Integrand size = 26, antiderivative size = 191 \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3} c}-\frac {a \log \left (-x+\sqrt [3]{x+x^3}\right )}{6 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{12 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{6 c d} \]
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Leaf count is larger than twice the leaf count of optimal. \(1609\) vs. \(2(191)=382\).
Time = 3.74 (sec) , antiderivative size = 1609, normalized size of antiderivative = 8.42, number of steps used = 61, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6857, 285, 335, 281, 337, 973, 477, 476, 495, 503, 524} \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\frac {a \sqrt [3]{x^3+x} x}{2 c}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {a \sqrt [3]{x^3+x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{d}-\sqrt [3]{-c} x^2\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{-1} \sqrt [3]{-c} x^2+\sqrt [3]{d}\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^2\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {a \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 c \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \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 )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}} \]
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Rule 281
Rule 285
Rule 335
Rule 337
Rule 476
Rule 477
Rule 495
Rule 503
Rule 524
Rule 973
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^6\right )}{d+c x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\sqrt [3]{x+x^3} \int \left (\frac {a \sqrt [3]{x} \sqrt [3]{1+x^2}}{c}+\frac {(b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}}{c \left (d+c x^6\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\left (a \sqrt [3]{x+x^3}\right ) \int \sqrt [3]{x} \sqrt [3]{1+x^2} \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{d+c x^6} \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{3 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x^3\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x^3\right )}\right ) \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt {d}-\sqrt {-c} x^3} \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt {d}+\sqrt {-c} x^3} \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}-\sqrt [6]{-c} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x\right )}\right ) \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}-\sqrt [6]{-c} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x\right )}\right ) \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}-\sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}-\sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}-\sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}-\sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}-\sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \left (-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-c} c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\left (\sqrt [3]{-c}+\sqrt [3]{d}\right ) (b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3} \left (\sqrt [3]{d}-\sqrt [3]{-c} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-c} c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\right )+2 \left (-\frac {\left ((-1)^{2/3} (b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-c} c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((-1)^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{-c}+\sqrt [3]{d}\right ) (b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-c} c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\right )+2 \left (\frac {\left (\sqrt [3]{-1} (b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{-c} c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\left (c+\sqrt [3]{-1} (-c)^{2/3} \sqrt [3]{d}\right ) (b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3} \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^3\right )} \, dx,x,x^{2/3}\right )}{4 c^2 d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\right ) \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\frac {\sqrt [3]{x+x^3} \left (a d \left (6 x^{4/3} \sqrt [3]{1+x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 \log \left (c \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )+(-2 b c+2 a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]\right )}{12 c d \sqrt [3]{x} \sqrt [3]{1+x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {x \left (\left (-a d +b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right ) \textit {\_R}}{\textit {\_R}^{3}-1}\right )+a d \left (-\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-3 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right )\right )}{6 d c \left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+x \left (x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}\right )\right ) \left (-{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}\) | \(206\) |
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Exception generated. \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\int { \frac {{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{c x^{6} + d} \,d x } \]
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Not integrable
Time = 3.38 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\int { \frac {{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{c x^{6} + d} \,d x } \]
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[Out]
Not integrable
Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx=\int \frac {\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}}{c\,x^6+d} \,d x \]
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