Integrand size = 34, antiderivative size = 199 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \left (b^2 x^2+a^3 x^3\right )^{2/3} \left (3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3+6545 a b^9 x^3+4860 a^{12} b^2 x^4-7854 a^4 b^7 x^4-7290 a^{15} x^5+11781 a^7 b^5 x^5\right )}{52360 b^{13} x^7}+\frac {a^2 \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^3} \]
Time = 0.00 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.67 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {9 \left (3080 b^{12}-220 a^3 b^{10} x+300 a^6 b^8 x^2-450 a^9 b^6 x^3+6545 a b^{11} x^3+810 a^{12} b^4 x^4-1309 a^4 b^9 x^4-2430 a^{15} b^2 x^5+3927 a^7 b^7 x^5-7290 a^{18} x^6+11781 a^{10} b^5 x^6\right )+52360 a b^{10} x^{17/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [-1+3 a^3 \text {$\#$1}^3-3 a^6 \text {$\#$1}^6+a^9 \text {$\#$1}^9+a b^5 \text {$\#$1}^9\&,\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right )-2 a^3 \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right ) \text {$\#$1}^3+a^6 \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 \text {$\#$1}^2-2 a^5 \text {$\#$1}^5+a^8 \text {$\#$1}^8+b^5 \text {$\#$1}^8}\&\right ]}{157080 b^{13} x^5 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(9*(3080*b^12 - 220*a^3*b^10*x + 300*a^6*b^8*x^2 - 450*a^9*b^6*x^3 + 6545* a*b^11*x^3 + 810*a^12*b^4*x^4 - 1309*a^4*b^9*x^4 - 2430*a^15*b^2*x^5 + 392 7*a^7*b^7*x^5 - 7290*a^18*x^6 + 11781*a^10*b^5*x^6) + 52360*a*b^10*x^(17/3 )*(b^2 + a^3*x)^(1/3)*RootSum[-1 + 3*a^3*#1^3 - 3*a^6*#1^6 + a^9*#1^9 + a* b^5*#1^9 & , (Log[x^(1/3)/(b^2 + a^3*x)^(1/3) - #1] - 2*a^3*Log[x^(1/3)/(b ^2 + a^3*x)^(1/3) - #1]*#1^3 + a^6*Log[x^(1/3)/(b^2 + a^3*x)^(1/3) - #1]*# 1^6)/(a^2*#1^2 - 2*a^5*#1^5 + a^8*#1^8 + b^5*#1^8) & ])/(157080*b^13*x^5*( x^2*(b^2 + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(3746\) vs. \(2(199)=398\).
Time = 6.92 (sec) , antiderivative size = 3746, normalized size of antiderivative = 18.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (a x^3-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int -\frac {1}{x^{20/3} \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{20/3} \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^6 \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {a^2}{b^2 \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}+\frac {a}{b^2 x^3 \sqrt [3]{x a^3+b^2}}+\frac {1}{b x^6 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \left (\frac {729 \left (x a^3+b^2\right )^{2/3} a^{15}}{5236 b^{13} x^{2/3}}-\frac {243 \left (x a^3+b^2\right )^{2/3} a^{12}}{2618 b^{11} x^{5/3}}+\frac {405 \left (x a^3+b^2\right )^{2/3} a^9}{5236 b^9 x^{8/3}}-\frac {9 \left (x a^3+b^2\right )^{2/3} a^7}{40 b^8 x^{2/3}}-\frac {90 \left (x a^3+b^2\right )^{2/3} a^6}{1309 b^7 x^{11/3}}+\frac {3 \left (x a^3+b^2\right )^{2/3} a^4}{20 b^6 x^{5/3}}+\frac {15 \left (x a^3+b^2\right )^{2/3} a^3}{238 b^5 x^{14/3}}+\frac {(-1)^{2/3} 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]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [3]{-1} 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]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}+\frac {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]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{7/9} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{4/9} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [9]{-1} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{8/9} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{5/9} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{2/9} 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} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right ) a^{19/9}}{18 b^{28/9} \sqrt [3]{x a^3+b^2}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) a^{17/9}}{3 \sqrt {3} b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) a^{17/9}}{3 \sqrt {3} b^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right ) a^{17/9}}{3 \sqrt {3} b^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}}}-\frac {(-1)^{7/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {(-1)^{4/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {(-1)^{8/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {(-1)^{5/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}+\frac {(-1)^{2/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {(-1)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}+b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}+b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}+b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{17/9}}{54 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{17/9}}{54 