\(\int \frac {1}{\sqrt [3]{x+x^3} (b+a x^6)} \, dx\) [805]

Optimal result

Integrand size = 19, antiderivative size = 61 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=-\frac {\text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3939\) vs. \(2(61)=122\).

Time = 3.83 (sec) , antiderivative size = 3939, normalized size of antiderivative = 64.57, number of steps used = 85, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {2081, 6847, 6857, 2181, 384, 524, 455, 57, 631, 210, 31} \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=-\frac {(-1)^{2/3} \sqrt [9]{a} \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} \sqrt [9]{a} \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{a} \sqrt [3]{x^2+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {\sqrt [3]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} \sqrt [9]{a} \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]{a} x^2}{\sqrt [3]{b}}\right ) x^{5/3}}{12 b^{10/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{2 \sqrt {3} \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x^2+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right ) \sqrt [3]{x}}{6 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \log \left (-\sqrt [3]{a} x^2-(-1)^{2/3} \sqrt [3]{b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{8/9} \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{a} x^2-(-1)^{2/3} \sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{5/9} \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{a} x^2-(-1)^{2/3} \sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/9} \sqrt [3]{x^2+1} \log \left (-\sqrt [3]{a} x^2-(-1)^{2/3} \sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{2/3} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{-1} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x^2+1} \log \left (\sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{9 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{7/9} \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{4/9} \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [9]{-1} \sqrt [3]{x^2+1} \log \left ((-1)^{2/3} \sqrt [3]{a} x^2+\sqrt [3]{b}\right ) \sqrt [3]{x}}{36 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/3} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{-1} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{7/9} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{4/9} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [9]{-1} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{8/9} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {(-1)^{5/9} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {(-1)^{2/9} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{12 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x^{2/3}-\sqrt [9]{b} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} x^{2/3}-\sqrt [9]{b} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x^2+1} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} x^{2/3}+\sqrt [9]{b} \sqrt [3]{x^2+1}\right ) \sqrt [3]{x}}{4 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9} \sqrt [3]{x^3+x}} \]

[In]

Int[1/((x + x^3)^(1/3)*(b + a*x^6)),x]

[Out]

-1/12*(a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((a^(1/3)*x^2)/b^(1/3))])/(b^(10/9)*(
x + x^3)^(1/3)) + ((-1)^(1/3)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -((a^(1/3)*x^2)
/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^(2/3)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1,
5/3, -x^2, -((a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(1/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1
/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^
(4/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*
b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, ((-1)
^(1/3)*a^(1/3)*x^2)/b^(1/3)])/(12*b^(10/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*Appe
llF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) + ((-1)^(5/9)
*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^
(10/9)*(x + x^3)^(1/3)) - ((-1)^(8/9)*a^(1/9)*x^(5/3)*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, -(((-1)
^(2/3)*a^(1/3)*x^2)/b^(1/3))])/(12*b^(10/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 - (2*(a^(1/3
) - b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*
(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*x^(2/3))/(b^(1
/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^
(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(1 + x^2)^(1/3)
))/Sqrt[3]])/(2*Sqrt[3]*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^
(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) - b^(1/
3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/3)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3
))/(a^(1/3) - b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^
(2/3)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - b^(1/3))^(1/3))/Sqrt[3]])/(6*
Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(1/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (
2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) + (-1)^(1/3)*b^
(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(4/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(
1/3))/(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*
(x + x^3)^(1/3)) - ((-1)^(7/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) + (-1)
^(1/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((
-1)^(2/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)
)/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(5/9)*x^(1/3)*(1
+ x^2)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3
]*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(8/9)*x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(
1 + (2*a^(1/9)*(1 + x^2)^(1/3))/(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(a^(1/3) - (-1)^(2/
3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2]
)/(12*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(2/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[
-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1
)^(5/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(
1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(8/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*x^2])/
(36*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + a^(
1/3)*x^2])/(9*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(1/3)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(
1/3) + a^(1/3)*x^2])/(36*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(2/3)*x^(1/3)*(1 + x^2)^(1
/3)*Log[b^(1/3) + a^(1/3)*x^2])/(36*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1
/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(12*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)
) + ((-1)^(1/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(36*(a^(1/3) + (-1)^(1/3)*b^(1/
3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(4/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2]
)/(36*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(7/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[
b^(1/3) + (-1)^(2/3)*a^(1/3)*x^2])/(36*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3
)*(1 + x^2)^(1/3)*Log[(a^(1/3) - b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) - b^(1/3))^(1/3)*b^(8
/9)*(x + x^3)^(1/3)) - ((-1)^(1/3)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) - b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(
1/3)])/(12*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(2/3)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/
3) - b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - ((-1)
^(1/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/
3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(4/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) + (
-1)^(1/3)*b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3
)^(1/3)) - ((-1)^(7/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1
/3)])/(12*(a^(1/3) + (-1)^(1/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(2/9)*x^(1/3)*(1 + x^2)^(1/3)*
Log[(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*
b^(8/9)*(x + x^3)^(1/3)) - ((-1)^(5/9)*x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3) - a^(1
/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + ((-1)^(8/9)*x^(1/3)*
(1 + x^2)^(1/3)*Log[(a^(1/3) - (-1)^(2/3)*b^(1/3))^(1/3) - a^(1/9)*(1 + x^2)^(1/3)])/(12*(a^(1/3) - (-1)^(2/3)
*b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*x
^(2/3) - b^(1/9)*(1 + x^2)^(1/3)])/(4*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)
*(1 + x^2)^(1/3)*Log[(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*x^(2/3) - b^(1/9)*(1 + x^2)^(1/3)])/(4*(-((-1)^(2
/3)*a^(1/3)) + b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[(a^(1/3) - b^(1/3))^(1/3
)*x^(2/3) + b^(1/9)*(1 + x^2)^(1/3)])/(4*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)*(x + x^3)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2181

Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x
^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ
[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a 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 {1}{\sqrt [3]{1+x^3} \left (b+a 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 {1}{9 b^{8/9} \left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = -\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x+x^3}} \]

[In]

Integrate[1/((x + x^3)^(1/3)*(b + a*x^6)),x]

[Out]

-1/6*(x^(1/3)*(1 + x^2)^(1/3)*RootSum[a - b + 3*b*#1^3 - 3*b*#1^6 + b*#1^9 & , (-2*Log[x^(1/3)] + Log[(1 + x^2
)^(1/3) - x^(2/3)*#1])/#1 & ])/(b*(x + x^3)^(1/3))

Maple [N/A] (verified)

Time = 2.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{9}-3 b \,\textit {\_Z}^{6}+3 b \,\textit {\_Z}^{3}+a -b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 b}\)	\(55\)

[In]

int(1/(x^3+x)^(1/3)/(a*x^6+b),x,method=_RETURNVERBOSE)

[Out]

-1/6*sum(ln((-_R*x+((x^2+1)*x)^(1/3))/x)/_R,_R=RootOf(_Z^9*b-3*_Z^6*b+3*_Z^3*b+a-b))/b

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(x^3+x)^(1/3)/(a*x^6+b),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [N/A]

Not integrable

Time = 7.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{6} + b\right )}\, dx \]

[In]

integrate(1/(x**3+x)**(1/3)/(a*x**6+b),x)

[Out]

Integral(1/((x*(x**2 + 1))**(1/3)*(a*x**6 + b)), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int { \frac {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(x^3+x)^(1/3)/(a*x^6+b),x, algorithm="maxima")

[Out]

-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((a*x^(19/3) + b*x^(1/3))*(x^2 + 1)^(1/3)) + integrate(9/40*(9*b*x^6 + 3*b*x
^4 - b*x^2 + 5*b)/((a^2*x^(37/3) + 2*a*b*x^(19/3) + b^2*x^(1/3))*(x^2 + 1)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int { \frac {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(x^3+x)^(1/3)/(a*x^6+b),x, algorithm="giac")

[Out]

integrate(1/((a*x^6 + b)*(x^3 + x)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int \frac {1}{\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]

[In]

int(1/((b + a*x^6)*(x + x^3)^(1/3)),x)

[Out]

int(1/((b + a*x^6)*(x + x^3)^(1/3)), x)