Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {\text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d} \]
Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{-b+a x^2} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d \sqrt [3]{-b x+a x^3}} \]
-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))
Leaf count is larger than twice the leaf count of optimal. \(3450\) vs. \(2(77)=154\).
Time = 6.03 (sec) , antiderivative size = 3450, normalized size of antiderivative = 44.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{a x^3-b x} \left (c x^6+d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{a x^2-b} \int \frac {1}{\sqrt [3]{x} \sqrt [3]{a x^2-b} \left (c x^6+d\right )}dx}{\sqrt [3]{a x^3-b x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a x^2-b} \int \frac {\sqrt [3]{x}}{\sqrt [3]{a x^2-b} \left (c x^6+d\right )}d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a x^2-b} \int \frac {1}{\sqrt [3]{a x-b} \left (c x^3+d\right )}dx^{2/3}}{2 \sqrt [3]{a x^3-b x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a x^2-b} \int \left (-\frac {1}{9 d^{8/9} \left (-\sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (\sqrt [9]{-1} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (-(-1)^{2/9} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (\sqrt [3]{-1} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (-(-1)^{4/9} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left ((-1)^{5/9} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (-(-1)^{2/3} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left ((-1)^{7/9} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}-\frac {1}{9 d^{8/9} \left (-(-1)^{8/9} \sqrt [9]{c} x^{2/3}-\sqrt [9]{d}\right ) \sqrt [3]{a x-b}}\right )dx^{2/3}}{2 \sqrt [3]{a x^3-b x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{a x^2-b} \left (-\frac {(-1)^{2/3} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}+\frac {\sqrt [3]{-1} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}-\frac {\sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{7/9} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{4/9} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}+\frac {\sqrt [9]{-1} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{8/9} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}+\frac {(-1)^{5/9} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}-\frac {(-1)^{2/9} \sqrt [9]{c} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{10/9} \sqrt [3]{a x-b}}-\frac {\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-b}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9}}-\frac {\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-b}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9}}+\frac {\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-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{7/9} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{4/9} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [9]{-1} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{8/9} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{5/9} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{2/9} \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {\log \left (-\sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{8/9} \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{5/9} \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{2/9} \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {\log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {\log \left (\sqrt [3]{-1} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9}}-\frac {\log \left (-(-1)^{2/3} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{7/9} \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{4/9} \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [9]{-1} \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}+\frac {\log \left (-\frac {\sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{8/9}}+\frac {\log \left (-\frac {\sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{8/9}}-\frac {\log \left (\frac {\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{2/3} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {\log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{8/9}}\right )}{2 \sqrt [3]{a x^3-b x}}\) |
(3*x^(1/3)*(-b + a*x^2)^(1/3)*(-1/18*(c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)* AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, -((c^(1/3)*x)/d^(1/3))])/(d^(10/9)*(-b + a*x)^(1/3)) + ((-1)^(1/3)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[ 2/3, 1/3, 1, 5/3, (a*x)/b, -((c^(1/3)*x)/d^(1/3))])/(18*d^(10/9)*(-b + a*x )^(1/3)) - ((-1)^(2/3)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1 /3, 1, 5/3, (a*x)/b, -((c^(1/3)*x)/d^(1/3))])/(18*d^(10/9)*(-b + a*x)^(1/3 )) + ((-1)^(1/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, ((-1)^(1/3)*c^(1/3)*x)/d^(1/3)])/(18*d^(10/9)*(-b + a*x)^(1 /3)) - ((-1)^(4/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, ((-1)^(1/3)*c^(1/3)*x)/d^(1/3)])/(18*d^(10/9)*(-b + a*x)^ (1/3)) + ((-1)^(7/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3 , 1, 5/3, (a*x)/b, ((-1)^(1/3)*c^(1/3)*x)/d^(1/3)])/(18*d^(10/9)*(-b + a*x )^(1/3)) - ((-1)^(2/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1 /3, 1, 5/3, (a*x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^(10/9)*(-b + a*x)^(1/3)) + ((-1)^(5/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2 /3, 1/3, 1, 5/3, (a*x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^(10/9) *(-b + a*x)^(1/3)) - ((-1)^(8/9)*c^(1/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*Appel lF1[2/3, 1/3, 1, 5/3, (a*x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^( 10/9)*(-b + a*x)^(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)^(1/3)))/Sqrt[3]]/(3*Sqrt[3]*((-1)^...
3.11.21.3.1 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 = 1.44 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 a d \,\textit {\_Z}^{6}+3 a^{2} d \,\textit {\_Z}^{3}-d \,a^{3}-b^{3} c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (a \,x^{2}-b \right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 d}\) | \(71\) |
-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
Exception generated. \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 6.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 9.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + d\right )} {\left (a x^{3} - b x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{-b x+a x^3} \left (d+c x^6\right )} \, dx=\int \frac {1}{{\left (a\,x^3-b\,x\right )}^{1/3}\,\left (c\,x^6+d\right )} \,d x \]