [3]{x^3+x} \left (\sqrt [3]{x^2+1} \left (27 x^6+3 x^4-4 x^2+20\right )-20 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^2\right )\right )}{216 \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.40, size = 112, normalized size = 1.00 \begin {gather*} \frac {1}{648} \sqrt [3]{x+x^3} \left (20 x-12 x^3+9 x^5+81 x^7\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{x+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 108, normalized size = 0.96 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{648} \, {\left (81 \, x^{7} + 9 \, x^{5} - 12 \, x^{3} + 20 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.00, size = 97, normalized size = 0.87 \begin {gather*} \frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{486} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.87, size = 17, normalized size = 0.15
method | result | size |
meijerg | \(\frac {3 x^{\frac {22}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x^{2}\right )}{22}\) | \(17\) |
trager | \(\frac {x \left (81 x^{6}+9 x^{4}-12 x^{2}+20\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{648}+\frac {5 \ln \left (-36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x +33 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+36 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}+51 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+15\right )}{243}+\frac {5 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+72 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {1}{3}} x -30 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+9 x \left (x^{3}+x \right )^{\frac {1}{3}}-25 x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10\right )}{81}\) | \(304\) |
risch | \(\frac {x \left (81 x^{6}+9 x^{4}-12 x^{2}+20\right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{648}+\frac {\left (\frac {5 \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}-38 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}+18 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-16 x^{4}+30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-70 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}+18 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+96 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-28 x^{2}-32 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12}{x^{2}+1}\right )}{243}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2} x^{4}+20 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{4}-48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+100 x^{4}+30 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+14 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) x^{2}-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )^{2}-48 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-36 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+140 x^{2}-6 \RootOf \left (\textit {\_Z}^{2}+2 \textit {\_Z} +4\right )-60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+40}{x^{2}+1}\right )}{486}\right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}+1\right )}\) | \(514\) |
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 {\left (x^{3} + x\right )}^{\frac {1}{3}} x^{6}\,{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 x^6\,{\left (x^3+x\right )}^{1/3} \,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 x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________