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {2 \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{17/9}}{27 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}+\frac {\log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}}}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {\log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}}}-\frac {(-1)^{2/3} \log \left (\sqrt [9]{b} \sqrt [3]{a^{8/3}+b^{5/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [9]{b} \sqrt [3]{a^{8/3}+b^{5/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\log \left (\sqrt [9]{b} \sqrt [3]{a^{8/3}+b^{5/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{6 b^3 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{6 b^3 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}}}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{6 b^3 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}}}-\frac {(-1)^{7/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {(-1)^{4/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}-\frac {\sqrt [9]{-1} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}-b^{5/3}}}+\frac {(-1)^{8/9} \log \left (\sqrt [9]{b} \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {(-1)^{5/9} \log \left (\sqrt [9]{b} \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}+\frac {(-1)^{2/9} \log \left (\sqrt [9]{b} \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}+\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {\left (x a^3+b^2\right )^{2/3} a}{8 b^4 x^{8/3}}-\frac {\left (x a^3+b^2\right )^{2/3}}{17 b^3 x^{17/3}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
(-3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(-1/17*(b^2 + a^3*x)^(2/3)/(b^3*x^(17/3)) + (15*a^3*(b^2 + a^3*x)^(2/3))/(238*b^5*x^(14/3)) - (90*a^6*(b^2 + a^3*x)^ (2/3))/(1309*b^7*x^(11/3)) + (405*a^9*(b^2 + a^3*x)^(2/3))/(5236*b^9*x^(8/ 3)) - (a*(b^2 + a^3*x)^(2/3))/(8*b^4*x^(8/3)) - (243*a^12*(b^2 + a^3*x)^(2 /3))/(2618*b^11*x^(5/3)) + (3*a^4*(b^2 + a^3*x)^(2/3))/(20*b^6*x^(5/3)) + (729*a^15*(b^2 + a^3*x)^(2/3))/(5236*b^13*x^(2/3)) - (9*a^7*(b^2 + a^3*x)^ (2/3))/(40*b^8*x^(2/3)) + (a^(19/9)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*Appell F1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*b^(28/9)*(b ^2 + a^3*x)^(1/3)) - ((-1)^(1/3)*a^(19/9)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)* AppellF1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*b^(28 /9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(2/3)*a^(19/9)*x^(2/3)*(1 + (a^3*x)/b^2)^ (1/3)*AppellF1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18 *b^(28/9)*(b^2 + a^3*x)^(1/3)) - ((-1)^(1/9)*a^(19/9)*x^(2/3)*(1 + (a^3*x) /b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*b^(28/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(4/9)*a^(19/9)* x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a ^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*b^(28/9)*(b^2 + a^3*x)^(1/3)) - ( (-1)^(7/9)*a^(19/9)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*b^(28/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(2/9)*a^(19/9)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)...
3.25.53.3.1 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 = 0.00 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {5236 b^{10} a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 \textit {\_Z}^{6} a^{3}+3 \textit {\_Z}^{3} a^{6}-a^{9}-b^{5} a \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{7}-6561 \left (a^{15} x^{5}-\frac {2}{3} a^{12} b^{2} x^{4}-\frac {1309}{810} a^{7} b^{5} x^{5}+\frac {5}{9} a^{9} b^{4} x^{3}+\frac {1309}{1215} a^{4} b^{7} x^{4}-\frac {40}{81} a^{6} b^{6} x^{2}-\frac {1309}{1458} a \,b^{9} x^{3}+\frac {110}{243} a^{3} b^{8} x -\frac {308}{729} b^{10}\right ) \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{15708 x^{7} b^{13}}\) | \(188\) |
1/15708*(5236*b^10*a^2*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=Roo tOf(_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9-a*b^5))*x^7-6561*(a^15*x^5-2/3*a^12*b^2 *x^4-1309/810*a^7*b^5*x^5+5/9*a^9*b^4*x^3+1309/1215*a^4*b^7*x^4-40/81*a^6* b^6*x^2-1309/1458*a*b^9*x^3+110/243*a^3*b^8*x-308/729*b^10)*(x^2*(a^3*x+b^ 2))^(2/3))/x^7/b^13
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.87 (sec) , antiderivative size = 22726, normalized size of antiderivative = 114.20 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 5.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{3} - b\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^6 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\int \frac {1}{x^6\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a\,x^3\right )} \,d x \]