Integrand size = 22, antiderivative size = 214 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {3 \left (x^2+x^3\right )^{2/3} \left (3080-3300 x+3600 x^2+2495 x^3-2994 x^4+4491 x^5\right )}{52360 x^7}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^3}+\sqrt [3]{2} \left (x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
[Out]
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 979, normalized size of antiderivative = 4.57, number of steps used = 30, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\frac {\left (8689+731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (8689-731 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}+\frac {2099 (x+1)}{13090 x \sqrt [3]{x^3+x^2}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {173 (x+1)}{5236 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (163+221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (163-221 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {107 (x+1)}{1309 x^3 \sqrt [3]{x^3+x^2}}-\frac {\left (47+17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}-\frac {\left (47-17 i \sqrt {3}\right ) (x+1)}{476 x^4 \sqrt [3]{x^3+x^2}}+\frac {x+1}{119 x^4 \sqrt [3]{x^3+x^2}}+\frac {3 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (113+23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (113-23987 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6793 (x+1)}{26180 \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log (1-x) \sqrt [3]{x+1}}{6 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (x+\sqrt [3]{-1}\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (x-(-1)^{2/3}\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}-\sqrt [3]{x+1}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}} \]
[In]
[Out]
Rule 12
Rule 129
Rule 384
Rule 491
Rule 597
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (-1+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \left (-\frac {1}{3 (1-x) x^{20/3} \sqrt [3]{1+x}}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )}-\frac {1}{3 x^{20/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{x^2+x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(1-x) x^{20/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{x^{18} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{x^{18} \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{x^{18} \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2+15 x^3}{x^{15} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{17 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-47-17 i \sqrt {3}\right )-15 \sqrt [3]{-1} x^3}{x^{15} \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{17 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-47+17 i \sqrt {3}\right )+15 (-1)^{2/3} x^3}{x^{15} \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{17 \sqrt [3]{x^2+x^3}} \\ & = \frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-214-24 x^3}{x^{12} \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-163-221 i \sqrt {3}+12 \left (1-16 i \sqrt {3}\right ) x^3}{x^{12} \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{238 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-163+221 i \sqrt {3}+12 \left (1+16 i \sqrt {3}\right ) x^3}{x^{12} \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{238 \sqrt [3]{x^2+x^3}} \\ & = \frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {692+1926 x^3}{x^9 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1151-1989 i \sqrt {3}+18 \left (125-96 i \sqrt {3}\right ) x^3}{x^9 \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1151+1989 i \sqrt {3}+18 \left (125+96 i \sqrt {3}\right ) x^3}{x^9 \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2618 \sqrt [3]{x^2+x^3}} \\ & = \frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-16792-4152 x^3}{x^6 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{20944 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2 \left (8689-731 i \sqrt {3}\right )+6 \left (3559-419 i \sqrt {3}\right ) x^3}{x^6 \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{20944 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {2 \left (8689+731 i \sqrt {3}\right )+6 \left (3559+419 i \sqrt {3}\right ) x^3}{x^6 \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{20944 \sqrt [3]{x^2+x^3}} \\ & = \frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {54344+50376 x^3}{x^3 \left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (113+23987 i \sqrt {3}\right )+6 \left (5441-3979 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {-2 \left (113-23987 i \sqrt {3}\right )+6 \left (5441+3979 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{104720 \sqrt [3]{x^2+x^3}} \\ & = \frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int -\frac {209440}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{209440 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int -\frac {209440}{\sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{209440 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int -\frac {209440}{\sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{209440 \sqrt [3]{x^2+x^3}} \\ & = \frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^3}} \\ & = \frac {6793 (1+x)}{26180 \sqrt [3]{x^2+x^3}}-\frac {\left (113-23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (113+23987 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}+\frac {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {1+x}{119 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47-17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (47+17 i \sqrt {3}\right ) (1+x)}{476 x^4 \sqrt [3]{x^2+x^3}}+\frac {107 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163-221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (163+221 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {173 (1+x)}{5236 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151-1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (1151+1989 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {2099 (1+x)}{13090 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689-731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (8689+731 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log (1-x)}{6 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{-1}+x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-(-1)^{2/3}+x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}-\sqrt [3]{1+x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {27720-1980 x+2700 x^2+54855 x^3-4491 x^4+13473 x^5+40419 x^6-26180\ 2^{2/3} \sqrt {3} x^{17/3} \sqrt [3]{1+x} \arctan \left (\frac {1+2 \sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}}{\sqrt {3}}\right )+26180\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (-1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}\right )-13090\ 2^{2/3} x^{17/3} \sqrt [3]{1+x} \log \left (1+\sqrt [3]{2} \sqrt [3]{\frac {x}{1+x}}+2^{2/3} \left (\frac {x}{1+x}\right )^{2/3}\right )+52360 x^{17/3} \sqrt [3]{1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right )+\log \left (\sqrt [3]{\frac {x}{1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]}{157080 x^5 \sqrt [3]{x^2 (1+x)}} \]
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Time = 6.77 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-13090 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{7}+52360 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{7}+26180 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{7}+26180 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right ) x^{7}+40419 \left (1+x \right ) \left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}} \left (x^{4}-\frac {5}{3} x^{3}+\frac {20}{9} x^{2}-\frac {6380}{4491} x +\frac {3080}{4491}\right )}{157080 x^{7}}\) | \(189\) |
trager | \(\text {Expression too large to display}\) | \(2151\) |
risch | \(\text {Expression too large to display}\) | \(2169\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.35 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.87 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {26180 \cdot 2^{\frac {2}{3}} x^{7} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 26180 \cdot 2^{\frac {2}{3}} x^{7} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 26180 \, \sqrt {6} 2^{\frac {1}{6}} x^{7} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 26180 \cdot 2^{\frac {2}{3}} x^{7} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 13090 \cdot 2^{\frac {2}{3}} x^{7} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} + x^{7}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} - x^{7}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} - x^{7}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 13090 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{7} + x^{7}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 9 \, {\left (4491 \, x^{5} - 2994 \, x^{4} + 2495 \, x^{3} + 3600 \, x^{2} - 3300 \, x + 3080\right )} {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{157080 \, x^{7}} \]
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Not integrable
Time = 0.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )} x^{6}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 41.26 (sec) , antiderivative size = 986, normalized size of antiderivative = 4.61 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx=-\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\right )} \,d x \]
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