Integrand size = 19, antiderivative size = 61 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=-\frac {\text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \]
Time = 7.96 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x+x^3}} \]
-1/6*(x^(1/3)*(1 + x^2)^(1/3)*RootSum[a - b + 3*b*#1^3 - 3*b*#1^6 + b*#1^9 & , (-2*Log[x^(1/3)] + Log[(1 + x^2)^(1/3) - x^(2/3)*#1])/#1 & ])/(b*(x + x^3)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(2940\) vs. \(2(61)=122\).
Time = 4.99 (sec) , antiderivative size = 2940, normalized size of antiderivative = 48.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, 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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x^3+x} \left (a x^6+b\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {1}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (a x^6+b\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {\sqrt [3]{x}}{\sqrt [3]{x^2+1} \left (a x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {1}{\sqrt [3]{x+1} \left (a x^3+b\right )}dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \left (-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (\sqrt [9]{-1} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (-(-1)^{2/9} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (\sqrt [3]{-1} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (-(-1)^{4/9} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left ((-1)^{5/9} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (-(-1)^{2/3} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left ((-1)^{7/9} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}-\frac {1}{9 b^{8/9} \left (-(-1)^{8/9} \sqrt [9]{a} x^{2/3}-\sqrt [9]{b}\right ) \sqrt [3]{x+1}}\right )dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \left (-\frac {(-1)^{2/3} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}+\frac {\sqrt [3]{-1} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}-\frac {\sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}+\frac {(-1)^{7/9} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}-\frac {(-1)^{4/9} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}+\frac {\sqrt [9]{-1} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}-\frac {(-1)^{8/9} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}+\frac {(-1)^{5/9} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}-\frac {(-1)^{2/9} \sqrt [9]{a} x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{18 b^{10/9}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} x^{2/3}}{\sqrt [9]{b} \sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9}}+\frac {(-1)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {\sqrt [3]{-1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{7/9} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{4/9} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{8/9} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{5/9} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{2/9} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}-\frac {\log \left (-\sqrt [3]{a} x-\sqrt [3]{b}\right )}{18 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{8/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{5/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{2/9} \log \left (-\sqrt [3]{a} x-(-1)^{2/3} \sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}+\frac {\log \left (\sqrt [3]{-1} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{18 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9}}+\frac {\log \left (-(-1)^{2/3} \sqrt [3]{a} x-\sqrt [3]{b}\right )}{18 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9}}+\frac {(-1)^{7/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{4/9} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {\sqrt [9]{-1} \log \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{54 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {\log \left (-\frac {\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {\log \left (\frac {\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} x^{2/3}}{\sqrt [9]{b}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b}} b^{8/9}}-\frac {\log \left (\frac {\sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} x^{2/3}}{\sqrt [9]{b}}-\sqrt [3]{x+1}\right )}{6 \sqrt [3]{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a}} b^{8/9}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}+\frac {\log \left (\sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-\sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{7/9} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{4/9} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}-\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{8/9} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}-\frac {(-1)^{5/9} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}+\frac {(-1)^{2/9} \log \left (\sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}-\sqrt [9]{a} \sqrt [3]{x+1}\right )}{18 \sqrt [3]{\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} b^{8/9}}\right )}{2 \sqrt [3]{x^3+x}}\) |
(3*x^(1/3)*(1 + x^2)^(1/3)*(-1/18*(a^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5 /3, -x, -((a^(1/3)*x)/b^(1/3))])/b^(10/9) + ((-1)^(1/3)*a^(1/9)*x^(2/3)*Ap pellF1[2/3, 1/3, 1, 5/3, -x, -((a^(1/3)*x)/b^(1/3))])/(18*b^(10/9)) - ((-1 )^(2/3)*a^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((a^(1/3)*x)/b^(1/ 3))])/(18*b^(10/9)) + ((-1)^(1/9)*a^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/ 3, -x, ((-1)^(1/3)*a^(1/3)*x)/b^(1/3)])/(18*b^(10/9)) - ((-1)^(4/9)*a^(1/9 )*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, ((-1)^(1/3)*a^(1/3)*x)/b^(1/3)])/ (18*b^(10/9)) + ((-1)^(7/9)*a^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, ((-1)^(1/3)*a^(1/3)*x)/b^(1/3)])/(18*b^(10/9)) - ((-1)^(2/9)*a^(1/9)*x^(2 /3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -(((-1)^(2/3)*a^(1/3)*x)/b^(1/3))])/(18 *b^(10/9)) + ((-1)^(5/9)*a^(1/9)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -( ((-1)^(2/3)*a^(1/3)*x)/b^(1/3))])/(18*b^(10/9)) - ((-1)^(8/9)*a^(1/9)*x^(2 /3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -(((-1)^(2/3)*a^(1/3)*x)/b^(1/3))])/(18 *b^(10/9)) - ArcTan[(1 - (2*(a^(1/3) - b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(1 + x)^(1/3)))/Sqrt[3]]/(3*Sqrt[3]*(a^(1/3) - b^(1/3))^(1/3)*b^(8/9)) + Arc Tan[(1 + (2*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*x^(2/3))/(b^(1/9)*(1 + x) ^(1/3)))/Sqrt[3]]/(3*Sqrt[3]*((-1)^(1/3)*a^(1/3) + b^(1/3))^(1/3)*b^(8/9)) + ArcTan[(1 + (2*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1/3)*x^(2/3))/(b^(1/9 )*(1 + x)^(1/3)))/Sqrt[3]]/(3*Sqrt[3]*(-((-1)^(2/3)*a^(1/3)) + b^(1/3))^(1 /3)*b^(8/9)) + ArcTan[(1 + (2*a^(1/9)*(1 + x)^(1/3))/(a^(1/3) - b^(1/3)...
3.9.4.3.1 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 = 5.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{9}-3 b \,\textit {\_Z}^{6}+3 b \,\textit {\_Z}^{3}+a -b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 b}\) | \(55\) |
Exception generated. \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Not integrable
Time = 7.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{6} + b\right )}\, dx \]
Not integrable
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int { \frac {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((a*x^(19/3) + b*x^(1/3))*(x^2 + 1)^(1/3 )) + integrate(9/40*(9*b*x^6 + 3*b*x^4 - b*x^2 + 5*b)/((a^2*x^(37/3) + 2*a *b*x^(19/3) + b^2*x^(1/3))*(x^2 + 1)^(1/3)), x)
Not integrable
Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int { \frac {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 6.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx=\int \frac {1}{\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]