Optimal. Leaf size=339 \[ \frac {\left (a x^2+x^3\right )^{2/3} \left (38038000 a^8-77157360 a^2 b^2+49533120 a^5 c-28528500 a^7 x+57868020 a b^2 x-37149840 a^4 c x+24453000 a^6 x^2-49601160 b^2 x^2+31842720 a^3 c x^2-22007700 a^5 x^3-28658448 a^2 c x^3+20314800 a^4 x^4+26453952 a c x^4-19045125 a^3 x^5-24800580 c x^5+18042750 a^2 x^6-17222625 a x^7+16533720 x^8\right )}{148803480 x}+\frac {\left (-135850 \sqrt {3} a^9+275562 \sqrt {3} a^3 b^2-176904 \sqrt {3} a^6 c+1594323 \sqrt {3} b^2 c\right ) \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a x^2+x^3}}\right )}{1594323}+\frac {\left (135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right ) \log \left (-x+\sqrt [3]{a x^2+x^3}\right )}{1594323}+\frac {\left (-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right ) \log \left (x^2+x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}\right )}{3188646} \]
[Out]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(339)=678\).
time = 0.72, antiderivative size = 993, normalized size of antiderivative = 2.93, number of steps
used = 28, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2078, 2036,
61, 2049} \begin {gather*} \frac {135850 x^{2/3} \sqrt [3]{a+x} \text {ArcTan}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^9}{531441 \sqrt {3} \sqrt [3]{x^3+a x^2}}+\frac {67925 x^{2/3} \sqrt [3]{a+x} \log (x) a^9}{1594323 \sqrt [3]{x^3+a x^2}}+\frac {67925 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^9}{531441 \sqrt [3]{x^3+a x^2}}+\frac {135850 \left (x^3+a x^2\right )^{2/3} a^8}{531441 x}-\frac {67925 \left (x^3+a x^2\right )^{2/3} a^7}{354294}+\frac {728 c x^{2/3} \sqrt [3]{a+x} \text {ArcTan}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^6}{2187 \sqrt {3} \sqrt [3]{x^3+a x^2}}+\frac {364 c x^{2/3} \sqrt [3]{a+x} \log (x) a^6}{6561 \sqrt [3]{x^3+a x^2}}+\frac {364 c x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^6}{2187 \sqrt [3]{x^3+a x^2}}+\frac {67925 x \left (x^3+a x^2\right )^{2/3} a^6}{413343}-\frac {13585 x^2 \left (x^3+a x^2\right )^{2/3} a^5}{91854}+\frac {728 c \left (x^3+a x^2\right )^{2/3} a^5}{2187 x}+\frac {2090 x^3 \left (x^3+a x^2\right )^{2/3} a^4}{15309}-\frac {182}{729} c \left (x^3+a x^2\right )^{2/3} a^4-\frac {14 b^2 x^{2/3} \sqrt [3]{a+x} \text {ArcTan}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^3}{27 \sqrt {3} \sqrt [3]{x^3+a x^2}}-\frac {7 b^2 x^{2/3} \sqrt [3]{a+x} \log (x) a^3}{81 \sqrt [3]{x^3+a x^2}}-\frac {7 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^3}{27 \sqrt [3]{x^3+a x^2}}-\frac {5225 x^4 \left (x^3+a x^2\right )^{2/3} a^3}{40824}+\frac {52}{243} c x \left (x^3+a x^2\right )^{2/3} a^3+\frac {275 x^5 \left (x^3+a x^2\right )^{2/3} a^2}{2268}-\frac {26}{135} c x^2 \left (x^3+a x^2\right )^{2/3} a^2-\frac {14 b^2 \left (x^3+a x^2\right )^{2/3} a^2}{27 x}-\frac {25}{216} x^6 \left (x^3+a x^2\right )^{2/3} a+\frac {8}{45} c x^3 \left (x^3+a x^2\right )^{2/3} a+\frac {7}{18} b^2 \left (x^3+a x^2\right )^{2/3} a-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \text {ArcTan}\left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3+a x^2}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{x^3+a x^2}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3+a x^2}}+\frac {1}{9} x^7 \left (x^3+a x^2\right )^{2/3}-\frac {1}{6} c x^4 \left (x^3+a x^2\right )^{2/3}-\frac {1}{3} b^2 x \left (x^3+a x^2\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 61
Rule 2036
Rule 2049
Rule 2078
Rubi steps
\begin {align*} \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx &=\int \left (\frac {b^2 c}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 x^3}{\sqrt [3]{a x^2+x^3}}-\frac {c x^6}{\sqrt [3]{a x^2+x^3}}+\frac {x^9}{\sqrt [3]{a x^2+x^3}}\right ) \, dx\\ &=-\left (b^2 \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx\right )-c \int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx+\left (b^2 c\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx+\int \frac {x^9}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {1}{27} (25 a) \int \frac {x^8}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{9} \left (7 a b^2\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{9} (8 a c) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {\left (b^2 c x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{\sqrt [3]{a x^2+x^3}}\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{324} \left (275 a^2\right ) \int \frac {x^7}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {1}{27} \left (14 a^2 b^2\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {1}{135} \left (104 a^2 c\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (5225 a^3\right ) \int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx}{6804}+\frac {1}{81} \left (14 a^3 b^2\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{81} \left (52 a^3 c\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (10450 a^4\right ) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx}{15309}-\frac {1}{729} \left (364 a^4 c\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {\left (14 a^3 b^2 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{81 \sqrt [3]{a x^2+x^3}}\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (27170 a^5\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx}{45927}+\frac {\left (728 a^5 c\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{2187}\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (67925 a^6\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx}{137781}-\frac {\left (728 a^6 c\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{6561}\\ &=\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (67925 a^7\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx}{177147}-\frac {\left (728 a^6 c x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{6561 \sqrt [3]{a x^2+x^3}}\\ &=-\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (135850 a^8\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{531441}\\ &=-\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (135850 a^9\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{1594323}\\ &=-\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (135850 a^9 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{1594323 \sqrt [3]{a x^2+x^3}}\\ &=-\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^9 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{531441 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {67925 a^9 x^{2/3} \sqrt [3]{a+x} \log (x)}{1594323 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {67925 a^9 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{531441 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 3.79, size = 401, normalized size = 1.18 \begin {gather*} \frac {x^{2/3} \left (3 \sqrt [3]{x} (a+x) \left (38038000 a^8-28528500 a^7 x+24453000 a^6 x^2+21060 a^5 \left (2352 c-1045 x^3\right )+19440 a^4 \left (-1911 c x+1045 x^4\right )+3645 a^3 \left (8736 c x^2-5225 x^5\right )-13122 a^2 \left (5880 b^2+2184 c x^3-1375 x^6\right )+137781 a \left (420 b^2 x+192 c x^4-125 x^7\right )-8266860 \left (6 b^2 x^2+3 c x^5-2 x^8\right )\right )+280 \sqrt {3} \left (182468 a^9-91854 a^3 b^2+58968 a^6 c+1594323 b^2 c\right ) \sqrt [3]{a+x} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{a+x}}\right )+11760 \sqrt {3} a^3 \left (7579 a^6-8748 b^2+5616 a^3 c\right ) \sqrt [3]{a+x} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+x}}{\sqrt [3]{x}}}{\sqrt {3}}\right )+280 \left (135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right ) \sqrt [3]{a+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{a+x}\right )+140 \left (-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right ) \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )\right )}{446410440 \sqrt [3]{x^2 (a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-b \right ) \left (x^{3}+b \right ) \left (x^{3}-c \right )}{\left (a \,x^{2}+x^{3}\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 325, normalized size = 0.96 \begin {gather*} \frac {280 \, \sqrt {3} {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 280 \, {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \log \left (-\frac {x - {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - 140 \, {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \log \left (\frac {x^{2} + {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}} x + {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (38038000 \, a^{8} + 18042750 \, a^{2} x^{6} - 17222625 \, a x^{7} + 16533720 \, x^{8} + 49533120 \, a^{5} c - 3645 \, {\left (5225 \, a^{3} + 6804 \, c\right )} x^{5} + 3888 \, {\left (5225 \, a^{4} + 6804 \, a c\right )} x^{4} - 77157360 \, a^{2} b^{2} - 4212 \, {\left (5225 \, a^{5} + 6804 \, a^{2} c\right )} x^{3} + 360 \, {\left (67925 \, a^{6} + 88452 \, a^{3} c - 137781 \, b^{2}\right )} x^{2} - 420 \, {\left (67925 \, a^{7} + 88452 \, a^{4} c - 137781 \, a b^{2}\right )} x\right )} {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{446410440 \, 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 {\left (- b + x^{3}\right ) \left (b + x^{3}\right ) \left (- c + x^{3}\right )}{\sqrt [3]{x^{2} \left (a + x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 572, normalized size = 1.69 \begin {gather*} \frac {280 \, \sqrt {3} {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 140 \, {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \log \left ({\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 280 \, {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \log \left ({\left | {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3 \, {\left (38038000 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 332832500 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 1289216500 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} + 49533120 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 2897952200 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} - 433414800 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 4158305800 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} + 1678818960 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} - 77157360 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 3938066825 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} - 3773716128 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} + 675126900 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 2448101425 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 5414949792 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} - 2615083380 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} - 952462700 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 5128154388 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} + 5833647540 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} + 204186220 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + 3164424732 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} - 8170413300 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} - 1170879948 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} + 7338216060 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} + 198438660 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} - 4119651900 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 1319941980 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 184626540 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{9}}{a^{9}}}{446410440 \, a} \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 {\left (x^3+b\right )\,\left (b-x^3\right )\,\left (c-x^3\right )}{{\left (x^3+a\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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