Integrand size = 32, antiderivative size = 200 \[ \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {3 \left (b^2 x^2+a^3 x^3\right )^{2/3} \left (3080 b^{10}-3300 a^3 b^8 x+3600 a^6 b^6 x^2-4050 a^9 b^4 x^3-6545 a b^9 x^3+4860 a^{12} b^2 x^4+7854 a^4 b^7 x^4-7290 a^{15} x^5-11781 a^7 b^5 x^5\right )}{52360 b^{13} x^7}-\frac {a^2 \text {RootSum}\left [a^9-a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^3} \]
Time = 0.27 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.68 \[ \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {9 \left (-3080 b^{12}+220 a^3 b^{10} x-300 a^6 b^8 x^2+450 a^9 b^6 x^3+6545 a b^{11} x^3-810 a^{12} b^4 x^4-1309 a^4 b^9 x^4+2430 a^{15} b^2 x^5+3927 a^7 b^7 x^5+7290 a^{18} x^6+11781 a^{10} b^5 x^6\right )-52360 a b^{10} x^{17/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [-1+3 a^3 \text {$\#$1}^3-3 a^6 \text {$\#$1}^6+a^9 \text {$\#$1}^9-a b^5 \text {$\#$1}^9\&,\frac {\log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right )-2 a^3 \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right ) \text {$\#$1}^3+a^6 \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 \text {$\#$1}^2-2 a^5 \text {$\#$1}^5+a^8 \text {$\#$1}^8-b^5 \text {$\#$1}^8}\&\right ]}{157080 b^{13} x^5 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(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 + 39 27*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. \(3786\) vs. \(2(200)=400\).
Time = 6.97 (sec) , antiderivative size = 3786, normalized size of antiderivative = 18.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2467, 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 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{20/3} \sqrt [3]{x a^3+b^2} \left (a x^3+b\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^6 \sqrt [3]{x a^3+b^2} \left (a x^3+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {a^2}{b^2 \sqrt [3]{x a^3+b^2} \left (a x^3+b\right )}-\frac {a}{b^2 x^3 \sqrt [3]{x a^3+b^2}}+\frac {1}{b x^6 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 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)^{2/3} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}-b^{5/3}}}}{\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 {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}-b^{5/3}}}}{\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 {\sqrt [9]{b}-\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^{8/3}-b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{a^{8/3}-b^{5/3}}}-\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 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]{\sqrt [3]{-1} a^{8/3}+b^{5/3}}}+\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 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]{\sqrt [3]{-1} a^{8/3}+b^{5/3}}}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 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]{\sqrt [3]{-1} a^{8/3}+b^{5/3}}}+\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}-\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right ) a^{17/9}}{9 \sqrt {3} b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\frac {\log \left (-\sqrt [3]{a} x-\sqrt [3]{b}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}-b^{5/3}}}-\frac {(-1)^{8/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\frac {(-1)^{5/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}-\frac {(-1)^{2/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{a^{8/3}-b^{5/3}}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{a^{8/3}-b^{5/3}}}+\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [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 {\log \left (-(-1)^{2/3} \sqrt [3]{a} x-\sqrt [3]{b}\right ) a^{17/9}}{18 b^3 \sqrt [3]{a^{8/3}-(-1)^{2/3} b^{5/3}}}+\frac {(-1)^{7/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}+b^{5/3}}}-\frac {(-1)^{4/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}+b^{5/3}}}+\frac {\sqrt [9]{-1} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right ) a^{17/9}}{54 b^3 \sqrt [3]{\sqrt [3]{-1} a^{8/3}+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]{b^{5/3}-(-1)^{2/3} a^{8/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}-\frac {(-1)^{5/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}+\frac {(-1)^{2/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}-\sqrt [3]{x a^3+b^2}\right ) a^{17/9}}{18 b^3 \sqrt [3]{b^{5/3}-(-1)^{2/3} a^{8/3}}}-\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)^{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 {\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)*AppellF 1[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^(1 9/9)*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)*...
3.25.60.3.1 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 = 0.65 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {-5236 b^{10} a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 \textit {\_Z}^{6} a^{3}+3 \textit {\_Z}^{3} a^{6}-a^{9}+b^{5} a \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{7}+6561 \left (a^{15} x^{5}-\frac {2}{3} a^{12} b^{2} x^{4}+\frac {1309}{810} a^{7} b^{5} x^{5}+\frac {5}{9} a^{9} b^{4} x^{3}-\frac {1309}{1215} a^{4} b^{7} x^{4}-\frac {40}{81} a^{6} b^{6} x^{2}+\frac {1309}{1458} a \,b^{9} x^{3}+\frac {110}{243} a^{3} b^{8} x -\frac {308}{729} b^{10}\right ) \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{15708 b^{13} x^{7}}\) | \(187\) |
1/15708*(-5236*b^10*a^2*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=Ro otOf(_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))/b^13/x^7
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.96 (sec) , antiderivative size = 24658, normalized size of antiderivative = 123.29 \[ \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 5.12 (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 = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} + b\right )} x^{6}} \,d x } \]
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 = 7.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \left (b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{x^6\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a\,x^3+b\right )} \,d x \]