Integrand size = 34, antiderivative size = 163 \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {3 \sqrt {3} x \sqrt [3]{-x+x^4}-3 x^2 \sqrt [3]{-x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{-x+x^4}+\sqrt {3} x^2 \sqrt [3]{-x+x^4}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{-x+x^4}\right )-\text {arctanh}\left (\frac {1+x \sqrt [3]{-x+x^4}}{1+x \sqrt [3]{-x+x^4}+2 x^2 \left (-x+x^4\right )^{2/3}}\right ) \]
-3^(1/2)*arctan((3*3^(1/2)*x*(x^4-x)^(1/3)-3*x^2*(x^4-x)^(1/3))/(-6+2*x*3^ (1/2)-3*x*(x^4-x)^(1/3)+3^(1/2)*x^2*(x^4-x)^(1/3)))-2*arctanh(-1+2*x*(x^4- x)^(1/3))-arctanh((1+x*(x^4-x)^(1/3))/(1+x*(x^4-x)^(1/3)+2*x^2*(x^4-x)^(2/ 3)))
\[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=\int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx \]
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.
| \(\displaystyle \int \frac {x^2 \left (7 x^3-4\right )}{\sqrt [3]{x^4-x} \left (x^7-x^4-1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^3-1} \int \frac {x^{5/3} \left (4-7 x^3\right )}{\sqrt [3]{x^3-1} \left (-x^7+x^4+1\right )}dx}{\sqrt [3]{x^4-x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^3-1} \int \frac {x^{7/3} \left (4-7 x^3\right )}{\sqrt [3]{x^3-1} \left (-x^7+x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4-x}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^3-1} \int \left (\frac {7 x^{16/3}}{\sqrt [3]{x^3-1} \left (x^7-x^4-1\right )}-\frac {4 x^{7/3}}{\sqrt [3]{x^3-1} \left (x^7-x^4-1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4-x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^3-1} \left (7 \int \frac {x^{16/3}}{\sqrt [3]{x^3-1} \left (x^7-x^4-1\right )}d\sqrt [3]{x}-4 \int \frac {x^{7/3}}{\sqrt [3]{x^3-1} \left (x^7-x^4-1\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x^4-x}}\) |
3.23.4.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 13.81 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.17
-ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^7+403142488045119745173 926*RootOf(_Z^2+_Z+1)*x^7+10427862756551384399808209*x^7-27509780243203968 05141*RootOf(_Z^2+_Z+1)^2*x^4-9624328758485465306265498*RootOf(_Z^2+_Z+1)* (x^4-x)^(2/3)*x^2-403142488045119745173926*RootOf(_Z^2+_Z+1)*x^4+108255032 88547863351371853*x^2*(x^4-x)^(2/3)-10427862756551384399808209*x^4-9624328 758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-308109538723884442175 792*RootOf(_Z^2+_Z+1)^2+10825503288547863351371853*x*(x^4-x)^(1/3)-1033282 9807230149096810075*RootOf(_Z^2+_Z+1)+92281971296914906192993)/(x^7-x^4-1) )*RootOf(_Z^2+_Z+1)+RootOf(_Z^2+_Z+1)*ln(-(2750978024320396805141*RootOf(_ Z^2+_Z+1)^2*x^7-397640531996478951563644*RootOf(_Z^2+_Z+1)*x^7+10027471246 530585051439424*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4+9624328 758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2+3976405319964789515 63644*RootOf(_Z^2+_Z+1)*x^4+20449832047033328657637351*x^2*(x^4-x)^(2/3)-1 0027471246530585051439424*x^4+9624328758485465306265498*RootOf(_Z^2+_Z+1)* (x^4-x)^(1/3)*x-308109538723884442175792*RootOf(_Z^2+_Z+1)^2+2044983204703 3328657637351*x*(x^4-x)^(1/3)+9716610729782380212458491*RootOf(_Z^2+_Z+1)+ 10117002239803179560827276)/(x^7-x^4-1))-ln(-(2750978024320396805141*RootO f(_Z^2+_Z+1)^2*x^7+403142488045119745173926*RootOf(_Z^2+_Z+1)*x^7+10427862 756551384399808209*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4-9624 328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2-4031424880451...
Time = 1.97 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} - x^{4}\right )}}{x^{7} - x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} - x^{4} - 3 \, {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x - 1}{x^{7} - x^{4} - 1}\right ) \]
-sqrt(3)*arctan((2*sqrt(3)*(x^4 - x)^(2/3)*x^2 - 4*sqrt(3)*(x^4 - x)^(1/3) *x - sqrt(3)*(x^7 - x^4))/(x^7 - x^4 + 8)) + 1/2*log((x^7 - x^4 - 3*(x^4 - x)^(2/3)*x^2 + 3*(x^4 - x)^(1/3)*x - 1)/(x^7 - x^4 - 1))
\[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=\int \frac {x^{2} \cdot \left (7 x^{3} - 4\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{7} - x^{4} - 1\right )}\, dx \]
\[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=\int { \frac {{\left (7 \, x^{3} - 4\right )} x^{2}}{{\left (x^{7} - x^{4} - 1\right )} {\left (x^{4} - x\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=\int { \frac {{\left (7 \, x^{3} - 4\right )} x^{2}}{{\left (x^{7} - x^{4} - 1\right )} {\left (x^{4} - x\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\int \frac {x^2\,\left (7\,x^3-4\right )}{{\left (x^4-x\right )}^{1/3}\,\left (-x^7+x^4+1\right )} \,d x \]