Integrand size = 32, antiderivative size = 174 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{-x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \]
Time = 16.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=-\frac {\sqrt [3]{-1+\frac {1}{x^2}} x \left (3 a d \left (-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+\frac {1}{x^2}}}{\sqrt {3}}\right )-2 \log \left (c \left (1+\sqrt [3]{-1+\frac {1}{x^2}}\right )\right )+\log \left (1-\sqrt [3]{-1+\frac {1}{x^2}}+\left (-1+\frac {1}{x^2}\right )^{2/3}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d-3 d \text {$\#$1}^3-3 d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{-1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x \left (-1+x^2\right )}} \]
-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))
Leaf count is larger than twice the leaf count of optimal. \(3520\) vs. \(2(174)=348\).
Time = 6.35 (sec) , antiderivative size = 3520, normalized size of antiderivative = 20.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 2035, 7266, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a x^6-b}{\sqrt [3]{x^3-x} \left (c x^6-d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {b-a x^6}{\sqrt [3]{x} \sqrt [3]{x^2-1} \left (d-c x^6\right )}dx}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {\sqrt [3]{x} \left (b-a x^6\right )}{\sqrt [3]{x^2-1} \left (d-c x^6\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {b-a x^3}{\sqrt [3]{x-1} \left (d-c x^3\right )}dx^{2/3}}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \left (\frac {a}{c \sqrt [3]{x-1}}+\frac {b c-a d}{c \sqrt [3]{x-1} \left (d-c x^3\right )}\right )dx^{2/3}}{2 \sqrt [3]{x^3-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \left (\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}-\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}+\frac {(b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}-\frac {(-1)^{7/9} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}+\frac {(-1)^{4/9} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}-\frac {\sqrt [9]{-1} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,-\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}+\frac {(-1)^{8/9} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}-\frac {(-1)^{5/9} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}+\frac {(-1)^{2/9} (b c-a d) \sqrt [3]{1-x} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x,\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9} \sqrt [3]{x-1}}+\frac {(-1)^{2/3} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [3]{-1} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{7/9} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{4/9} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [9]{-1} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{8/9} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{5/9} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{2/9} (b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{x-1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {a \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3} c}-\frac {(b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x-1}}}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \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-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \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-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{2/3} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [3]{-1} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{7/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{4/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [9]{-1} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{8/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{5/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{2/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{x-1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {a \log \left (\sqrt [3]{x-1}-x^{2/3}\right )}{2 c}+\frac {(b c-a d) \log \left (-\frac {\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x-1}\right )}{6 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (\frac {\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x-1}\right )}{6 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (\frac {\sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x-1}\right )}{6 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{2/3} (b c-a d) \log \left (\sqrt [3]{d}-\sqrt [3]{c} x\right )}{54 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [3]{-1} (b c-a d) \log \left (\sqrt [3]{d}-\sqrt [3]{c} x\right )}{54 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {2 (b c-a d) \log \left (\sqrt [3]{d}-\sqrt [3]{c} x\right )}{27 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (\sqrt [3]{-1} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{18 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{8/9} (b c-a d) \log \left (\sqrt [3]{-1} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{5/9} (b c-a d) \log \left (\sqrt [3]{-1} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{2/9} (b c-a d) \log \left (\sqrt [3]{-1} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{18 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{7/9} (b c-a d) \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{4/9} (b c-a d) \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [9]{-1} (b c-a d) \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}\right )}{2 \sqrt [3]{x^3-x}}\) |
(3*x^(1/3)*(-1 + x^2)^(1/3)*(((b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2 /3, 1/3, 1, 5/3, x, (c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1 /3)) - ((-1)^(1/3)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, (c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) + ((-1 )^(2/3)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, (c ^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) - ((-1)^(1/9)*(b* c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -(((-1)^(1/3) *c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) + ((-1)^(4/9)* (b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -(((-1)^(1 /3)*c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) - ((-1)^(7/ 9)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, -(((-1) ^(1/3)*c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) + ((-1)^ (2/9)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, ((-1 )^(2/3)*c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) - ((-1)^ (5/9)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, ((-1 )^(2/3)*c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) + ((-1)^ (8/9)*(b*c - a*d)*(1 - x)^(1/3)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, x, ((-1 )^(2/3)*c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)*(-1 + x)^(1/3)) + ((b*c - a*d)*ArcTan[(1 - (2*c^(1/9)*(-1 + x)^(1/3))/(c^(1/3) - d^(1/3))^(1/3))/S qrt[3]])/(9*Sqrt[3]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)) - ((-1)^(1/3)*...
3.23.88.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.45 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {\left (a d -b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )-3 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) d a}{6 d c}\) | \(148\) |
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
Exception generated. \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Not integrable
Time = 28.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.15 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int \frac {a x^{6} - b}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (c x^{6} - d\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.18 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int { \frac {a x^{6} - b}{{\left (c x^{6} - d\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 3.60 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int { \frac {a x^{6} - b}{{\left (c x^{6} - d\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 6.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.17 \[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx=\int \frac {b-a\,x^6}{{\left (x^3-x\right )}^{1/3}\,\left (d-c\,x^6\right )} \,d x \]