\(\int \frac {x (x^2 c_3-c_4)}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} (x+3 x^2 c_3+3 c_4) (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\) [2728]

Optimal result

Integrand size = 90, antiderivative size = 250 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{2 \sqrt {1-c_0}}-\frac {\arctan \left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{4 \sqrt {-1-c_0}}+\frac {3 \arctan \left (\frac {\sqrt {1-3 c_0} \sqrt {-1+3 c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+3 c_0}\right ) \sqrt {-1+3 c_1}}{4 \sqrt {1-3 c_0}} \] Output:

-1/2*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)-1/4*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)+3/4*arctan((1-3*_C0)^(1/2)*(-1+3*_C1)^(1/2 
)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(-1+3*_C0))*(-1+3*_C1)^( 
1/2)/(1-3*_C0)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(57413\) vs. \(2(250)=500\).

Time = 6.90 (sec) , antiderivative size = 57413, normalized size of antiderivative = 229.65 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Result too large to show} \] Input:

Integrate[(x*(x^2*C[3] - C[4]))/(Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + 
 x^2*C[3] + C[4])]*(x + 3*x^2*C[3] + 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.

\(\displaystyle \int \frac {x \left (c_3 x^2-c_4\right )}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \left (3 c_3 x^2+x+3 c_4\right ) \left (c_3{}^2 x^4-x^2+2 c_3 c_4 x^2+c_4{}^2\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {x \left (c_3 x^2-c_4\right )}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \left (3 c_3 x^2+x+3 c_4\right ) \left (c_3{}^2 x^4+(-1+2 c_3 c_4) x^2+c_4{}^2\right )}dx\)

\(\Big \downarrow \) 7270

\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \int \frac {x \left (x^2 c_3-c_4\right ) \sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (3 c_3 x^2+x+3 c_4\right ) \left (c_3{}^2 x^4-(1-2 c_3 c_4) x^2+c_4{}^2\right )}dx}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\)

\(\Big \downarrow \) 7279

\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \int \left (\frac {\sqrt {c_3 x^2+c_1 x+c_4} (2 x c_3-1)}{8 \left (c_3 x^2-x+c_4\right ) \sqrt {c_3 x^2+c_0 x+c_4}}+\frac {(2 x c_3+1) \sqrt {c_3 x^2+c_1 x+c_4}}{4 \left (c_3 x^2+x+c_4\right ) \sqrt {c_3 x^2+c_0 x+c_4}}-\frac {3 (6 x c_3+1) \sqrt {c_3 x^2+c_1 x+c_4}}{8 \sqrt {c_3 x^2+c_0 x+c_4} \left (3 c_3 x^2+x+3 c_4\right )}\right )dx}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c_3 x^2+c_0 x+c_4} \left (-\frac {9}{4} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (6 x c_3-\sqrt {1-36 c_3 c_4}+1\right )}dx-\frac {9}{4} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (6 x c_3+\sqrt {1-36 c_3 c_4}+1\right )}dx+\frac {1}{4} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}-1\right )}dx+\frac {1}{2} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3-\sqrt {1-4 c_3 c_4}+1\right )}dx+\frac {1}{4} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}-1\right )}dx+\frac {1}{2} c_3 \int \frac {\sqrt {c_3 x^2+c_1 x+c_4}}{\sqrt {c_3 x^2+c_0 x+c_4} \left (2 x c_3+\sqrt {1-4 c_3 c_4}+1\right )}dx\right )}{\sqrt {\frac {c_3 x^2+c_0 x+c_4}{c_3 x^2+c_1 x+c_4}} \sqrt {c_3 x^2+c_1 x+c_4}}\)

Input:

Int[(x*(x^2*C[3] - C[4]))/(Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C 
[3] + C[4])]*(x + 3*x^2*C[3] + 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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs. \(2(181)=362\).

