Integrand size = 90, antiderivative size = 195 \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=6 \text {arctanh}\left (\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}\right )-\frac {2 \arctan \left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{\sqrt {1-c_0}}-\frac {4 \arctan \left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{\sqrt {-1-c_0}} \] Output:
6*arctanh(((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2))-2*arctan((1-_C0 )^(1/2)*(-1+_C1)^(1/2)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(-1 +_C0))*(-1+_C1)^(1/2)/(1-_C0)^(1/2)-4*arctan((-1-_C0)^(1/2)*(1+_C1)^(1/2)* ((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(1+_C0))*(1+_C1)^(1/2)/(-1 -_C0)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(43491\) vs. \(2(195)=390\).
Time = 6.53 (sec) , antiderivative size = 43491, normalized size of antiderivative = 223.03 \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Result too large to show} \] Input:
Integrate[((x^2*C[3] - C[4])*(x + 3*x^2*C[3] + 3*C[4]))/(x*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(-x^2 + x^4*C[3]^2 + 2*x^2*C[ 3]*C[4] + C[4]^2)),x]
Output:
Result too large to show
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.
\(\displaystyle \int \frac {\left (c_3 x^2-c_4\right ) \left (3 c_3 x^2+x+3 c_4\right )}{x \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \left (c_3{}^2 x^4-x^2+2 c_3 c_4 x^2+c_4{}^2\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (c_3 x^2-c_4\right ) \left (3 c_3 x^2+x+3 c_4\right )}{x \sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \left (c_3{}^2 x^4+(-1+2 c_3 c_4) x^2+c_4{}^2\right )}dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \int \frac {\left (x^2 c_3-c_4\right ) \sqrt {c_3 x^2+c_1 x+c_4} \left (3 c_3 x^2+x+3 c_4\right )}{x \sqrt {c_3 x^2+c_0 x+c_4} \left (c_3{}^2 x^4-(1-2 c_3 c_4) x^2+c_4{}^2\right )}dx}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \int \left (\frac {2 \sqrt {c_3 x^2+c_1 x+c_4} (2 x c_3-1)}{\left (c_3 x^2-x+c_4\right ) \sqrt {c_3 x^2+c_0 x+c_4}}-\frac {3 \sqrt {c_3 x^2+c_1 x+c_4}}{x \sqrt {c_3 x^2+c_0 x+c_4}}+\frac {(2 x c_3+1) \sqrt {c_3 x^2+c_1 x+c_4}}{\left (c_3 x^2+x+c_4\right ) \sqrt {c_3 x^2+c_0 x+c_4}}\right )dx}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \left (-3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{x \sqrt {c_3 x^2+c_0 x+c_4}}dx+4 c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}-1\right )}dx+2 c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}+1\right )}dx+4 c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}-1\right )}dx+2 c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}+1\right )}dx\right )}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\) |
Input:
Int[((x^2*C[3] - C[4])*(x + 3*x^2*C[3] + 3*C[4]))/(x*Sqrt[(x*C[0] + x^2*C[ 3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4 ] + C[4]^2)),x]
Output:
$Aborted
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 18.15 (sec) , antiderivative size = 35680305, normalized size of antiderivative = 182975.92
Input:
int((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1* x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \] Input:
integrate((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^ 2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="fric as")
Output:
Timed out
Timed out. \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \] Input:
integrate((_C3*x**2-_C4)*(3*_C3*x**2+3*_C4+x)/x/((_C3*x**2+_C0*x+_C4)/(_C3 *x**2+_C1*x+_C4))**(1/2)/(_C3**2*x**4+2*_C3*_C4*x**2+_C4**2-x**2),x)
Output:
Timed out
\[ \text {Unable to generate Latex} \] Input:
integrate((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^ 2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="maxi ma")
Output:
integrate((3*_C3*x^2 + 3*_C4 + x)*(_C3*x^2 - _C4)/((_C3^2*x^4 + 2*_C3*_C4* x^2 + _C4^2 - x^2)*x*sqrt((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4)) ), x)
\[ \text {Unable to generate Latex} \] Input:
integrate((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^ 2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="giac ")
Output:
integrate((3*_C3*x^2 + 3*_C4 + x)*(_C3*x^2 - _C4)/((_C3^2*x^4 + 2*_C3*_C4* x^2 + _C4^2 - x^2)*x*sqrt((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4)) ), x)
Timed out. \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int -\frac {\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )\,\left (3\,_{\mathrm {C3}}\,x^2+x+3\,_{\mathrm {C4}}\right )}{x\,\sqrt {\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}}\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2-x^2\right )} \,d x \] Input:
int(-((_C4 - _C3*x^2)*(3*_C4 + x + 3*_C3*x^2))/(x*((_C4 + _C0*x + _C3*x^2) /(_C4 + _C1*x + _C3*x^2))^(1/2)*(_C4^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)) ,x)
Output:
int(-((_C4 - _C3*x^2)*(3*_C4 + x + 3*_C3*x^2))/(x*((_C4 + _C0*x + _C3*x^2) /(_C4 + _C1*x + _C3*x^2))^(1/2)*(_C4^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)) , x)
\[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {\left (\textit {\_C3} \,x^{2}-\textit {\_C4} \right ) \left (3 \textit {\_C3} \,x^{2}+3 \textit {\_C4} +x \right )}{x \sqrt {\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}}\, \left (\textit {\_C3}^{2} x^{4}+2 \textit {\_C3} \textit {\_C4} \,x^{2}+\textit {\_C4}^{2}-x^{2}\right )}d x \] Input:
int((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1* x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x)
Output:
int((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1* x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x)