Integrand size = 33, antiderivative size = 107 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\frac {\text {RootSum}\left [a^{18}-a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \]
Time = 0.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^{18}-a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(x^(2/3)*(b^2 + a^3*x)^(1/3)*RootSum[a^18 - a^6*b^11 - 6*a^15*#1^3 + 15*a^ 12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Log[x^(1/ 3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(6*b*(x^2*(b^2 + a^3* x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(6596\) vs. \(2(107)=214\).
Time = 12.20 (sec) , antiderivative size = 6596, normalized size of antiderivative = 61.64, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2467, 25, 2035, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt [3]{a^3 x^3+b^2 x^2} \left (a^6 x^6-b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int -\frac {1}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (b-a^6 x^6\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (b-a^6 x^6\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (b-a^6 x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-a^3 x^3\right ) \sqrt [3]{x a^3+b^2}}+\frac {1}{2 \sqrt {b} \left (a^3 x^3+\sqrt {b}\right ) \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \left (-\frac {(-1)^{2/3} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{2/3} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [3]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {\sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{7/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{4/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [9]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{7/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{4/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {\sqrt [9]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{-1} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{8/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{5/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{2/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{8/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{5/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{2/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{\frac {x a^3}{b^2}+1} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [6]{b}},-\frac {a^3 x}{b^2}\right )}{36 b^{19/18} \sqrt [3]{x a^3+b^2}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2+b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} b^{11/6}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} b^{11/6}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} b^{11/6}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} b^{11/6}}}-\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {\arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {(-1)^{7/9} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {(-1)^{4/9} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {(-1)^{8/9} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}-\frac {(-1)^{5/9} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}+\frac {(-1)^{2/9} \arctan \left (\frac {\sqrt [18]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}-\frac {(-1)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}-\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\sqrt [18]{b}}{\sqrt {3} \sqrt [18]{b}}\right )}{18 \sqrt {3} \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\frac {\log \left (-a x-\sqrt [6]{b}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {(-1)^{2/3} \log \left (\sqrt [6]{b}-a x\right )}{108 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [6]{b}-a x\right )}{108 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}+\frac {\log \left (\sqrt [6]{b}-a x\right )}{27 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {(-1)^{8/9} \log \left (-a x-(-1)^{2/3} \sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\frac {(-1)^{5/9} \log \left (-a x-(-1)^{2/3} \sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}-\frac {(-1)^{2/9} \log \left (-a x-(-1)^{2/3} \sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\frac {(-1)^{2/3} \log \left (a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {\sqrt [3]{-1} \log \left (a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {\log \left (a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {\log \left (\sqrt [3]{-1} a x-\sqrt [6]{b}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} b^{11/6}}}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{-1} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{-1} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{-1} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}+\frac {\log \left (\sqrt [3]{-1} a x+\sqrt [6]{b}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} b^{11/6}}}+\frac {\log \left (-(-1)^{2/3} a x-\sqrt [6]{b}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} b^{11/6}}}+\frac {(-1)^{7/9} \log \left (\sqrt [6]{b}-(-1)^{2/3} a x\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}-\frac {(-1)^{4/9} \log \left (\sqrt [6]{b}-(-1)^{2/3} a x\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [6]{b}-(-1)^{2/3} a x\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {\log \left (\sqrt [6]{b}-(-1)^{2/3} a x\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} b^{11/6}}}+\frac {(-1)^{7/9} \log \left ((-1)^{2/3} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}-\frac {(-1)^{4/9} \log \left ((-1)^{2/3} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\frac {\sqrt [9]{-1} \log \left ((-1)^{2/3} a x+\sqrt [6]{b}\right )}{108 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}-\frac {(-1)^{2/3} \log \left (\sqrt [18]{b} \sqrt [3]{a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [18]{b} \sqrt [3]{a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {\log \left (\sqrt [18]{b} \sqrt [3]{a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {(-1)^{7/9} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\frac {(-1)^{4/9} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}-\frac {\sqrt [9]{-1} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2+b^{11/6}}}+\frac {(-1)^{8/9} \log \left (\sqrt [18]{b} \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}-\frac {(-1)^{5/9} \log \left (\sqrt [18]{b} \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}+\frac {(-1)^{2/9} \log \left (\sqrt [18]{b} \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}-\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{b^{11/6}-(-1)^{2/3} a^2}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2-b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2+b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+b^{11/6}}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-\sqrt [3]{-1} b^{11/6}}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+\sqrt [3]{-1} b^{11/6}}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2-(-1)^{2/3} b^{11/6}}}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [3]{a} b \sqrt [3]{a^2+(-1)^{2/3} b^{11/6}}}-\frac {(-1)^{2/3} \log \left (\sqrt [18]{b} \sqrt [3]{a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [18]{b} \sqrt [3]{a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {\log \left (\sqrt [18]{b} \sqrt [3]{a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{a^2-b^{11/6}}}-\frac {(-1)^{7/9} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {(-1)^{4/9} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}-\frac {\sqrt [9]{-1} \log \left (\sqrt [18]{b} \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{\sqrt [3]{-1} a^2-b^{11/6}}}+\frac {(-1)^{8/9} \log \left (\sqrt [18]{b} \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}-\frac {(-1)^{5/9} \log \left (\sqrt [18]{b} \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}+\frac {(-1)^{2/9} \log \left (\sqrt [18]{b} \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}+\sqrt [3]{x a^3+b^2}\right )}{36 \sqrt [3]{a} b \sqrt [3]{-(-1)^{2/3} a^2-b^{11/6}}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
(-3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(-1/36*(a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^ (1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((a*x)/b^(1/6)), -((a^3*x)/b^2)])/(b^(19 /18)*(b^2 + a^3*x)^(1/3)) + ((-1)^(1/3)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^ (1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((a*x)/b^(1/6)), -((a^3*x)/b^2)])/(36*b^ (19/18)*(b^2 + a^3*x)^(1/3)) - ((-1)^(2/3)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^ 2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((a*x)/b^(1/6)), -((a^3*x)/b^2)])/(36 *b^(19/18)*(b^2 + a^3*x)^(1/3)) + (a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3) *AppellF1[2/3, 1, 1/3, 5/3, (a*x)/b^(1/6), -((a^3*x)/b^2)])/(36*b^(19/18)* (b^2 + a^3*x)^(1/3)) - ((-1)^(1/3)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3) *AppellF1[2/3, 1, 1/3, 5/3, (a*x)/b^(1/6), -((a^3*x)/b^2)])/(36*b^(19/18)* (b^2 + a^3*x)^(1/3)) + ((-1)^(2/3)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3) *AppellF1[2/3, 1, 1/3, 5/3, (a*x)/b^(1/6), -((a^3*x)/b^2)])/(36*b^(19/18)* (b^2 + a^3*x)^(1/3)) - ((-1)^(1/9)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3) *AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a*x)/b^(1/6)), -((a^3*x)/b^2)])/ (36*b^(19/18)*(b^2 + a^3*x)^(1/3)) + ((-1)^(4/9)*a^(1/3)*x^(2/3)*(1 + (a^3 *x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a*x)/b^(1/6)), -(( a^3*x)/b^2)])/(36*b^(19/18)*(b^2 + a^3*x)^(1/3)) - ((-1)^(7/9)*a^(1/3)*x^( 2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a*x) /b^(1/6)), -((a^3*x)/b^2)])/(36*b^(19/18)*(b^2 + a^3*x)^(1/3)) + ((-1)^(1/ 9)*a^(1/3)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (...
3.16.63.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{18}-6 a^{3} \textit {\_Z}^{15}+15 a^{6} \textit {\_Z}^{12}-20 a^{9} \textit {\_Z}^{9}+15 a^{12} \textit {\_Z}^{6}-6 a^{15} \textit {\_Z}^{3}+a^{18}-a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 b}\) | \(94\) |
1/6*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(_Z^18-6*_Z^15*a ^3+15*_Z^12*a^6-20*_Z^9*a^9+15*_Z^6*a^12-6*_Z^3*a^15+a^18-a^6*b^11))/b
Exception generated. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> System error: Heap exhausted (no more space for allocation).44695552 bytes available, 46036080 requested.PROCEE D WITH CAUTION.
Not integrable
Time = 2.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a^{6} x^{6} - b\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} - b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.69 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} - b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-b+a^6 x^6\right )} \, dx=-\int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a^6\,x^6\right )} \,d x \]