Integrand size = 35, antiderivative size = 156 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\frac {\text {RootSum}\left [a^3-a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b}-\frac {\text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b} \]
Time = 0.72 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a x} \left (\text {RootSum}\left [a^3-a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-\text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{6 a b \sqrt [3]{x^2 (-b+a x)}} \]
(x^(2/3)*(-b + a*x)^(1/3)*(RootSum[a^3 - a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - # 1^9 & , (-Log[x^(1/3)] + Log[(-b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ] - Root Sum[a^3 + a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x^(1/3)] + Log[(- b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(6*a*b*(x^2*(-b + a*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(6787\) vs. \(2(156)=312\).
Time = 12.55 (sec) , antiderivative size = 6787, normalized size of antiderivative = 43.51, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt [3]{a x^3-b x^2} \left (a^2 x^6-b^2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a x-b} \int -\frac {x^{7/3}}{\sqrt [3]{a x-b} \left (b^2-a^2 x^6\right )}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a x-b} \int \frac {x^{7/3}}{\sqrt [3]{a x-b} \left (b^2-a^2 x^6\right )}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a x-b} \int \frac {x^3}{\sqrt [3]{a x-b} \left (b^2-a^2 x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a x-b} \int \left (-\frac {1}{2 a \left (a x^3-b\right ) \sqrt [3]{a x-b}}-\frac {1}{2 a \left (a x^3+b\right ) \sqrt [3]{a x-b}}\right )d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a x-b} \left (\frac {(-1)^{2/3} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {\sqrt [3]{-1} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{2/3} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {\sqrt [3]{-1} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {\sqrt [3]{a} x}{\sqrt [3]{b}},\frac {a x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{7/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{4/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {\sqrt [9]{-1} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{7/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{4/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {\sqrt [9]{-1} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{8/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{5/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{2/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{8/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{5/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{2/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \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 x}{b}\right )}{36 a^{8/9} b^{10/9} \sqrt [3]{a x-b}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}-\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b}-\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b}-\frac {(-1)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {\sqrt [3]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}+\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}+\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}+\frac {(-1)^{2/3} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {\sqrt [3]{-1} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}+\frac {\arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {(-1)^{7/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {(-1)^{4/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {(-1)^{8/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}-\frac {(-1)^{5/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}+\frac {(-1)^{2/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{18 \sqrt {3} a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}-\frac {\log \left (-\sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{108 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{108 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{27 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {(-1)^{8/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{5/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}+\frac {(-1)^{2/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {\log \left (\sqrt [3]{-1} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}+\frac {\log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b}-\frac {\log \left (-(-1)^{2/3} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {\log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b}-\frac {(-1)^{7/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}+\frac {(-1)^{4/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {\sqrt [9]{-1} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{108 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{2/3} \log \left (\sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{-(-1)^{2/3} a^{2/3}-b^{2/3}} b}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b}+\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}+\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b}-\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b}+\frac {\log \left (\sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a x-b}\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b}+\frac {(-1)^{2/3} \log \left (\sqrt [9]{b} \sqrt [3]{a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {\sqrt [3]{-1} \log \left (\sqrt [9]{b} \sqrt [3]{a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}+\frac {\log \left (\sqrt [9]{b} \sqrt [3]{a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b}-\frac {(-1)^{7/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {(-1)^{4/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}-\frac {\sqrt [9]{-1} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{\sqrt [3]{-1} a^{2/3}+b^{2/3}} b}+\frac {(-1)^{8/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}-\frac {(-1)^{5/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}+\frac {(-1)^{2/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}}+\sqrt [3]{a x-b}\right )}{36 a^{10/9} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^{2/3}} b}\right )}{\sqrt [3]{a x^3-b x^2}}\) |
(-3*x^(2/3)*(-b + a*x)^(1/3)*((x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1 , 1/3, 5/3, -((a^(1/3)*x)/b^(1/3)), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a *x)^(1/3)) - ((-1)^(1/3)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((a^(1/3)*x)/b^(1/3)), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1 /3)) + ((-1)^(2/3)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -((a^(1/3)*x)/b^(1/3)), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) + (x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/ 3), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) - ((-1)^(1/3)*x^(2/3) *(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/3), (a*x) /b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) + ((-1)^(2/3)*x^(2/3)*(1 - (a* x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (a^(1/3)*x)/b^(1/3), (a*x)/b])/(36* a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) - ((-1)^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/ 3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a^(1/3)*x)/b^(1/3)), (a*x)/b]) /(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) + ((-1)^(4/9)*x^(2/3)*(1 - (a*x)/b )^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a^(1/3)*x)/b^(1/3)), (a*x )/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) - ((-1)^(7/9)*x^(2/3)*(1 - (a *x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(((-1)^(1/3)*a^(1/3)*x)/b^(1/3)), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) - ((-1)^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, ((-1)^(1/3)*a^(1/3)*x)/b^(1/3 ), (a*x)/b])/(36*a^(8/9)*b^(10/9)*(-b + a*x)^(1/3)) + ((-1)^(4/9)*x^(2/...
3.22.41.3.1 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 = 0.66 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a \,\textit {\_Z}^{6}+3 a^{2} \textit {\_Z}^{3}-a^{3}-a \,b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a \,\textit {\_Z}^{6}+3 a^{2} \textit {\_Z}^{3}-a^{3}+a \,b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6 a b}\) | \(131\) |
1/6*(-sum(ln((-_R*x+(x^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^9-3*_Z^6*a+3*_ Z^3*a^2-a^3-a*b^2))+sum(ln((-_R*x+(x^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^ 9-3*_Z^6*a+3*_Z^3*a^2-a^3+a*b^2)))/a/b
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.36 (sec) , antiderivative size = 48833, normalized size of antiderivative = 313.03 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 12.58 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.22 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 5.77 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.23 \[ \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx=-\int \frac {x^3}{\left (b^2-a^2\,x^6\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \]