Integrand size = 101, antiderivative size = 685 \[ \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\frac {\arctan \left (\frac {\frac {\sqrt [4]{1+c_0}}{\sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\sqrt [4]{-1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {2} \sqrt [4]{1+c_0}}}{\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{1+c_0}}{2 \sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+c_0} \sqrt [4]{-1-c_1} \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {1+c_0}+\sqrt {-1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{1+c_0}}{2 \sqrt {2} \sqrt [4]{-1-c_1}}-\frac {\arctan \left (\frac {\frac {\sqrt [4]{-1+c_0}}{\sqrt {2} \sqrt [4]{1-c_1}}-\frac {\sqrt [4]{1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {2} \sqrt [4]{-1+c_0}}}{\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{-1+c_0}}{2 \sqrt {2} \sqrt [4]{1-c_1}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+c_0} \sqrt [4]{1-c_1} \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\sqrt {-1+c_0}+\sqrt {1-c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}\right ) \sqrt [4]{-1+c_0}}{2 \sqrt {2} \sqrt [4]{1-c_1}}+\frac {1}{4} (c_0-c_1) \text {RootSum}\left [1+c_0{}^2-2 \text {$\#$1}^4-2 c_0 c_1 \text {$\#$1}^4+\text {$\#$1}^8+c_1{}^2 \text {$\#$1}^8\&,\frac {\log \left (\sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}}{-1-c_0 c_1+\text {$\#$1}^4+c_1{}^2 \text {$\#$1}^4}\&\right ] \]
\[ \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \]
Integrate[(x^2*(x^2*C[3] - C[4])*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2 *C[3] + C[4]))^(1/4))/((-x + x^2*C[3] + C[4])*(x + x^2*C[3] + C[4])*(x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]
Integrate[(x^2*(x^2*C[3] - C[4])*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2 *C[3] + C[4]))^(1/4))/((-x + x^2*C[3] + C[4])*(x + x^2*C[3] + C[4])*(x^2 + x^4*C[3]^2 + 2*x^2*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.
| \(\displaystyle \int \frac {x^2 \left (c_3 x^2-c_4\right ) \sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\left (c_3 x^2-x+c_4\right ) \left (c_3 x^2+x+c_4\right ) \left (c_3{}^2 x^4+x^2+2 c_3 c_4 x^2+c_4{}^2\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^2 \left (c_3 x^2-c_4\right ) \sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}}}{\left (c_3 x^2-x+c_4\right ) \left (c_3 x^2+x+c_4\right ) \left (c_3{}^2 x^4+(1+2 c_3 c_4) x^2+c_4{}^2\right )}dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt [4]{c_3 x^2+c_1 x+c_4} \int -\frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{c_3 x^2+c_0 x+c_4}}{\left (-c_3 x^2+x-c_4\right ) \left (c_3 x^2+x+c_4\right ) \sqrt [4]{c_3 x^2+c_1 x+c_4} \left (c_3{}^2 x^4+(2 c_3 c_4+1) x^2+c_4{}^2\right )}dx}{\sqrt [4]{c_3 x^2+c_0 x+c_4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt [4]{c_3 x^2+c_1 x+c_4} \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{c_3 x^2+c_0 x+c_4}}{\left (-c_3 x^2+x-c_4\right ) \left (c_3 x^2+x+c_4\right ) \sqrt [4]{c_3 x^2+c_1 x+c_4} \left (c_3{}^2 x^4+(2 c_3 c_4+1) x^2+c_4{}^2\right )}dx}{\sqrt [4]{c_3 x^2+c_0 x+c_4}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt [4]{c_3 x^2+c_1 x+c_4} \int \left (\frac {\sqrt [4]{c_3 x^2+c_0 x+c_4} (1-2 x c_3)}{4 \left (c_3 x^2-x+c_4\right ) \sqrt [4]{c_3 x^2+c_1 x+c_4}}+\frac {(2 x c_3+1) \sqrt [4]{c_3 x^2+c_0 x+c_4}}{4 \left (c_3 x^2+x+c_4\right ) \sqrt [4]{c_3 x^2+c_1 x+c_4}}+\frac {\left (x^2 c_3-c_4\right ) \sqrt [4]{c_3 x^2+c_0 x+c_4}}{2 \sqrt [4]{c_3 x^2+c_1 x+c_4} \left (c_3{}^2 x^4+(2 c_3 c_4+1) x^2+c_4{}^2\right )}\right )dx}{\sqrt [4]{c_3 x^2+c_0 x+c_4}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt [4]{c_3 x^2+c_1 x+c_4} \left (-\frac {1}{2} c_3 \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}-1\right )}dx+\frac {1}{2} c_3 \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}+1\right )}dx-\frac {1}{2} c_3 \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}-1\right )}dx+\frac {1}{2} c_3 \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}+1\right )}dx-\frac {c_3 \left (\sqrt {4 c_3 c_4+1}+1\right ) \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (\sqrt {-2 c_3 c_4-\sqrt {4 c_3 