\(\int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} (-d+c x^6)} \, dx\) [2288]

Optimal result

Integrand size = 32, antiderivative size = 174 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{-x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4581\) vs. \(2(174)=348\).

Time = 5.26 (sec) , antiderivative size = 4581, normalized size of antiderivative = 26.33, number of steps used = 97, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2081, 6847, 6857, 245, 2181, 384, 525, 524, 455, 58, 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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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]{1-x^2} \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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\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 {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x^2-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\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)^{2/3} (b c-a d) \sqrt [3]{x^2-1} \log \left (\sqrt [3]{d}-\sqrt [3]{c} x^2\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]{d}-\sqrt [3]{c} x^2\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]{d}-\sqrt [3]{c} x^2\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 (\sqrt [3]{-1} \sqrt [3]{c} x^2+\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]{-1} \sqrt [3]{c} x^2+\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]{-1} \sqrt [3]{c} x^2+\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]{-1} \sqrt [3]{c} x^2+\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 {(b c-a d) \sqrt [3]{x^2-1} \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x^2\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 (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x^2\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 (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x^2\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 (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x^2\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}} \]

[In]

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

[Out]

((b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10
/9)*(-x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/
3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*App
ellF1[2/3, 1/3, 1, 5/3, x^2, (c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c
- a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8
/9)*d^(10/9)*(-x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^
2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(5
/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, -(((-1)^(1/3)*c^(1/3)*x^2)/d^(1/3))])/(12*c^(8/9)*d^(10/9)
*(-x + x^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/
3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(
1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/3)*c^(1/3)*x^2)/d^(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3))
+ ((-1)^(8/9)*(b*c - a*d)*x^(5/3)*(1 - x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^2, ((-1)^(2/3)*c^(1/3)*x^2)/d^
(1/3)])/(12*c^(8/9)*d^(10/9)*(-x + x^3)^(1/3)) + (Sqrt[3]*a*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(
-1 + x^2)^(1/3))/Sqrt[3]])/(2*c*(-x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*(c^(1
/3) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(
8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*((-1)^(1/3)*c^(1/3) + d^(1/3))^(
1/3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*
(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/
3)*x^(2/3))/(d^(1/9)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*c*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9)
*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) -
d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c
- a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])
/(6*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)
^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - d
^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1
/9)*(-1 + x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/
3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*
(-1 + x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^
(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(7/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1
+ x^2)^(1/3))/(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3
)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^
2)^(1/3))/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^
(8/9)*(-x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(
1/3))/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9
)*(-x + x^3)^(1/3)) + ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*c^(1/9)*(-1 + x^2)^(1/3)
)/(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3))/Sqrt[3]])/(6*Sqrt[3]*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-
x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - c^(1/3)*x^2])/(9*c*(c^(1/3) - d^(1/3))^(
1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - c^(1/3)*x^2])/
(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(2/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*L
og[d^(1/3) - c^(1/3)*x^2])/(36*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-
1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(12*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*(-x +
x^3)^(1/3)) - ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(
c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/
3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3))
- ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) - (-
1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - (-1)^(
2/3)*c^(1/3)*x^2])/(12*c*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(1/9)*(b*c
- a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - (-1)^(2/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1
/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(4/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - (-1)^(2/3)*c^(1
/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(7/9)*(b*c - a*d)*x^(1/
3)*(-1 + x^2)^(1/3)*Log[d^(1/3) - (-1)^(2/3)*c^(1/3)*x^2])/(36*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*
(-x + x^3)^(1/3)) - (3*a*x^(1/3)*(-1 + x^2)^(1/3)*Log[x^(2/3) - (-1 + x^2)^(1/3)])/(4*c*(-x + x^3)^(1/3)) + ((
b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3)
- d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - d
^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(
2/3)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^
(1/3) - d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(1/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/
3) + (-1)^(1/3)*d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)
*(-x + x^3)^(1/3)) + ((-1)^(4/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)
+ c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(7/
9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/
(12*c*(c^(1/3) + (-1)^(1/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) + ((-1)^(2/9)*(b*c - a*d)*x^(1/3)*(-1 + x
^2)^(1/3)*Log[(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) - (-1)^(2/3)*d^
(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((-1)^(5/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - (-1)^
(2/3)*d^(1/3))^(1/3) + c^(1/9)*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3
)^(1/3)) + ((-1)^(8/9)*(b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3) + c^(1/9)
*(-1 + x^2)^(1/3)])/(12*c*(c^(1/3) - (-1)^(2/3)*d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3
)*(-1 + x^2)^(1/3)*Log[((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*x^(2/3) - d^(1/9)*(-1 + x^2)^(1/3)])/(4*c*((-1)^(1
/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)*(-x + x^3)^(1/3)) - ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(-((-1)^(2/
3)*c^(1/3)) + d^(1/3))^(1/3)*x^(2/3) - d^(1/9)*(-1 + x^2)^(1/3)])/(4*c*(-((-1)^(2/3)*c^(1/3)) + d^(1/3))^(1/3)
*d^(8/9)*(-x + x^3)^(1/3)) + ((b*c - a*d)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(c^(1/3) - d^(1/3))^(1/3)*x^(2/3) + d^(
1/9)*(-1 + x^2)^(1/3)])/(4*c*(c^(1/3) - d^(1/3))^(1/3)*d^(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 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 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

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]{-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*}

Mathematica [A] (verified)

Time = 16.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=-\frac {\sqrt [3]{-1+\frac {1}{x^2}} x \left (3 a d \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+\frac {1}{x^2}}}{\sqrt {3}}\right )-2 \log \left (c \left (1+\sqrt [3]{-1+\frac {1}{x^2}}\right )\right )+\log \left (1-\sqrt [3]{-1+\frac {1}{x^2}}+\left (-1+\frac {1}{x^2}\right )^{2/3}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d-3 d \text {$\#$1}^3-3 d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{-1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x \left (-1+x^2\right )}} \]

[In]

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

[Out]

-1/12*((-1 + x^(-2))^(1/3)*x*(3*a*d*(-2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^(-2))^(1/3))/Sqrt[3]] - 2*Log[c*(1 + (-1
+ x^(-2))^(1/3))] + Log[1 - (-1 + x^(-2))^(1/3) + (-1 + x^(-2))^(2/3)]) + 2*(b*c - a*d)*RootSum[c - d - 3*d*#
1^3 - 3*d*#1^6 - d*#1^9 & , Log[(-1 + x^(-2))^(1/3) - #1]/#1 & ]))/(c*d*(x*(-1 + x^2))^(1/3))

Maple [N/A] (verified)

Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(\frac {\left (a d -b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )-3 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) d a}{6 d c}\)	\(148\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [N/A]

Not integrable

Time = 28.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.15 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int \frac {a x^{6} - b}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (c x^{6} - d\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.18 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int { \frac {a x^{6} - b}{{\left (c x^{6} - d\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.60 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int { \frac {a x^{6} - b}{{\left (c x^{6} - d\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 6.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.17 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int \frac {b-a\,x^6}{{\left (x^3-x\right )}^{1/3}\,\left (d-c\,x^6\right )} \,d x \]

[In]

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

[Out]

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