Integrand size = 39, antiderivative size = 147 \[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\frac {\sqrt [3]{b-a x^6}}{x}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} x}{\sqrt [3]{c} x+2 \sqrt [3]{b-a x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{c} \log \left (-\sqrt [3]{c} x+\sqrt [3]{b-a x^6}\right )-\frac {1}{6} \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}\right ) \]
[Out]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 2.50 (sec) , antiderivative size = 1003, normalized size of antiderivative = 6.82, number of steps used = 38, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6860, 372, 371, 1576, 476, 441, 440, 525, 524} \[ \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=-\frac {2 a^2 c \sqrt [3]{b-a x^6} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {Hypergeometric2F1}\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} \]
[In]
[Out]
Rule 371
Rule 372
Rule 440
Rule 441
Rule 476
Rule 524
Rule 525
Rule 1576
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \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} \operatorname {Hypergeometric2F1}\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} \operatorname {Hypergeometric2F1}\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} \operatorname {Hypergeometric2F1}\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} \operatorname {Hypergeometric2F1}\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} \operatorname {Hypergeometric2F1}\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} \operatorname {Hypergeometric2F1}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {Hypergeometric2F1}\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {AppellF1}\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} \operatorname {Hypergeometric2F1}\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*}
Time = 1.99 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\frac {\sqrt [3]{b-a x^6}}{x}+\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} x}{\sqrt [3]{c} x+2 \sqrt [3]{b-a x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \sqrt [3]{c} \log \left (-\sqrt [3]{c} x+\sqrt [3]{b-a x^6}\right )-\frac {1}{6} \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x \sqrt [3]{b-a x^6}+\left (b-a x^6\right )^{2/3}\right ) \]
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[Out]
Time = 0.88 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-2 c^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (c^{\frac {1}{3}} x +2 \left (-a \,x^{6}+b \right )^{\frac {1}{3}}\right )}{3 c^{\frac {1}{3}} x}\right ) x -c^{\frac {1}{3}} \ln \left (\frac {c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} x \left (-a \,x^{6}+b \right )^{\frac {1}{3}}+\left (-a \,x^{6}+b \right )^{\frac {2}{3}}}{x^{2}}\right ) x +2 c^{\frac {1}{3}} \ln \left (\frac {-c^{\frac {1}{3}} x +\left (-a \,x^{6}+b \right )^{\frac {1}{3}}}{x}\right ) x +6 \left (-a \,x^{6}+b \right )^{\frac {1}{3}}}{6 x}\) | \(130\) |
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Timed out. \[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}} \,d x } \]
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\[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\int { \frac {{\left (a x^{6} + b\right )} {\left (-a x^{6} + b\right )}^{\frac {1}{3}}}{{\left (a x^{6} + c x^{3} - b\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{b-a x^6} \left (b+a x^6\right )}{x^2 \left (-b+c x^3+a x^6\right )} \, dx=\int \frac {\left (a\,x^6+b\right )\,{\left (b-a\,x^6\right )}^{1/3}}{x^2\,\left (a\,x^6+c\,x^3-b\right )} \,d x \]
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