Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {\text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(4634\) vs. \(2(77)=154\).
Time = 7.99 (sec) , antiderivative size = 4634, normalized size of antiderivative = 60.18, number of steps used = 94, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6847, 6857, 2181, 384, 525, 524, 455, 58, 631, 210, 31} \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {(-1)^{2/3} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{-1} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{7/9} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{4/9} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [9]{-1} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{8/9} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{5/9} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{2/9} \sqrt [9]{c} \sqrt [3]{1-\frac {a x^2}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x^2}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 d^{10/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{a x^2-b}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{a x^2-b}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{a x^2-b} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{a x^2-b}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{2/3} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{-1} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{7/9} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{4/9} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [9]{-1} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{8/9} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{5/9} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{2/9} \sqrt [3]{a x^2-b} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x^2-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{8/9} \sqrt [3]{a x^2-b} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{5/9} \sqrt [3]{a x^2-b} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{2/9} \sqrt [3]{a x^2-b} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{2/3} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{-1} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{a x^2-b} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{9 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{12 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{7/9} \sqrt [3]{a x^2-b} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{4/9} \sqrt [3]{a x^2-b} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [9]{-1} \sqrt [3]{a x^2-b} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{2/3} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{-1} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{7/9} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{4/9} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [9]{-1} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{8/9} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {(-1)^{5/9} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {(-1)^{2/9} \sqrt [3]{a x^2-b} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}-\frac {\sqrt [3]{a x^2-b} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{4 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{a x^2-b} \log \left (\sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{4 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9} \sqrt [3]{a x^3-b x}}+\frac {\sqrt [3]{a x^2-b} \log \left (\sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{a x^2-b}\right ) \sqrt [3]{x}}{4 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9} \sqrt [3]{a x^3-b x}} \]
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Rule 31
Rule 58
Rule 210
Rule 384
Rule 455
Rule 524
Rule 525
Rule 631
Rule 2081
Rule 2181
Rule 6847
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-b+a x^2} \left (d+c x^6\right )} \, dx}{\sqrt [3]{-b x+a x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-b x+a x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-b x+a x^3}} \\ & = -\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{-b+a x^3}} \, dx,x,x^{2/3}\right )}{6 d^{8/9} \sqrt [3]{-b x+a x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{-b+a x^2} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d \sqrt [3]{-b x+a x^3}} \]
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Time = 1.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 a d \,\textit {\_Z}^{6}+3 a^{2} d \,\textit {\_Z}^{3}-d \,a^{3}-b^{3} c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (a \,x^{2}-b \right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 d}\) | \(71\) |
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Exception generated. \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 6.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 9.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{{\left (a\,x^3-b\,x\right )}^{1/3}\,\left (c\,x^6+d\right )} \,d x \]
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