\(\int \frac {1}{\sqrt [3]{-b x+a x^3} (d+c x^6)} \, dx\) [1020]

Optimal result

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]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4634\) vs. \(2(77)=154\).

Time = 8.83 (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}} \]

[In]

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

[Out]

-1/12*(c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/
(d^(10/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/3)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/
3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/3)*c^(1/9)*x^(5/3)*(1
- (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -((c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*
x^3)^(1/3)) + ((-1)^(1/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1
/3)*c^(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(4/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^
(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(
1/3)) + ((-1)^(7/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, ((-1)^(1/3)*c^
(1/3)*x^2)/d^(1/3)])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*
AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3
)) + ((-1)^(5/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(
1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(8/9)*c^(1/9)*x^(5/3)*(1 - (a*x^2)/b)^(1/3)*
AppellF1[2/3, 1/3, 1, 5/3, (a*x^2)/b, -(((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3))])/(12*d^(10/9)*(-(b*x) + a*x^3)^(1/3
)) - (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-b
+ a*x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)
) - (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(
-b + a*x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^
(1/3)) + (x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 + (2*(b*c^(1/3) + a*d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-b + a*x^
2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - (x^(1/3)*(-b +
a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(
b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/3)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(
1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))
^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(2/3)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a
*x^2)^(1/3))/(b*c^(1/3) + a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(1/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(4/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(7/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) - (-1)^(1/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(2/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) + ((-1)^(5/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - ((-1)^(8/9)*x^(1/3)*(-b + a*x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-b + a*x^2)^(1/3))/(b*c^(1/3
) + (-1)^(2/3)*a*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x)
+ a*x^3)^(1/3)) - (x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(12*((-1)^(1/3)*b*c^(
1/3) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(2/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2
/3)*d^(1/3)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((
-1)^(5/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d
^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(8/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[-((-1)^(2/3)*d^(1/3
)) - c^(1/3)*x^2])/(36*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + (x^(1/3)*(-b
+ a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(9*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) -
((-1)^(1/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*
(-(b*x) + a*x^3)^(1/3)) + ((-1)^(2/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + c^(1/3)*x^2])/(36*(b*c^(1/3) +
a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - (x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3
)*x^2])/(12*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(1/9)*x^(1/3)*
(-b + a*x^2)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(36*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)
*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(4/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(36*(
b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(7/9)*x^(1/3)*(-b + a*x^2)^(1/
3)*Log[d^(1/3) + (-1)^(2/3)*c^(1/3)*x^2])/(36*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3
)^(1/3)) - (x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) + a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b
*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^
(1/3) + a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*
x^3)^(1/3)) - ((-1)^(2/3)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) + a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^
(1/3)])/(12*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(1/9)*x^(1/3)*(-b + a*x^2)^(
1/3)*Log[(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) - (-1)^(1/3)*a
*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(4/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) - (-1)
^(1/3)*a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-
(b*x) + a*x^3)^(1/3)) + ((-1)^(7/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3) +
c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) - (-1)^(1/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((
-1)^(2/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)
])/(12*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + ((-1)^(5/9)*x^(1/3)*(-b + a*
x^2)^(1/3)*Log[(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) + (-1)^(
2/3)*a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) - ((-1)^(8/9)*x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3)
+ (-1)^(2/3)*a*d^(1/3))^(1/3) + c^(1/9)*(-b + a*x^2)^(1/3)])/(12*(b*c^(1/3) + (-1)^(2/3)*a*d^(1/3))^(1/3)*d^(8
/9)*(-(b*x) + a*x^3)^(1/3)) - (x^(1/3)*(-b + a*x^2)^(1/3)*Log[(b*c^(1/3) + a*d^(1/3))^(1/3)*x^(2/3) - d^(1/9)*
(-b + a*x^2)^(1/3)])/(4*(b*c^(1/3) + a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + (x^(1/3)*(-b + a*x^2)^
(1/3)*Log[((-1)^(1/3)*b*c^(1/3) - a*d^(1/3))^(1/3)*x^(2/3) + d^(1/9)*(-b + a*x^2)^(1/3)])/(4*((-1)^(1/3)*b*c^(
1/3) - a*d^(1/3))^(1/3)*d^(8/9)*(-(b*x) + a*x^3)^(1/3)) + (x^(1/3)*(-b + a*x^2)^(1/3)*Log[(-((-1)^(2/3)*b*c^(1
/3)) - a*d^(1/3))^(1/3)*x^(2/3) + d^(1/9)*(-b + a*x^2)^(1/3)])/(4*(-((-1)^(2/3)*b*c^(1/3)) - a*d^(1/3))^(1/3)*
d^(8/9)*(-(b*x) + a*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 58

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] && NegQ
[(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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

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]{-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*}

Mathematica [A] (verified)

Time = 8.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{-b+a x^2} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d \sqrt [3]{-b x+a x^3}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 4.46 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 a d \,\textit {\_Z}^{6}+3 a^{2} d \,\textit {\_Z}^{3}-d \,a^{3}-b^{3} c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (a \,x^{2}-b \right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 d}\)	\(71\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 6.56 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 8.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{{\left (a\,x^3-b\,x\right )}^{1/3}\,\left (c\,x^6+d\right )} \,d x \]

[In]

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

[Out]

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