Integrand size = 32, antiderivative size = 119 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\frac {3 \left (1+4 x^2\right ) \sqrt [3]{x+2 x^3}}{16 x^3}+\frac {1}{8} \text {RootSum}\left [11-9 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {11 \log (x)-11 \log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-9 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Time = 3.01 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\frac {\sqrt [3]{x+2 x^3} \left (9 \sqrt [3]{1+2 x^2} \left (1+4 x^2\right )+2 x^{8/3} \text {RootSum}\left [11-9 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {22 \log (x)-33 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-9 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{48 x^3 \sqrt [3]{1+2 x^2}} \] Input:
Integrate[((x + 2*x^3)^(1/3)*(-1 + x^4))/(x^4*(2 - x^2 + x^4)),x]
Output:
((x + 2*x^3)^(1/3)*(9*(1 + 2*x^2)^(1/3)*(1 + 4*x^2) + 2*x^(8/3)*RootSum[11 - 9*#1^3 + 2*#1^6 & , (22*Log[x] - 33*Log[(1 + 2*x^2)^(1/3) - x^(2/3)*#1] + 2*Log[x]*#1^3 - 3*Log[(1 + 2*x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-9*#1^2 + 4*#1^5) & ]))/(48*x^3*(1 + 2*x^2)^(1/3))
Result contains complex when optimal does not.
Time = 3.72 (sec) , antiderivative size = 1540, normalized size of antiderivative = 12.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 25, 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.
| \(\displaystyle \int \frac {\sqrt [3]{2 x^3+x} \left (x^4-1\right )}{x^4 \left (x^4-x^2+2\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{2 x^3+x} \int -\frac {\sqrt [3]{2 x^2+1} \left (1-x^4\right )}{x^{11/3} \left (x^4-x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{2 x^3+x} \int \frac {\sqrt [3]{2 x^2+1} \left (1-x^4\right )}{x^{11/3} \left (x^4-x^2+2\right )}dx}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 \sqrt [3]{2 x^3+x} \int \frac {\sqrt [3]{2 x^2+1} \left (1-x^4\right )}{x^3 \left (x^4-x^2+2\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 \sqrt [3]{2 x^3+x} \int \left (-\frac {x \sqrt [3]{2 x^2+1} \left (x^2+5\right )}{4 \left (x^4-x^2+2\right )}+\frac {\sqrt [3]{2 x^2+1}}{4 x}+\frac {\sqrt [3]{2 x^2+1}}{2 x^3}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt [3]{2 x^3+x} \left (-\frac {\left (2 x^2+1\right )^{4/3}}{16 x^{8/3}}-\frac {\sqrt [3]{2 x^2+1}}{8 x^{2/3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}}{\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{\sqrt {21} \sqrt [3]{-i+\sqrt {7}} \left (2 \left (-2 i+\sqrt {7}\right )\right )^{2/3}}+\frac {i \sqrt {\frac {3}{7}} \left (\frac {i-\sqrt {7}}{2 i-\sqrt {7}}\right )^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}}{\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{8\ 2^{2/3}}+\frac {5 i \arctan \left (\frac {\frac {2 \sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}}{\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {21} \sqrt [3]{\frac {i-\sqrt {7}}{2 i-\sqrt {7}}}}-\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {7}}{2 \left (2 i+\sqrt {7}\right )}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{\sqrt {21} \sqrt [3]{i+\sqrt {7}} \left (2 \left (2 i+\sqrt {7}\right )\right )^{2/3}}-\frac {i \sqrt {\frac {3}{7}} \left (\frac {i+\sqrt {7}}{2 i+\sqrt {7}}\right )^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {7}}{2 \left (2 i+\sqrt {7}\right )}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{8\ 2^{2/3}}-\frac {5 i \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {7}}{2 \left (2 i+\sqrt {7}\right )}} \sqrt [3]{2 x^2+1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {21} \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}}}+\frac {\log \left (-2 x^2-i \sqrt {7}+1\right )}{6 \sqrt {7} \sqrt [3]{i+\sqrt {7}} \left (2 \left (2 i+\sqrt {7}\right )\right )^{2/3}}+\frac {i \left (\frac {i+\sqrt {7}}{2 i+\sqrt {7}}\right )^{2/3} \log \left (-2 x^2-i \sqrt {7}+1\right )}{16\ 2^{2/3} \sqrt {7}}+\frac {5 i \log \left (-2 x^2-i \sqrt {7}+1\right )}{24\ 2^{2/3} \sqrt {7} \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}}}+\frac {\log \left (-2 x^2+i \sqrt {7}+1\right )}{6 \sqrt {7} \sqrt [3]{-i+\sqrt {7}} \left (2 \left (-2 i+\sqrt {7}\right )\right )^{2/3}}-\frac {i \left (\frac {i-\sqrt {7}}{2 i-\sqrt {7}}\right )^{2/3} \log \left (-2 x^2+i \sqrt {7}+1\right )}{16\ 2^{2/3} \sqrt {7}}-\frac {5 i \log \left (-2 x^2+i \sqrt {7}+1\right )}{24\ 2^{2/3} \sqrt {7} \sqrt [3]{\frac {i-\sqrt {7}}{2 i-\sqrt {7}}}}-\frac {\log \left (\sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}-\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}\right )}{2 \sqrt {7} \sqrt [3]{-i+\sqrt {7}} \left (2 \left (-2 i+\sqrt {7}\right )\right )^{2/3}}+\frac {3 i \left (\frac {i-\sqrt {7}}{2 i-\sqrt {7}}\right )^{2/3} \log \left (\sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}-\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}\right )}{16\ 2^{2/3} \sqrt {7}}+\frac {5 i \log \left (\sqrt [3]{2 \left (-2 i+\sqrt {7}\right )} x^{2/3}-\sqrt [3]{-i+\sqrt {7}} \sqrt [3]{2 x^2+1}\right )}{8\ 2^{2/3} \sqrt {7} \sqrt [3]{\frac {i-\sqrt {7}}{2 i-\sqrt {7}}}}-\frac {\log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{2 x^2+1}\right )}{2 \sqrt {7} \sqrt [3]{i+\sqrt {7}} \left (2 \left (2 i+\sqrt {7}\right )\right )^{2/3}}-\frac {3 i \left (\frac {i+\sqrt {7}}{2 i+\sqrt {7}}\right )^{2/3} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{2 x^2+1}\right )}{16\ 2^{2/3} \sqrt {7}}-\frac {5 i \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}} \sqrt [3]{2 x^2+1}\right )}{8\ 2^{2/3} \sqrt {7} \sqrt [3]{\frac {i+\sqrt {7}}{2 i+\sqrt {7}}}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}\) |
Input:
Int[((x + 2*x^3)^(1/3)*(-1 + x^4))/(x^4*(2 - x^2 + x^4)),x]
Output:
(-3*(x + 2*x^3)^(1/3)*(-1/8*(1 + 2*x^2)^(1/3)/x^(2/3) - (1 + 2*x^2)^(4/3)/ (16*x^(8/3)) + (((5*I)/4)*ArcTan[(1 + (2*(2*(-2*I + Sqrt[7]))^(1/3)*x^(2/3 ))/((-I + Sqrt[7])^(1/3)*(1 + 2*x^2)^(1/3)))/Sqrt[3]])/(2^(2/3)*Sqrt[21]*( (I - Sqrt[7])/(2*I - Sqrt[7]))^(1/3)) + ((I/8)*Sqrt[3/7]*((I - Sqrt[7])/(2 *I - Sqrt[7]))^(2/3)*ArcTan[(1 + (2*(2*(-2*I + Sqrt[7]))^(1/3)*x^(2/3))/(( -I + Sqrt[7])^(1/3)*(1 + 2*x^2)^(1/3)))/Sqrt[3]])/2^(2/3) - ArcTan[(1 + (2 *(2*(-2*I + Sqrt[7]))^(1/3)*x^(2/3))/((-I + Sqrt[7])^(1/3)*(1 + 2*x^2)^(1/ 3)))/Sqrt[3]]/(Sqrt[21]*(-I + Sqrt[7])^(1/3)*(2*(-2*I + Sqrt[7]))^(2/3)) - (((5*I)/4)*ArcTan[(1 + (2*x^(2/3))/(((I + Sqrt[7])/(2*(2*I + Sqrt[7])))^( 1/3)*(1 + 2*x^2)^(1/3)))/Sqrt[3]])/(2^(2/3)*Sqrt[21]*((I + Sqrt[7])/(2*I + Sqrt[7]))^(1/3)) - ((I/8)*Sqrt[3/7]*((I + Sqrt[7])/(2*I + Sqrt[7]))^(2/3) *ArcTan[(1 + (2*x^(2/3))/(((I + Sqrt[7])/(2*(2*I + Sqrt[7])))^(1/3)*(1 + 2 *x^2)^(1/3)))/Sqrt[3]])/2^(2/3) - ArcTan[(1 + (2*x^(2/3))/(((I + Sqrt[7])/ (2*(2*I + Sqrt[7])))^(1/3)*(1 + 2*x^2)^(1/3)))/Sqrt[3]]/(Sqrt[21]*(I + Sqr t[7])^(1/3)*(2*(2*I + Sqrt[7]))^(2/3)) + (((5*I)/24)*Log[1 - I*Sqrt[7] - 2 *x^2])/(2^(2/3)*Sqrt[7]*((I + Sqrt[7])/(2*I + Sqrt[7]))^(1/3)) + ((I/16)*( (I + Sqrt[7])/(2*I + Sqrt[7]))^(2/3)*Log[1 - I*Sqrt[7] - 2*x^2])/(2^(2/3)* Sqrt[7]) + Log[1 - I*Sqrt[7] - 2*x^2]/(6*Sqrt[7]*(I + Sqrt[7])^(1/3)*(2*(2 *I + Sqrt[7]))^(2/3)) - (((5*I)/24)*Log[1 + I*Sqrt[7] - 2*x^2])/(2^(2/3)*S qrt[7]*((I - Sqrt[7])/(2*I - Sqrt[7]))^(1/3)) - ((I/16)*((I - Sqrt[7])/...