c_4+1}-1}-\sqrt {2} x c_3\right )}dx}{4 \sqrt {-2 c_3 c_4-\sqrt {4 c_3 c_4+1}-1}}-\frac {c_3 \left (\sqrt {4 c_3 c_4+1}+1\right ) \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (\sqrt {2} x c_3+\sqrt {-2 c_3 c_4-\sqrt {4 c_3 c_4+1}-1}\right )}dx}{4 \sqrt {-2 c_3 c_4-\sqrt {4 c_3 c_4+1}-1}}-\frac {c_3 \left (1-\sqrt {4 c_3 c_4+1}\right ) \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (\sqrt {-2 c_3 c_4+\sqrt {4 c_3 c_4+1}-1}-\sqrt {2} x c_3\right )}dx}{4 \sqrt {-2 c_3 c_4+\sqrt {4 c_3 c_4+1}-1}}-\frac {c_3 \left (1-\sqrt {4 c_3 c_4+1}\right ) \int \frac {\sqrt [4]{c_3 x^2+c_0 x+c_4}}{\sqrt [4]{c_3 x^2+c_1 x+c_4} \left (\sqrt {2} x c_3+\sqrt {-2 c_3 c_4+\sqrt {4 c_3 c_4+1}-1}\right )}dx}{4 \sqrt {-2 c_3 c_4+\sqrt {4 c_3 c_4+1}-1}}\right )}{\sqrt [4]{c_3 x^2+c_0 x+c_4}}\) |
Int[(x^2*(x^2*C[3] - C[4])*((x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]))^(1/4))/((-x + x^2*C[3] + C[4])*(x + x^2*C[3] + C[4])*(x^2 + x^4*C [3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]
3.32.19.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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]
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]
Not integrable
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.12
\[\int \frac {x^{2} \left (\textit {\_C3} \,x^{2}-\textit {\_C4} \right ) \left (\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}\right )^{\frac {1}{4}}}{\left (\textit {\_C3} \,x^{2}+\textit {\_C4} -x \right ) \left (\textit {\_C3} \,x^{2}+\textit {\_C4} +x \right ) \left (\textit {\_C3}^{2} x^{4}+2 \textit {\_C3} \textit {\_C4} \,x^{2}+\textit {\_C4}^{2}+x^{2}\right )}d x\]
int(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3 *x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x)
int(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4)/(_C3 *x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x)
Timed out. \[ \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \]
integrate(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4 )/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, a lgorithm="fricas")
Timed out. \[ \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \]
integrate(x**2*(_C3*x**2-_C4)*((_C3*x**2+_C0*x+_C4)/(_C3*x**2+_C1*x+_C4))* *(1/4)/(_C3*x**2+_C4-x)/(_C3*x**2+_C4+x)/(_C3**2*x**4+2*_C3*_C4*x**2+_C4** 2+x**2),x)
Not integrable
Time = 0.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.12 \[ \text {Unable to display latex} \]
integrate(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4 )/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, a lgorithm="maxima")
integrate((_C3*x^2 - _C4)*x^2*((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))^(1/4)/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 + x^2)*(_C3*x^2 + _C4 + x) *(_C3*x^2 + _C4 - x)), x)
Not integrable
Time = 0.70 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.12 \[ \text {Unable to display latex} \]
integrate(x^2*(_C3*x^2-_C4)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/4 )/(_C3*x^2+_C4-x)/(_C3*x^2+_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2+x^2),x, a lgorithm="giac")
integrate((_C3*x^2 - _C4)*x^2*((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))^(1/4)/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 + x^2)*(_C3*x^2 + _C4 + x) *(_C3*x^2 + _C4 - x)), x)
Not integrable
Time = 9.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.12 \[ \int \frac {x^2 \left (x^2 c_3-c_4\right ) \sqrt [4]{\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{\left (-x+x^2 c_3+c_4\right ) \left (x+x^2 c_3+c_4\right ) \left (x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int -\frac {x^2\,\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )\,{\left (\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}\right )}^{1/4}}{\left (_{\mathrm {C3}}\,x^2-x+_{\mathrm {C4}}\right )\,\left (_{\mathrm {C3}}\,x^2+x+_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2+x^2\right )} \,d x \]
int(-(x^2*(_C4 - _C3*x^2)*((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2) )^(1/4))/((_C4 - x + _C3*x^2)*(_C4 + x + _C3*x^2)*(_C4^2 + x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)),x)