Optimal. Leaf size=101 \[ \frac {\left (x^3+x^2\right )^{2/3} \left (109573 x^6+19071 x^5-6357 x^4+20985 x^3-900 x^2+660 x-9240\right )}{52360 x^7 (x+1)}-\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {\log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [C] time = 0.94, antiderivative size = 844, normalized size of antiderivative = 8.36, number of steps used = 27, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2056, 6725, 21, 45, 37, 129, 155, 12, 91} \begin {gather*} -\frac {\left (1511+4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (x+1)}{52360 x \sqrt [3]{x^3+x^2}}-\frac {2187 (x+1)}{1309 x \sqrt [3]{x^3+x^2}}+\frac {\left (2249+153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (2249-153 i \sqrt {3}\right ) (x+1)}{20944 x^2 \sqrt [3]{x^3+x^2}}+\frac {3645 (x+1)}{2618 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (41+17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}+\frac {\left (41-17 i \sqrt {3}\right ) (x+1)}{2618 x^3 \sqrt [3]{x^3+x^2}}-\frac {1620 (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 {135 (x+1)}{119 x^4 \sqrt [3]{x^3+x^2}}-\frac {20 (x+1)}{17 x^5 \sqrt [3]{x^3+x^2}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (x+1)}{104720 \sqrt [3]{x^3+x^2}}+\frac {6561 (x+1)}{2618 \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 \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)^{2/3} x-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 (\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}}+\frac {1}{x^5 \sqrt [3]{x^3+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 37
Rule 45
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 {2 (1+x)}{17 x^5 \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 {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} (1+x)^{4/3}} \, dx}{3 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {2 (1+x)}{17 x^5 \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 {1}{9} \left (-41+17 i \sqrt {3}\right )-\frac {4}{3} \left (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 (-41-17 i \sqrt {3}\right )-\frac {4}{3} \left (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 {\left (6 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{20/3} \sqrt [3]{1+x}} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \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 (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 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 {1}{27} \left (-2249+153 i \sqrt {3}\right )-\frac {2}{3} \left (23-6 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 {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {1}{27} \left (-2249-153 i \sqrt {3}\right )-\frac {2}{3} \left (23+6 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 (90 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{17/3} \sqrt [3]{1+x}} \, dx}{17 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 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 {2}{81} \left (1511+4777 i \sqrt {3}\right )-\frac {2}{27} \left (895-1201 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 {\left (27 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{81} \left (1511-4777 i \sqrt {3}\right )-\frac {2}{27} \left (895+1201 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 (540 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{14/3} \sqrt [3]{1+x}} \, dx}{119 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 {2}{243} \left (21647-11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921-1633 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 {\left (81 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{243} \left (21647+11849 i \sqrt {3}\right )+\frac {2}{81} \left (7921+1633 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 (4860 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 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 {\left (3645 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{8/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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}{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 {\left (2187 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{5/3} \sqrt [3]{1+x}} \, dx}{1309 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{x^5 \sqrt [3]{x^2+x^3}}+\frac {6561 (1+x)}{2618 \sqrt [3]{x^2+x^3}}-\frac {\left (21647-11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {\left (21647+11849 i \sqrt {3}\right ) (1+x)}{104720 \sqrt [3]{x^2+x^3}}-\frac {20 (1+x)}{17 x^5 \sqrt [3]{x^2+x^3}}+\frac {135 (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 {1620 (1+x)}{1309 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41-17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {\left (41+17 i \sqrt {3}\right ) (1+x)}{2618 x^3 \sqrt [3]{x^2+x^3}}+\frac {3645 (1+x)}{2618 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249-153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}+\frac {\left (2249+153 i \sqrt {3}\right ) (1+x)}{20944 x^2 \sqrt [3]{x^2+x^3}}-\frac {2187 (1+x)}{1309 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511-4777 i \sqrt {3}\right ) (1+x)}{52360 x \sqrt [3]{x^2+x^3}}-\frac {\left (1511+4777 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 \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 \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]{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 = 0.15, size = 116, normalized size = 1.15 \begin {gather*} \frac {52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (3-i \sqrt {3}\right ) x}{2 (x+1)}\right )+52360 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (3+i \sqrt {3}\right ) x}{2 (x+1)}\right )+109573 x^6+19071 x^5-6357 x^4+20985 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 = 101, normalized size = 1.00 \begin {gather*} \frac {\left (x^2+x^3\right )^{2/3} \left (-9240+660 x-900 x^2+20985 x^3-6357 x^4+19071 x^5+109573 x^6\right )}{52360 x^7 (1+x)}-\frac {1}{3} \text {RootSum}\left [3-3 \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 = 0.65, size = 1412, normalized size = 13.98
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 8.18, size = 2059, normalized size = 20.39
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2059\) |
trager | \(\text {Expression too large to display}\) | \(2119\) |
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 = -0.01 \begin {gather*} \int \frac {1}{x^6\,{\left (x^3+x^2\right )}^{1/3}\,\left (x^3+1\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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