Integrand size = 30, antiderivative size = 103 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\frac {1}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2 x^3+\sqrt [3]{-x+x^5}}\right )+\frac {1}{4} \log \left (x^3+\sqrt [3]{-x+x^5}\right )-\frac {1}{8} \log \left (x^6-x^3 \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \] Output:
1/4*3^(1/2)*arctan(3^(1/2)*(x^5-x)^(1/3)/(-2*x^3+(x^5-x)^(1/3)))+1/4*ln(x^ 3+(x^5-x)^(1/3))-1/8*ln(x^6-x^3*(x^5-x)^(1/3)+(x^5-x)^(2/3))
Time = 10.44 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\frac {1}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2 x^3+\sqrt [3]{-x+x^5}}\right )+\frac {1}{4} \log \left (x^3+\sqrt [3]{-x+x^5}\right )-\frac {1}{8} \log \left (x^6-x^3 \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \] Input:
Integrate[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]
Output:
(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2*x^3 + (-x + x^5)^(1/3))])/4 + Log[x^3 + (-x + x^5)^(1/3)]/4 - Log[x^6 - x^3*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/8
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.16 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2467, 2035, 7279, 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.
\(\displaystyle \int \frac {x^2 \left (x^4-2\right )}{\sqrt [3]{x^5-x} \left (x^8+x^4-1\right )} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {x^{5/3} \left (2-x^4\right )}{\sqrt [3]{x^4-1} \left (-x^8-x^4+1\right )}dx}{\sqrt [3]{x^5-x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \frac {x^{7/3} \left (2-x^4\right )}{\sqrt [3]{x^4-1} \left (-x^8-x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \int \left (\frac {x^{19/3}}{\sqrt [3]{x^4-1} \left (x^8+x^4-1\right )}-\frac {2 x^{7/3}}{\sqrt [3]{x^4-1} \left (x^8+x^4-1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x^5-x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^4-1} \left (\frac {\sqrt [3]{1-x^4} x^{8/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{2 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [3]{x^4-1}}+\frac {\sqrt [3]{1-x^4} x^{8/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt {5} \sqrt [3]{x^4-1}}-\frac {\sqrt [3]{1-x^4} x^{8/3} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {2 x^4}{1-\sqrt {5}},x^4\right )}{2 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [3]{x^4-1}}-\frac {\sqrt [3]{1-x^4} x^{8/3} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {2 x^4}{1-\sqrt {5}},x^4\right )}{8 \sqrt {5} \sqrt [3]{x^4-1}}\right )}{\sqrt [3]{x^5-x}}\) |
Input:
Int[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]
Output:
(3*x^(1/3)*(-1 + x^4)^(1/3)*((x^(8/3)*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1 , 5/3, x^4, (-2*x^4)/(1 + Sqrt[5])])/(8*Sqrt[5]*(-1 + x^4)^(1/3)) + (x^(8/ 3)*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (-2*x^4)/(1 + Sqrt[5])] )/(2*Sqrt[5]*(1 + Sqrt[5])*(-1 + x^4)^(1/3)) - (x^(8/3)*(1 - x^4)^(1/3)*Ap pellF1[2/3, 1, 1/3, 5/3, (-2*x^4)/(1 - Sqrt[5]), x^4])/(8*Sqrt[5]*(-1 + x^ 4)^(1/3)) - (x^(8/3)*(1 - x^4)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (-2*x^4)/( 1 - Sqrt[5]), x^4])/(2*Sqrt[5]*(1 - Sqrt[5])*(-1 + x^4)^(1/3))))/(-x + x^5 )^(1/3)
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_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 31.52 (sec) , antiderivative size = 626, normalized size of antiderivative = 6.08
Input:
int(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x,method=_RETURNVERBOSE)
Output:
-1/4*ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8+ 4529039623257700492643576695162024*RootOf(16*_Z^2+4*_Z+1)*x^8+610884417613 243258599747967858176*x^8-6458608904506033300021952070673408*RootOf(16*_Z^ 2+4*_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16* _Z^2+4*_Z+1)^2*x^4-25262880761481365661024699935681*(x^5-x)^(1/3)*x^5+6458 608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(2/3)*x^2-65 72417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4+25262880761481 365661024699935681*(x^5-x)^(2/3)*x^2-575090408768717286416168985366486*x^4 +17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2+657241706373 0249524797944388667308*RootOf(16*_Z^2+4*_Z+1)+5750904087687172864161689853 66486)/(x^8+x^4-1))-ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2 +4*_Z+1)^2*x^8+4529039623257700492643576695162024*RootOf(16*_Z^2+4*_Z+1)*x ^8+610884417613243258599747967858176*x^8-645860890450603330002195207067340 8*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(1/3)*x^5-1754345435151838641563704306078 1056*RootOf(16*_Z^2+4*_Z+1)^2*x^4-25262880761481365661024699935681*(x^5-x) ^(1/3)*x^5+6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5- x)^(2/3)*x^2-6572417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4 +25262880761481365661024699935681*(x^5-x)^(2/3)*x^2-5750904087687172864161 68985366486*x^4+17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1) ^2+6572417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)+5750904087...
Time = 2.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{8} + 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + 4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2}}{x^{8} - 8 \, x^{4} + 8}\right ) + \frac {1}{8} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + x^{4} + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2} - 1}{x^{8} + x^{4} - 1}\right ) \] Input:
integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="fricas")
Output:
-1/4*sqrt(3)*arctan((sqrt(3)*x^8 + 2*sqrt(3)*(x^5 - x)^(1/3)*x^5 + 4*sqrt( 3)*(x^5 - x)^(2/3)*x^2)/(x^8 - 8*x^4 + 8)) + 1/8*log((x^8 + 3*(x^5 - x)^(1 /3)*x^5 + x^4 + 3*(x^5 - x)^(2/3)*x^2 - 1)/(x^8 + x^4 - 1))
Timed out. \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate(x**2*(x**4-2)/(x**5-x)**(1/3)/(x**8+x**4-1),x)
Output:
Timed out
\[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="maxima")
Output:
integrate((x^4 - 2)*x^2/((x^8 + x^4 - 1)*(x^5 - x)^(1/3)), x)
\[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int { \frac {{\left (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="giac")
Output:
integrate((x^4 - 2)*x^2/((x^8 + x^4 - 1)*(x^5 - x)^(1/3)), x)
Timed out. \[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int \frac {x^2\,\left (x^4-2\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^8+x^4-1\right )} \,d x \] Input:
int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)),x)
Output:
int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)), x)
\[ \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx=\int \frac {x^{2} \left (x^{4}-2\right )}{\left (x^{5}-x \right )^{\frac {1}{3}} \left (x^{8}+x^{4}-1\right )}d x \] Input:
int(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x)
Output:
int(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x)