\(\int \frac {(x^2 c_3-c_4) (x+3 x^2 c_3+3 c_4)}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\) [2420]

Optimal result

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)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

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
 

Rubi [F]

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
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 18.15 (sec) , antiderivative size = 35680305, normalized size of antiderivative = 182975.92

method	result	size
default	\(\text {Expression too large to display}\)	\(35680305\)

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
 

Fricas [F(-1)]

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
 

Sympy [F(-1)]

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
 

Maxima [F]

\[ \text {Unable to generate Latex} \] 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)
 

Giac [F]

\[ \text {Unable to generate Latex} \] 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)
 

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {\left (\textit {\_C3} \,x^{2}-\textit {\_C4} \right ) \left (3 \textit {\_C3} \,x^{2}+3 \textit {\_C4} +x \right )}{x \sqrt {\frac {\textit {\_C3} \,x^{2}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{2}+\textit {\_C1} x +\textit {\_C4}}}\, \left (\textit {\_C3}^{2} x^{4}+2 \textit {\_C3} \textit {\_C4} \,x^{2}+\textit {\_C4}^{2}-x^{2}\right )}d x \] 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)