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]
Time = 208.89 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69
| method | result | size |
| pseudoelliptic | \(\frac {\left (12 x^{2}+3\right ) \left (2 x^{3}+x \right )^{\frac {1}{3}}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+11\right )}{\sum }\frac {\left (\textit {\_R}^{3}+11\right ) \ln \left (\frac {-\textit {\_R} x +\left (2 x^{3}+x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-9\right )}\right ) x^{3}}{16 x^{3}}\) | \(82\) |
| trager | \(\text {Expression too large to display}\) | \(11115\) |
| risch | \(\text {Expression too large to display}\) | \(13838\) |
Input:
int((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x,method=_RETURNVERBOSE)
Output:
1/16*((12*x^2+3)*(2*x^3+x)^(1/3)-2*sum((_R^3+11)*ln((-_R*x+(2*x^3+x)^(1/3) )/x)/_R^2/(4*_R^3-9),_R=RootOf(2*_Z^6-9*_Z^3+11))*x^3)/x^3
Exception generated. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (trace 0)
Timed out. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((2*x**3+x)**(1/3)*(x**4-1)/x**4/(x**4-x**2+2),x)
Output:
Timed out
Not integrable
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{2} + 2\right )} x^{4}} \,d x } \] Input:
integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="maxima")
Output:
integrate((x^4 - 1)*(2*x^3 + x)^(1/3)/((x^4 - x^2 + 2)*x^4), x)
Exception generated. \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Not integrable
Time = 8.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\int \frac {{\left (2\,x^3+x\right )}^{1/3}\,\left (x^4-1\right )}{x^4\,\left (x^4-x^2+2\right )} \,d x \] Input:
int(((x + 2*x^3)^(1/3)*(x^4 - 1))/(x^4*(x^4 - x^2 + 2)),x)
Output:
int(((x + 2*x^3)^(1/3)*(x^4 - 1))/(x^4*(x^4 - x^2 + 2)), x)
Not integrable
Time = 0.48 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [3]{x+2 x^3} \left (-1+x^4\right )}{x^4 \left (2-x^2+x^4\right )} \, dx=\frac {90 \left (2 x^{2}+1\right )^{\frac {1}{3}} x^{2}+3 \left (2 x^{2}+1\right )^{\frac {1}{3}}+104 x^{\frac {8}{3}} \left (\int \frac {\left (2 x^{2}+1\right )^{\frac {1}{3}}}{2 x^{\frac {23}{3}}-x^{\frac {17}{3}}+3 x^{\frac {11}{3}}+2 x^{\frac {5}{3}}}d x \right )+104 x^{\frac {8}{3}} \left (\int \frac {\left (2 x^{2}+1\right )^{\frac {1}{3}} x^{3}}{2 x^{\frac {20}{3}}-x^{\frac {14}{3}}+3 x^{\frac {8}{3}}+2 x^{\frac {2}{3}}}d x \right )-48 x^{\frac {8}{3}} \left (\int \frac {\left (2 x^{2}+1\right )^{\frac {1}{3}} x}{2 x^{\frac {20}{3}}-x^{\frac {14}{3}}+3 x^{\frac {8}{3}}+2 x^{\frac {2}{3}}}d x \right )}{16 x^{\frac {8}{3}}} \] Input:
int((2*x^3+x)^(1/3)*(x^4-1)/x^4/(x^4-x^2+2),x)
Output:
(90*(2*x**2 + 1)**(1/3)*x**2 + 3*(2*x**2 + 1)**(1/3) + 104*x**(2/3)*int((2 *x**2 + 1)**(1/3)/(2*x**(2/3)*x**7 - x**(2/3)*x**5 + 3*x**(2/3)*x**3 + 2*x **(2/3)*x),x)*x**2 + 104*x**(2/3)*int(((2*x**2 + 1)**(1/3)*x**3)/(2*x**(2/ 3)*x**6 - x**(2/3)*x**4 + 3*x**(2/3)*x**2 + 2*x**(2/3)),x)*x**2 - 48*x**(2 /3)*int(((2*x**2 + 1)**(1/3)*x)/(2*x**(2/3)*x**6 - x**(2/3)*x**4 + 3*x**(2 /3)*x**2 + 2*x**(2/3)),x)*x**2)/(16*x**(2/3)*x**2)