Integrand size = 98, antiderivative size = 726 \[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}}{\sqrt {3} \sqrt [3]{1+c_0}}\right ) \sqrt [3]{-1-c_1}}{\sqrt {3} \sqrt [3]{1+c_0}}-\frac {\sqrt [3]{-1-c_1} \log \left (\sqrt [3]{1+c_0}+\sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}\right )}{3 \sqrt [3]{1+c_0}}+\frac {\sqrt [3]{-1-c_1} \log \left ((1+c_0){}^{2/3}-\sqrt [3]{1+c_0} \sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}+(-1-c_1){}^{2/3} \left (\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}\right ){}^{2/3}\right )}{6 \sqrt [3]{1+c_0}}-\frac {1}{3} \text {RootSum}\left [1-c_0+c_0{}^2-2 \text {$\#$1}^3+c_0 \text {$\#$1}^3+c_1 \text {$\#$1}^3-2 c_0 c_1 \text {$\#$1}^3+\text {$\#$1}^6-c_1 \text {$\#$1}^6+c_1{}^2 \text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )-c_0 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )+2 c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )-c_0 c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )+\log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3-c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3+c_1{}^2 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}+c_0 \text {$\#$1}+c_1 \text {$\#$1}-2 c_0 c_1 \text {$\#$1}+2 \text {$\#$1}^4-2 c_1 \text {$\#$1}^4+2 c_1{}^2 \text {$\#$1}^4}\&\right ] \]
\[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx \]
Integrate[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C [3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]
Integrate[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C [3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)), x]
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.
| \(\displaystyle \int \frac {x \left (2 c_3 x^3-c_4\right )}{\left (c_3 x^3-x+c_4\right ) \sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{c_3 x^3+c_0 x+c_4} \int -\frac {x \left (2 x^3 c_3-c_4\right ) \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\left (-c_3 x^3+x-c_4\right ) \sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx}{\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}} \sqrt [3]{c_3 x^3+c_1 x+c_4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{c_3 x^3+c_0 x+c_4} \int \frac {x \left (2 x^3 c_3-c_4\right ) \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\left (-c_3 x^3+x-c_4\right ) \sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx}{\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}} \sqrt [3]{c_3 x^3+c_1 x+c_4}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [3]{c_3 x^3+c_0 x+c_4} \int \left (\frac {\sqrt [3]{c_3 x^3+c_1 x+c_4} \left (1-3 x^2 c_3\right )}{3 \left (c_3 x^3-x+c_4\right ) \sqrt [3]{c_3 x^3+c_0 x+c_4}}+\frac {\sqrt [3]{c_3 x^3+c_1 x+c_4} \left (3 c_3{}^2 x^5+5 c_3 x^3+3 c_3 c_4 x^2+x-c_4\right )}{3 \sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}\right )dx}{\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}} \sqrt [3]{c_3 x^3+c_1 x+c_4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (\frac {1}{3} \int \frac {\sqrt [3]{c_3 x^3+c_1 x+c_4}}{\left (c_3 x^3-x+c_4\right ) \sqrt [3]{c_3 x^3+c_0 x+c_4}}dx-c_3 \int \frac {x^2 \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\left (c_3 x^3-x+c_4\right ) \sqrt [3]{c_3 x^3+c_0 x+c_4}}dx+c_4 c_3 \int \frac {x^2 \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx+\frac {5}{3} c_3 \int \frac {x^3 \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx-\frac {1}{3} c_4 \int \frac {\sqrt [3]{c_3 x^3+c_1 x+c_4}}{\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx+\frac {1}{3} \int \frac {x \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx+c_3{}^2 \int \frac {x^5 \sqrt [3]{c_3 x^3+c_1 x+c_4}}{\sqrt [3]{c_3 x^3+c_0 x+c_4} \left (c_3{}^2 x^6+c_3 x^4+2 c_3 c_4 x^3+x^2+c_4 x+c_4{}^2\right )}dx\right )}{\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}} \sqrt [3]{c_3 x^3+c_1 x+c_4}}\) |
Int[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C [4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]
3.32.25.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Not integrable
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.11
\[\int \frac {x \left (2 \textit {\_C3} \,x^{3}-\textit {\_C4} \right )}{\left (\textit {\_C3} \,x^{3}+\textit {\_C4} -x \right ) \left (\frac {\textit {\_C3} \,x^{3}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{3}+\textit {\_C1} x +\textit {\_C4}}\right )^{\frac {1}{3}} \left (\textit {\_C3}^{2} x^{6}+2 \textit {\_C3} \textit {\_C4} \,x^{3}+\textit {\_C3} \,x^{4}+\textit {\_C4}^{2}+\textit {\_C4} x +x^{2}\right )}d x\]
int(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+ _C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x)
int(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+ _C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x)
Timed out. \[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \]
integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+ _C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, alg orithm="fricas")
Timed out. \[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \]
integrate(x*(2*_C3*x**3-_C4)/(_C3*x**3+_C4-x)/((_C3*x**3+_C0*x+_C4)/(_C3*x **3+_C1*x+_C4))**(1/3)/(_C3**2*x**6+2*_C3*_C4*x**3+_C3*x**4+_C4**2+_C4*x+x **2),x)
Not integrable
Time = 0.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.11 \[ \text {Unable to display latex} \]
integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+ _C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, alg orithm="maxima")
integrate((2*_C3*x^3 - _C4)*x/((_C3^2*x^6 + 2*_C3*_C4*x^3 + _C3*x^4 + _C4^ 2 + _C4*x + x^2)*(_C3*x^3 + _C4 - x)*((_C3*x^3 + _C0*x + _C4)/(_C3*x^3 + _ C1*x + _C4))^(1/3)), x)
Exception generated. \[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=\text {Exception raised: AttributeError} \]
integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+ _C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, alg orithm="giac")
Not integrable
Time = 9.52 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.11 \[ \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx=\int -\frac {x\,\left (_{\mathrm {C4}}-2\,_{\mathrm {C3}}\,x^3\right )}{{\left (\frac {_{\mathrm {C3}}\,x^3+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^3+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}\right )}^{1/3}\,\left (_{\mathrm {C3}}\,x^3-x+_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^6+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^3+_{\mathrm {C3}}\,x^4+{_{\mathrm {C4}}}^2+_{\mathrm {C4}}\,x+x^2\right )} \,d x \]
int(-(x*(_C4 - 2*_C3*x^3))/(((_C4 + _C0*x + _C3*x^3)/(_C4 + _C1*x + _C3*x^ 3))^(1/3)*(_C4 - x + _C3*x^3)*(_C4*x + _C3*x^4 + _C4^2 + x^2 + _C3^2*x^6 + 2*_C3*_C4*x^3)),x)