Optimal. Leaf size=214 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x^2}-2 x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3+x^2} x+\sqrt [3]{2} \left (x^3+x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {3 \left (x^3+x^2\right )^{2/3} \left (4491 x^5-2994 x^4+2495 x^3+3600 x^2-3300 x+3080\right )}{52360 x^7} \]
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Rubi [C] time = 1.09, antiderivative size = 983, normalized size of antiderivative = 4.59, number of steps used = 27, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2056, 6725, 129, 155, 12, 91} \begin {gather*} -\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 (15-17 (-1)^{2/3}\right ) (x+1)}{238 x^4 \sqrt [3]{x^3+x^2}}-\frac {\left (15+17 \sqrt [3]{-1}\right ) (x+1)}{238 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} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x+1}}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}+\frac {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} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}+\frac {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 (\sqrt [3]{-1} x+1\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \log \left (1-(-1)^{2/3} x\right ) \sqrt [3]{x+1}}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x+1}}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 91
Rule 129
Rule 155
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (-1+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx &=\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 {3 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{3}-5 x}{(1-x) x^{17/3} \sqrt [3]{1+x}} \, dx}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (15+17 \sqrt [3]{-1}\right )+5 \sqrt [3]{-1} x}{x^{17/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{17 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{3} \left (15-17 (-1)^{2/3}\right )-5 (-1)^{2/3} x}{x^{17/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {214}{9}+\frac {8 x}{3}}{(1-x) x^{14/3} \sqrt [3]{1+x}} \, dx}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (163+221 i \sqrt {3}\right )-\frac {4}{3} \left (1-16 i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{238 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{9} \left (163-221 i \sqrt {3}\right )-\frac {4}{3} \left (1+16 i \sqrt {3}\right ) x}{x^{14/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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 (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {692}{27}-\frac {214 x}{3}}{(1-x) x^{11/3} \sqrt [3]{1+x}} \, dx}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-1151+1989 i \sqrt {3}\right )-\frac {2}{3} \left (125-96 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{2618 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-1151-1989 i \sqrt {3}\right )-\frac {2}{3} \left (125+96 i \sqrt {3}\right ) x}{x^{11/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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 (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {16792}{81}+\frac {1384 x}{27}}{(1-x) x^{8/3} \sqrt [3]{1+x}} \, dx}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{81} \left (8689-731 i \sqrt {3}\right )-\frac {2}{27} \left (3559-419 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{20944 \sqrt [3]{x^2+x^3}}-\frac {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{81} \left (8689+731 i \sqrt {3}\right )-\frac {2}{27} \left (3559+419 i \sqrt {3}\right ) x}{x^{8/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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 (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {54344}{243}-\frac {16792 x}{81}}{(1-x) x^{5/3} \sqrt [3]{1+x}} \, dx}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (113+23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441-3979 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{104720 \sqrt [3]{x^2+x^3}}+\frac {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (113-23987 i \sqrt {3}\right )-\frac {2}{81} \left (5441+3979 i \sqrt {3}\right ) x}{x^{5/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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 (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 (1-x) x^{2/3} \sqrt [3]{1+x}} \, dx}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{209440 \sqrt [3]{x^2+x^3}}-\frac {\left (243 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {209440}{729 x^{2/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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 ) \int \frac {1}{(1-x) x^{2/3} \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^{2/3} \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^{2/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 \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 (15+17 \sqrt [3]{-1}\right ) (1+x)}{238 x^4 \sqrt [3]{x^2+x^3}}-\frac {\left (15-17 (-1)^{2/3}\right ) (1+x)}{238 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} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\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 (1+\sqrt [3]{-1} 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 (1-(-1)^{2/3} 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 (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}\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]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1+(-1)^{2/3}}}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 135, normalized size = 0.63 \begin {gather*} \frac {-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x}{x+1}\right )-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-i \sqrt {3} x}{2 x+2}\right )-52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {i \sqrt {3} x+x}{2 x+2}\right )+13473 x^6+4491 x^5-1497 x^4+18285 x^3+900 x^2-660 x+9240}{52360 x^5 \sqrt [3]{x^2 (x+1)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 214, normalized size = 1.00 \begin {gather*} \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 {\tan ^{-1}\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 ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 870, normalized size = 4.07
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 69.05, size = 986, normalized size = 4.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 33.91, size = 2069, normalized size = 9.67
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2069\) |
trager | \(\text {Expression too large to display}\) | \(4312\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^6-x^9\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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