Optimal. Leaf size=147 \[ -\frac {1}{6} \sqrt [3]{c} \log \left (\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}+c^{2/3} x^2\right )+\frac {1}{3} \sqrt [3]{c} \log \left (\sqrt [3]{b-a x^6}-\sqrt [3]{c} x\right )+\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{2 \sqrt [3]{b-a x^6}+\sqrt [3]{c} x}\right )}{\sqrt {3}}+\frac {\sqrt [3]{b-a x^6}}{x} \]
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Rubi [C] time = 3.84, antiderivative size = 1003, normalized size of antiderivative = 6.82, number of steps used = 38, number of rules used = 9, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6728, 365, 364, 1562, 465, 430, 429, 511, 510} \begin {gather*} -\frac {2 a^2 c \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (c^2-\sqrt {c^2+4 a b} c+2 a b\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^5}{5 \sqrt {c^2+4 a b} \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1-\frac {c}{\sqrt {c^2+4 a b}}\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{c^2-\sqrt {c^2+4 a b} c+2 a b},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c-\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {a \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{\sqrt {c^2+4 a b} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (\frac {c}{\sqrt {c^2+4 a b}}+1\right ) \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {c^2+4 a b}\right )},\frac {a x^6}{b}\right ) x^2}{2 \left (2 a b+c \left (c+\sqrt {c^2+4 a b}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{\sqrt [3]{1-\frac {a x^6}{b}} x} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 364
Rule 365
Rule 429
Rule 430
Rule 465
Rule 510
Rule 511
Rule 1562
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx &=\int \left (-\frac {\sqrt [3]{b-a x^6}}{x^2}+\frac {x \left (c+2 a x^3\right ) \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}\right ) \, dx\\ &=-\int \frac {\sqrt [3]{b-a x^6}}{x^2} \, dx+\int \frac {x \left (c+2 a x^3\right ) \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx\\ &=-\frac {\sqrt [3]{b-a x^6} \int \frac {\sqrt [3]{1-\frac {a x^6}{b}}}{x^2} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}+\int \left (\frac {c x \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}+\frac {2 a x^4 \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+(2 a) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx+c \int \frac {x \sqrt [3]{b-a x^6}}{-b+c x^3+a x^6} \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+(2 a) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c-\sqrt {4 a b+c^2}+2 a x^3\right )}+\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c+\sqrt {4 a b+c^2}+2 a x^3\right )}\right ) \, dx+c \int \left (\frac {2 a x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c-\sqrt {4 a b+c^2}+2 a x^3\right )}-\frac {2 a x \sqrt [3]{b-a x^6}}{\sqrt {4 a b+c^2} \left (c+\sqrt {4 a b+c^2}+2 a x^3\right )}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {(2 a c) \int \frac {x \sqrt [3]{b-a x^6}}{c-\sqrt {4 a b+c^2}+2 a x^3} \, dx}{\sqrt {4 a b+c^2}}-\frac {(2 a c) \int \frac {x \sqrt [3]{b-a x^6}}{c+\sqrt {4 a b+c^2}+2 a x^3} \, dx}{\sqrt {4 a b+c^2}}+\left (2 a \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{c-\sqrt {4 a b+c^2}+2 a x^3} \, dx+\left (2 a \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{c+\sqrt {4 a b+c^2}+2 a x^3} \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {(2 a c) \int \left (\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx}{\sqrt {4 a b+c^2}}+\frac {(2 a c) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx}{\sqrt {4 a b+c^2}}+\left (2 a \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \left (\frac {\left (-c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx+\left (2 a \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \left (\frac {\left (c+\sqrt {4 a b+c^2}\right ) x \sqrt [3]{b-a x^6}}{2 \left (2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6\right )}+\frac {a x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {\left (2 a^2 c\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\frac {\left (2 a^2 c\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\left (2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx+\left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx+\left (2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x^4 \sqrt [3]{b-a x^6}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx-\left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \int \frac {x \sqrt [3]{b-a x^6}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6} \, dx+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2\right ) \int \frac {x \sqrt [3]{b-a x^6}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2\right ) \int \frac {x \sqrt [3]{b-a x^6}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2}}\\ &=\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {1}{2} \left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )-\frac {1}{2} \left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b-a x^3}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2}}-\frac {\left (2 a^2 c \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 c \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \int \frac {x^4 \sqrt [3]{1-\frac {a x^6}{b}}}{-2 a b-c^2-c \sqrt {4 a b+c^2}+2 a^2 x^6} \, dx}{\sqrt [3]{1-\frac {a x^6}{b}}}\\ &=-\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c^2-c \sqrt {4 a b+c^2}\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {\left (a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a \left (c-\sqrt {4 a b+c^2}\right )^2 \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{-2 a b-c^2+c \sqrt {4 a b+c^2}+2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\left (a \left (c+\sqrt {4 a b+c^2}\right )^2 \sqrt [3]{b-a x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-\frac {a x^3}{b}}}{2 a b+c^2+c \sqrt {4 a b+c^2}-2 a^2 x^3} \, dx,x,x^2\right )}{2 \sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}\\ &=-\frac {a x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{2 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {a x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{\sqrt {4 a b+c^2} \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {a c \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^2 \sqrt [3]{b-a x^6} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{2 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c^2-c \sqrt {4 a b+c^2}\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1-\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c^2-c \sqrt {4 a b+c^2}},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c-\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {2 a^2 c x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \sqrt {4 a b+c^2} \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}-\frac {2 a^2 \left (1+\frac {c}{\sqrt {4 a b+c^2}}\right ) x^5 \sqrt [3]{b-a x^6} F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {2 a^2 x^6}{2 a b+c \left (c+\sqrt {4 a b+c^2}\right )},\frac {a x^6}{b}\right )}{5 \left (2 a b+c \left (c+\sqrt {4 a b+c^2}\right )\right ) \sqrt [3]{1-\frac {a x^6}{b}}}+\frac {\sqrt [3]{b-a x^6} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};\frac {a x^6}{b}\right )}{x \sqrt [3]{1-\frac {a x^6}{b}}}\\ \end {align*}
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Mathematica [F] time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.10, size = 147, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{b-a x^6}}{x}+\frac {\sqrt [3]{c} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{\sqrt [3]{c} x+2 \sqrt [3]{b-a x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{c} \log \left (-\sqrt [3]{c} x+\sqrt [3]{b-a x^6}\right )-\frac {1}{6} \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-a \,x^{6}+b \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}{x^{2} \left (a \,x^{6}+x^{3} c -b \right )}\, dx\]
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*} \int \frac {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}}\,{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 {\left (a\,x^6+b\right )\,{\left (b-a\,x^6\right )}^{1/3}}{x^2\,\left (a\,x^6+c\,x^3-b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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