Integrand size = 26, antiderivative size = 164 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(4506\) vs. \(2(164)=328\).
Time = 5.53 (sec) , antiderivative size = 4506, normalized size of antiderivative = 27.48, number of steps used = 88, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6847, 6857, 245, 2181, 384, 524, 455, 57, 631, 210, 31} \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,\frac {\sqrt [3]{-1} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {(-1)^{2/3} \sqrt [3]{c} x^2}{\sqrt [3]{d}}\right ) x^{5/3}}{12 c^{8/9} d^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt {3} a \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 c \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{c} x^2-(-1)^{2/3} \sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{9 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{c} x^2+\sqrt [3]{d}\right ) \sqrt [3]{x}}{36 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {3 a \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}}-\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9} \sqrt [3]{x^3+x}}+\frac {(b c-a d) \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9} \sqrt [3]{x^3+x}} \]
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Rule 31
Rule 57
Rule 210
Rule 245
Rule 384
Rule 455
Rule 524
Rule 631
Rule 2081
Rule 2181
Rule 6847
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {b+a x^6}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (d+c x^6\right )} \, dx}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {b+a x^9}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {a}{c \sqrt [3]{1+x^3}}+\frac {b c-a d}{c \sqrt [3]{1+x^3} \left (d+c x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = \frac {\left (3 a \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (d+c x^9\right )} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}+\frac {\left (3 (b c-a d) \sqrt [3]{x} \sqrt [3]{1+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]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt {3} a \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c \sqrt [3]{x+x^3}}-\frac {3 a \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-\sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [9]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+\sqrt [3]{-1} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{4/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{5/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{2/3} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}+(-1)^{7/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}}-\frac {\left ((b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{d}-(-1)^{8/9} \sqrt [9]{c} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 c d^{8/9} \sqrt [3]{x+x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Result contains complex when optimal does not.
Time = 16.52 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.17 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{1+\frac {1}{x^2}} x \left (3 a d \left (i \left (i+\sqrt {3}\right ) \log \left (\sqrt {2-2 i \sqrt {3}}-2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+\left (-1-i \sqrt {3}\right ) \log \left (\sqrt {2+2 i \sqrt {3}}+2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+2 \log \left (-1+\sqrt [3]{1+\frac {1}{x^2}}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x+x^3}} \]
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Time = 0.60 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {\left (a d -b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )-3 \left (\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) d a}{6 d c}\) | \(148\) |
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Exception generated. \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 25.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.15 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a x^{6} + b}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (c x^{6} + d\right )}\, dx \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 3.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 6.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a\,x^6+b}{\left (c\,x^6+d\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]
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