\(\int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} (1-x^6)} \, dx\) [2946]

Optimal result

Integrand size = 37, antiderivative size = 352 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\frac {\sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{6 (2 a+2 b+c)}+\frac {\arctan \left (\frac {\sqrt {a-b-c} x}{\sqrt {a}-\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a-b-c}}-\frac {\arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a}+\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a+b-c}}-\frac {\sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{6 (2 a-2 b+c)} \] Output:

(-2*a-2*b-c)^(1/2)*arctan((-2*a-2*b-c)^(1/2)*x/(a^(1/2)-2*a^(1/2)*x+a^(1/2 
)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(12*a+12*b+6*c)+1/3*arctan((a-b-c) 
^(1/2)*x/(a^(1/2)-a^(1/2)*x+a^(1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/ 
(a-b-c)^(1/2)-1/3*arctan((a+b-c)^(1/2)*x/(a^(1/2)+a^(1/2)*x+a^(1/2)*x^2-(a 
*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(a+b-c)^(1/2)-(-2*a+2*b-c)^(1/2)*arctan((- 
2*a+2*b-c)^(1/2)*x/(a^(1/2)+2*a^(1/2)*x+a^(1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x 
+a)^(1/2)))/(12*a-12*b+6*c)
 

Mathematica [A] (verified)

Time = 2.82 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\frac {1}{6} \left (-\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {\arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}+\frac {2 \arctan \left (\frac {\sqrt {a-b-c} x}{\sqrt {a} \left (1-x+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {a-b-c}}-\frac {2 \arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a} \left (1+x+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {a+b-c}}\right ) \] Input:

Integrate[x^3/(Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]*(1 - x^6)),x]
 

Output:

(-(ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a]*(-1 + x)^2 - Sqrt[a + b*x + c* 
x^2 + b*x^3 + a*x^4])]/Sqrt[-2*a - 2*b - c]) + ArcTan[(Sqrt[-2*a + 2*b - c 
]*x)/(Sqrt[a]*(1 + x)^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/Sqrt[-2* 
a + 2*b - c] + (2*ArcTan[(Sqrt[a - b - c]*x)/(Sqrt[a]*(1 - x + x^2) - Sqrt 
[a + b*x + c*x^2 + b*x^3 + a*x^4])])/Sqrt[a - b - c] - (2*ArcTan[(Sqrt[a + 
 b - c]*x)/(Sqrt[a]*(1 + x + x^2) - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]) 
])/Sqrt[a + b - c])/6
 

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.

\(\displaystyle \int \frac {x^3}{\left (1-x^6\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {x-2}{6 \left (x^2-x+1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}-\frac {x}{3 \left (x^2-1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}+\frac {x+2}{6 \left (x^2+x+1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{6} \int \frac {1}{(x-1) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {1}{6} \int \frac {1}{(x+1) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}-1\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}+1\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}-1\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx\)

Input:

Int[x^3/(Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]*(1 - x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.50 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b +4 a \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {-a +b +c}\, \sqrt {-a -b +c}\, \sqrt {2 a -2 b +c}+2 \ln \left (\frac {2 \sqrt {-a -b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (-2 a +b \right ) x^{2}+\left (-4 a -b +2 c \right ) x -2 a +b}{x^{2}+x +1}\right ) \sqrt {2 a +2 b +c}\, \sqrt {-a +b +c}\, \sqrt {2 a -2 b +c}-\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {2 a +2 b +c}\, \sqrt {-a +b +c}\, \sqrt {-a -b +c}-2 \ln \left (\frac {2 \sqrt {-a +b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (2 a +b \right ) x^{2}+\left (-4 a +b +2 c \right ) x +2 a +b}{x^{2}-x +1}\right ) \sqrt {2 a +2 b +c}\, \sqrt {-a -b +c}\, \sqrt {2 a -2 b +c}}{12 \sqrt {2 a +2 b +c}\, \sqrt {-a +b +c}\, \sqrt {-a -b +c}\, \sqrt {2 a -2 b +c}}\)	\(435\)

Input:

int(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x,method=_RETURNVERBOSE)
 

Output:

1/12*(ln((2*(2*a+2*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(b+4*a)*x^2+ 
(-4*a+2*b+2*c)*x+4*a+b)/(-1+x)^2)*(-a+b+c)^(1/2)*(-a-b+c)^(1/2)*(2*a-2*b+c 
)^(1/2)+2*ln((2*(-a-b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(-2*a+b)*x^ 
2+(-4*a-b+2*c)*x-2*a+b)/(x^2+x+1))*(2*a+2*b+c)^(1/2)*(-a+b+c)^(1/2)*(2*a-2 
*b+c)^(1/2)-ln((2*(2*a-2*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(b-4*a 
)*x^2+(-4*a-2*b+2*c)*x-4*a+b)/(1+x)^2)*(2*a+2*b+c)^(1/2)*(-a+b+c)^(1/2)*(- 
a-b+c)^(1/2)-2*ln((2*(-a+b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(2*a+b 
)*x^2+(-4*a+b+2*c)*x+2*a+b)/(x^2-x+1))*(2*a+2*b+c)^(1/2)*(-a-b+c)^(1/2)*(2 
*a-2*b+c)^(1/2))/(2*a+2*b+c)^(1/2)/(-a+b+c)^(1/2)/(-a-b+c)^(1/2)/(2*a-2*b+ 
c)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (296) = 592\).

Time = 12.24 (sec) , antiderivative size = 15809, normalized size of antiderivative = 44.91 \[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x, algorithm="frica 
s")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=- \int \frac {x^{3}}{x^{6} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} - \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \] Input:

integrate(x**3/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2)/(-x**6+1),x)
 

Output:

-Integral(x**3/(x**6*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) - sqrt(a*x** 
4 + a + b*x**3 + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\int { -\frac {x^{3}}{{\left (x^{6} - 1\right )} \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a}} \,d x } \] Input:

integrate(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x, algorithm="maxim 
a")
 

Output:

-integrate(x^3/((x^6 - 1)*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x, algorithm="giac" 
)
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=-\int \frac {x^3}{\left (x^6-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \] Input:

int(-x^3/((x^6 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)
 

Output:

-int(x^3/((x^6 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx=\int \frac {x^{3}}{\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}\, \left (-x^{6}+1\right )}d x \] Input:

int(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x)
 

Output:

int(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x)