Time = 60.66 (sec) , antiderivative size = 943, normalized size of antiderivative = 3.77

method	result	size
default	\(\text {Expression too large to display}\)	\(943\)

Input:

int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3 
*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(_C3*x^2+_C0*x+_C4)*(((1+_C1)*(1+_C0))^(1/2)*((-1+3*_C1)*(-1+3*_C0))^ 
(1/2)*ln((x^2*_C0*_C3+x^2*_C1*_C3+2*x^2*_C3+2*x*_C0*_C1+2*((1+_C1)*(1+_C0) 
)^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)+x*_C0+x*_C1+_C0*_C 
4+_C1*_C4+2*_C4)*_C3/(_C3*x^2+_C4-x))*_C0+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)* 
ln((x^2*_C0*_C3+x^2*_C1*_C3-2*x^2*_C3+2*x*_C0*_C1+2*((-1+_C1)*(-1+_C0))^(1 
/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-x*_C1+_C0*_C4+_C 
1*_C4-2*_C4)*_C3/(_C3*x^2+_C4+x))*((-1+_C1)*(-1+_C0))^(1/2)*_C0-9*ln((3*x^ 
2*_C0*_C3+3*x^2*_C1*_C3-2*x^2*_C3+6*x*_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0))^(1 
/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-x*_C1+3*_C0*_C4+ 
3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x))*_C0^2*_C1-((1+_C1)*(1+_C0))^(1/2 
)*((-1+3*_C1)*(-1+3*_C0))^(1/2)*ln((x^2*_C0*_C3+x^2*_C1*_C3+2*x^2*_C3+2*x* 
_C0*_C1+2*((1+_C1)*(1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4) 
)^(1/2)+x*_C0+x*_C1+_C0*_C4+_C1*_C4+2*_C4)*_C3/(_C3*x^2+_C4-x))+2*((-1+3*_ 
C1)*(-1+3*_C0))^(1/2)*ln((x^2*_C0*_C3+x^2*_C1*_C3-2*x^2*_C3+2*x*_C0*_C1+2* 
((-1+_C1)*(-1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)- 
x*_C0-x*_C1+_C0*_C4+_C1*_C4-2*_C4)*_C3/(_C3*x^2+_C4+x))*((-1+_C1)*(-1+_C0) 
)^(1/2)+3*ln((3*x^2*_C0*_C3+3*x^2*_C1*_C3-2*x^2*_C3+6*x*_C0*_C1+2*((-1+3*_ 
C1)*(-1+3*_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C 
0-x*_C1+3*_C0*_C4+3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x))*_C0^2+9*ln((3* 
x^2*_C0*_C3+3*x^2*_C1*_C3-2*x^2*_C3+6*x*_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0...
 

Fricas [A] (verification not implemented)

Time = 33.43 (sec) , antiderivative size = 6425, normalized size of antiderivative = 25.70 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/ 
(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x*(_C3*x**2-_C4)/((_C3*x**2+_C0*x+_C4)/(_C3*x**2+_C1*x+_C4))**(1 
/2)/(3*_C3*x**2+3*_C4+x)/(_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(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/ 
(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="maxi 
ma")
 

Output:

integrate((_C3*x^2 - _C4)*x/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*(3* 
_C3*x^2 + 3*_C4 + x)*sqrt((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4)) 
), x)
 

Giac [F]

\[ \text {Unable to generate Latex} \] Input:

integrate(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/ 
(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x, algorithm="giac 
")
 

Output:

integrate((_C3*x^2 - _C4)*x/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*(3* 
_C3*x^2 + 3*_C4 + x)*sqrt((_C3*x^2 + _C0*x + _C4)/(_C3*x^2 + _C1*x + _C4)) 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=-\int \frac {x\,\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )}{\sqrt {\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}}\,\left (3\,_{\mathrm {C3}}\,x^2+x+3\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2-x^2\right )} \,d x \] Input:

int(-(x*(_C4 - _C3*x^2))/(((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2) 
)^(1/2)*(3*_C4 + x + 3*_C3*x^2)*(_C4^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)) 
,x)
 

Output:

-int((x*(_C4 - _C3*x^2))/(((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2) 
)^(1/2)*(3*_C4 + x + 3*_C3*x^2)*(_C4^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)) 
, x)
 

Reduce [F]

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

int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3 
*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x)
 

Output:

int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3 
*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x^2+_C4^2-x^2),x)