Integrand size = 34, antiderivative size = 133 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \left (5 b^4-6 a^3 b^2 x+9 a^6 x^2\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{40 b^7 x^4}+\frac {a \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^2} \]
Time = 0.67 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {9 \left (5 b^6-a^3 b^4 x+3 a^6 b^2 x^2+9 a^9 x^3\right )+40 a b^5 x^{8/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 b^7 x^2 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(9*(5*b^6 - a^3*b^4*x + 3*a^6*b^2*x^2 + 9*a^9*x^3) + 40*a*b^5*x^(8/3)*(b^2 + a^3*x)^(1/3)*RootSum[a^9 + a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , ( -Log[x^(1/3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*b^7*x^ 2*(x^2*(b^2 + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(3580\) vs. \(2(133)=266\).
Time = 7.63 (sec) , antiderivative size = 3580, normalized size of antiderivative = 26.92, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a x^3-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int -\frac {1}{x^{11/3} \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{11/3} \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^3 \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {a}{b \sqrt [3]{x a^3+b^2} \left (b-a x^3\right )}+\frac {1}{b x^3 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \left (-\frac {9 \left (x a^3+b^2\right )^{2/3} a^6}{40 b^7 x^{2/3}}+\frac {3 \left (x a^3+b^2\right )^{2/3} a^3}{20 b^5 x^{5/3}}+\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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{10/9}}{18 b^{19/9} \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^{8/9}}{3 \sqrt {3} b^2 \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^{8/9}}{3 \sqrt {3} b^2 \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^{8/9}}{3 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \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^{8/9}}{9 \sqrt {3} b^2 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{8/9}}{54 b^2 \sqrt [3]{a^{8/3}+b^{5/3}}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{8/9}}{54 b^2 \sqrt [3]{a^{8/3}+b^{5/3}}}+\frac {2 \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) a^{8/9}}{27 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{54 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{6 b^2 \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^{8/9}}{6 b^2 \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^{8/9}}{6 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \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^{8/9}}{18 b^2 \sqrt [3]{-(-1)^{2/3} a^{8/3}-b^{5/3}}}-\frac {\left (x a^3+b^2\right )^{2/3}}{8 b^3 x^{8/3}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
(-3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(-1/8*(b^2 + a^3*x)^(2/3)/(b^3*x^(8/3)) + (3*a^3*(b^2 + a^3*x)^(2/3))/(20*b^5*x^(5/3)) - (9*a^6*(b^2 + a^3*x)^(2/3)) /(40*b^7*x^(2/3)) + (a^(10/9)*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^(19/9)*(b^2 + a ^3*x)^(1/3)) - ((-1)^(1/3)*a^(10/9)*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^(19/9)*(b ^2 + a^3*x)^(1/3)) + ((-1)^(2/3)*a^(10/9)*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^(19 /9)*(b^2 + a^3*x)^(1/3)) - ((-1)^(1/9)*a^(10/9)*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^(19/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(4/9)*a^(10/9)*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^(19/9)*(b^2 + a^3*x)^(1/3)) - ((-1)^( 7/9)*a^(10/9)*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^(19/9)*(b^2 + a^3 *x)^(1/3)) + ((-1)^(2/9)*a^(10/9)*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1 [2/3, 1, 1/3, 5/3, ((-1)^(2/3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*b^ (19/9)*(b^2 + a^3*x)^(1/3)) - ((-1)^(5/9)*a^(10/9)*x^(2/3)*(1 + (a^3*x)/b^ 2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, ((-1)^(2/3)*a^(1/3)*x)/b^(1/3), -((a^3 *x)/b^2)])/(18*b^(19/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(8/9)*a^(10/9)*x^(...
3.20.7.3.1 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 = 0.57 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(\frac {40 b^{5} a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}-b^{5} a \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+81 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}} a^{6} x^{2}-54 a^{3} b^{2} x \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}+45 b^{4} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{120 x^{4} b^{7}}\) | \(150\) |
1/120/x^4*(40*b^5*a*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf (_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9-a*b^5))*x^4+81*(x^2*(a^3*x+b^2))^(2/3)*a^6 *x^2-54*a^3*b^2*x*(x^2*(a^3*x+b^2))^(2/3)+45*b^4*(x^2*(a^3*x+b^2))^(2/3))/ b^7
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.30 (sec) , antiderivative size = 22662, normalized size of antiderivative = 170.39 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 2.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{3} - b\right )}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}} \,d x } \]
Not integrable
Time = 3.17 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}} \,d x } \]
Not integrable
Time = 6.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (-b+a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\int \frac {1}{x^3\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-a\,x^3\right )} \,d x \]