Integrand size = 41, antiderivative size = 226 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {1}{3} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-a b^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 a^3 \log (x) \text {$\#$1}^3-2 a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^6+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a^6 \text {$\#$1}-2 a^3 \text {$\#$1}^4+\text {$\#$1}^7}\&\right ] \]
Time = 10.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x \left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {a^3 x+\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
(x*(((-2)^(1/3)*a^(1/3) + 2*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/ 3)*(a^(8/3) - (-2)^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (-((-1)^(2/3)*2^(1/3 )*a^(1/3)) + 2*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (-(2^(1/3)*a^(1/3)) + 2*b ^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^3*x + 2^(1/3)*a^(1/3)*b^(5/3)*x) /(b^2 + a^3*x)]))/(2*b^(1/3)*(x^2*(b^2 + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(5155\) vs. \(2(226)=452\).
Time = 11.10 (sec) , antiderivative size = 5155, normalized size of antiderivative = 22.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, 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.
| \(\displaystyle \int \frac {a x^2-b}{\left (2 a x^3-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {b-a x^2}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {b-a x^2}{\sqrt [3]{x a^3+b^2} \left (b-2 a x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {b^{10/9}-\frac {\sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{b}-\sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}-\frac {\sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [3]{-1} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}-\frac {\sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}+\frac {\sqrt [3]{-1} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{-2} \sqrt [9]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}-\frac {(-1)^{2/3} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}+\frac {\sqrt [3]{-1} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}-\frac {(-1)^{2/3} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left ((-1)^{5/9} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}+\frac {\sqrt [3]{-1} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left ((-1)^{7/9} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{x a^3+b^2}}+\frac {b^{10/9}-\frac {(-1)^{2/3} \sqrt [3]{a} b^{7/9}}{2^{2/3}}}{9 b \left (\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{2} \sqrt [9]{a} \sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \left (-\frac {(-1)^{4/9} \sqrt [9]{a} \left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) 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]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{7/9} \sqrt [9]{a} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) 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]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{4/9} \sqrt [9]{a} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) 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]{-2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{2/3} \sqrt [9]{a} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}+\frac {\sqrt [3]{-1} \sqrt [9]{a} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}-\frac {\sqrt [9]{a} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}+\frac {(-1)^{5/9} \sqrt [9]{a} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} b^{4/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{2/9} \sqrt [9]{a} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{8/9} b^{4/9} \sqrt [3]{x a^3+b^2}}-\frac {(-1)^{2/9} \sqrt [9]{a} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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]{2} \sqrt [3]{a} x}{\sqrt [3]{b}},-\frac {a^3 x}{b^2}\right )}{18\ 2^{5/9} b^{4/9} \sqrt [3]{x a^3+b^2}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}+\frac {(-1)^{5/9} \left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {(-1)^{8/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{5/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {(-1)^{7/9} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{4/9} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\sqrt [9]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {(-1)^{2/3} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9\ 2^{4/9} \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {(-1)^{5/9} \left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{8/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {(-1)^{5/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{36 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{36 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}+\frac {(-1)^{4/9} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{7/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\sqrt [9]{-1} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{7/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\sqrt [9]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{54\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{36 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}+\frac {(-1)^{2/3} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}-\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{2^{2/3} a^{8/3}+2 b^{5/3}}}+\frac {(-1)^{5/9} \left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {(-1)^{8/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{5/9} \left (\sqrt [3]{-1} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{-(-2)^{2/3} a^{8/3}-2 b^{5/3}}}-\frac {(-1)^{4/9} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\sqrt [9]{-1} \left (\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{7/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}+\frac {\sqrt [9]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}+\sqrt [3]{2} \sqrt [3]{x a^3+b^2}\right )}{18\ 2^{4/9} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{\sqrt [3]{-1} 2^{2/3} a^{8/3}-2 b^{5/3}}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
(3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(-1/18*((-1)^(4/9)*a^(1/9)*(a^(1/3) - (-2)^ (2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-2)^(1/3)*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(2^(5/9)*b^(4/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(4/9)*a^(1/9)*((-1)^(1/3)*a^(1/3) + 2^(2/3)*b^(1/3 ))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-2)^(1/3 )*a^(1/3)*x)/b^(1/3)), -((a^3*x)/b^2)])/(18*2^(5/9)*b^(4/9)*(b^2 + a^3*x)^ (1/3)) - ((-1)^(7/9)*a^(1/9)*((-1)^(1/3)*a^(1/3) + 2^(2/3)*b^(1/3))*x^(2/3 )*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-2)^(1/3)*a^(1/3) *x)/b^(1/3)), -((a^3*x)/b^2)])/(18*2^(5/9)*b^(4/9)*(b^2 + a^3*x)^(1/3)) - (a^(1/9)*(a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*Appel lF1[2/3, 1, 1/3, 5/3, (2^(1/3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*2^ (5/9)*b^(4/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(1/3)*a^(1/9)*(a^(1/3) - 2^(2/3 )*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (2^( 1/3)*a^(1/3)*x)/b^(1/3), -((a^3*x)/b^2)])/(18*2^(5/9)*b^(4/9)*(b^2 + a^3*x )^(1/3)) - ((-1)^(2/3)*a^(1/9)*(a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(1 + (a ^3*x)/b^2)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (2^(1/3)*a^(1/3)*x)/b^(1/3), - ((a^3*x)/b^2)])/(18*2^(5/9)*b^(4/9)*(b^2 + a^3*x)^(1/3)) - ((-1)^(2/9)*a^( 1/9)*(2^(1/3)*a^(1/3) + 2*(-1)^(1/3)*b^(1/3))*x^(2/3)*(1 + (a^3*x)/b^2)^(1 /3)*AppellF1[2/3, 1, 1/3, 5/3, ((-1)^(2/3)*2^(1/3)*a^(1/3)*x)/b^(1/3), -(( a^3*x)/b^2)])/(18*2^(8/9)*b^(4/9)*(b^2 + a^3*x)^(1/3)) + ((-1)^(5/9)*a^...
3.27.3.3.1 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 = 0.37 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.46
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}-2 a \,b^{5}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{3} a^{3}+a^{6}-a \,b^{3}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R} \left (\textit {\_R}^{6}-2 \textit {\_R}^{3} a^{3}+a^{6}\right )}\right )}{3}\) | \(105\) |
-1/3*sum(1/_R*(_R^6-2*_R^3*a^3+a^6-a*b^3)*ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3 ))/x)/(_R^6-2*_R^3*a^3+a^6),_R=RootOf(_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9-2*a*b ^5))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.42 (sec) , antiderivative size = 68055, normalized size of antiderivative = 301.13 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 10.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.14 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} - b\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.18 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} - b\right )}} \,d x } \]
Not integrable
Time = 3.68 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.01 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} - b\right )}} \,d x } \]
Not integrable
Time = 7.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.17 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {b-a\,x^2}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a\,x^3\right )} \,d x \]