Integrand size = 30, antiderivative size = 147 \[ \int \frac {x^2 \left (4+7 x^3\right )}{\sqrt [3]{x+x^4} \left (-1+x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {3 \sqrt {3} x \sqrt [3]{x+x^4}-3 x^2 \sqrt [3]{x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{x+x^4}+\sqrt {3} x^2 \sqrt [3]{x+x^4}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{x+x^4}\right )-\text {arctanh}\left (\frac {1+x \sqrt [3]{x+x^4}}{1+x \sqrt [3]{x+x^4}+2 x^2 \left (x+x^4\right )^{2/3}}\right ) \]
-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 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^3+1} \int -\frac {x^{5/3} \left (7 x^3+4\right )}{\sqrt [3]{x^3+1} \left (-x^7-x^4+1\right )}dx}{\sqrt [3]{x^4+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^3+1} \int \frac {x^{5/3} \left (7 x^3+4\right )}{\sqrt [3]{x^3+1} \left (-x^7-x^4+1\right )}dx}{\sqrt [3]{x^4+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^3+1} \int \frac {x^{7/3} \left (7 x^3+4\right )}{\sqrt [3]{x^3+1} \left (-x^7-x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4+x}}\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^3+1} \left (-4 \int \frac {x^{7/3}}{\sqrt [3]{x^3+1} \left (x^7+x^4-1\right )}d\sqrt [3]{x}-7 \int \frac {x^{16/3}}{\sqrt [3]{x^3+1} \left (x^7+x^4-1\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{x^4+x}}\) |
3.21.54.3.1 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 = 6.57 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30
| method | result | size |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-17080755 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}+134838374 x^{7}-25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+168096051 \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-17080755 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+311926719 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}+134838374 x^{4}+168096051 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x +311926719 x \left (x^{4}+x \right )^{\frac {1}{3}}+50338432 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+210346022 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+202257561}{x^{7}+x^{4}-1}\right )-\ln \left (-\frac {25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+33257677 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-126749913 x^{7}+25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+168096051 \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+33257677 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-143830668 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}-126749913 x^{4}+168096051 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -143830668 x \left (x^{4}+x \right )^{\frac {1}{3}}-50338432 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+109669158 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-42249971}{x^{7}+x^{4}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+33257677 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-126749913 x^{7}+25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+168096051 \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+33257677 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-143830668 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}-126749913 x^{4}+168096051 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -143830668 x \left (x^{4}+x \right )^{\frac {1}{3}}-50338432 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+109669158 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-42249971}{x^{7}+x^{4}-1}\right )\) | \(485\) |
RootOf(_Z^2+_Z+1)*ln((-25169216*RootOf(_Z^2+_Z+1)^2*x^7-17080755*RootOf(_Z ^2+_Z+1)*x^7+134838374*x^7-25169216*RootOf(_Z^2+_Z+1)^2*x^4+168096051*(x^4 +x)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-17080755*RootOf(_Z^2+_Z+1)*x^4+311926719*x ^2*(x^4+x)^(2/3)+134838374*x^4+168096051*RootOf(_Z^2+_Z+1)*(x^4+x)^(1/3)*x +311926719*x*(x^4+x)^(1/3)+50338432*RootOf(_Z^2+_Z+1)^2+210346022*RootOf(_ Z^2+_Z+1)+202257561)/(x^7+x^4-1))-ln(-(25169216*RootOf(_Z^2+_Z+1)^2*x^7+33 257677*RootOf(_Z^2+_Z+1)*x^7-126749913*x^7+25169216*RootOf(_Z^2+_Z+1)^2*x^ 4+168096051*(x^4+x)^(2/3)*RootOf(_Z^2+_Z+1)*x^2+33257677*RootOf(_Z^2+_Z+1) *x^4-143830668*x^2*(x^4+x)^(2/3)-126749913*x^4+168096051*RootOf(_Z^2+_Z+1) *(x^4+x)^(1/3)*x-143830668*x*(x^4+x)^(1/3)-50338432*RootOf(_Z^2+_Z+1)^2+10 9669158*RootOf(_Z^2+_Z+1)-42249971)/(x^7+x^4-1))*RootOf(_Z^2+_Z+1)-ln(-(25 169216*RootOf(_Z^2+_Z+1)^2*x^7+33257677*RootOf(_Z^2+_Z+1)*x^7-126749913*x^ 7+25169216*RootOf(_Z^2+_Z+1)^2*x^4+168096051*(x^4+x)^(2/3)*RootOf(_Z^2+_Z+ 1)*x^2+33257677*RootOf(_Z^2+_Z+1)*x^4-143830668*x^2*(x^4+x)^(2/3)-12674991 3*x^4+168096051*RootOf(_Z^2+_Z+1)*(x^4+x)^(1/3)*x-143830668*x*(x^4+x)^(1/3 )-50338432*RootOf(_Z^2+_Z+1)^2+109669158*RootOf(_Z^2+_Z+1)-42249971)/(x^7+ x^4-1))
Time = 2.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (4+7 x^3\right )}{\sqrt [3]{x+x^4} \left (-1+x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} + x^{4}\right )}}{x^{7} + x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} + x^{4} - 3 \, {\left (x^{4} + x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} + x\right )}^{\frac {1}{3}} x - 1}{x^{7} + x^{4} - 1}\right ) \]
-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. \[ \int \frac {x^2 \left (4+7 x^3\right )}{\sqrt [3]{x+x^4} \left (-1+x^4+x^7\right )} \, dx=\int \frac {x^2\,\left (7\,x^3+4\right )}{{\left (x^4+x\right )}^{1/3}\,\left (x^7+x^4-1\right )} \,d x \]