Integrand size = 37, antiderivative size = 174 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {2 \sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 (2 a+2 b+c)}+\frac {4 \arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a}+\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a+b-c}} \]
-2*(-2*a-2*b-c)^(1/2)*arctan((-2*a-2*b-c)^(1/2)*x/(a^(1/2)-2*x*a^(1/2)+a^( 1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(6*a+6*b+3*c)+4/3*arctan((a+b-c )^(1/2)*x/(a^(1/2)+x*a^(1/2)+a^(1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2))) /(a+b-c)^(1/2)
Time = 1.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2}{3} \left (\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {2 \arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a} \left (1+x+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {a+b-c}}\right ) \]
(2*(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]))/3
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.
\(\displaystyle \int \frac {x^3+1}{\left (x^3-1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2}{\left (x^3-1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}+\frac {1}{\sqrt {a x^4+a+b x^3+b x+c x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{\sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {2}{3} \int \frac {1}{(1-x) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {2}{3} \int \frac {1}{\left (\sqrt [3]{-1} x+1\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {2}{3} \int \frac {1}{\left (1-(-1)^{2/3} x\right ) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx\) |
3.23.87.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.91 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {2 \ln \left (\frac {2 \sqrt {-a -b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (-2 a +b \right ) x^{2}+\left (-4 a -b +2 c \right ) x -2 a +b}{x^{2}+x +1}\right ) \sqrt {2 a +2 b +c}+\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {-a -b +c}}{3 \sqrt {2 a +2 b +c}\, \sqrt {-a -b +c}}\) | \(184\) |
-1/3*(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)+ln((2*(2*a+2*b+c)^(1/2) *(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(4*a+b)*x^2+(-4*a+2*b+2*c)*x+4*a+b)/(-1+x )^2)*(-a-b+c)^(1/2))/(2*a+2*b+c)^(1/2)/(-a-b+c)^(1/2)
Time = 1.84 (sec) , antiderivative size = 1497, normalized size of antiderivative = 8.60 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]
[-1/6*(2*(2*a + 2*b + c)*sqrt(-a - b + c)*log(-((8*a*b - b^2 - 4*a*c)*x^4 - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a - b)*c)*x^3 - (24*a^2 + 3*b^2 - 4*(5*a + 2*b)*c + 8*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((2*a - b)* x^2 + (4*a + b - 2*c)*x + 2*a - b)*sqrt(-a - b + c) + 8*a*b - b^2 - 4*a*c - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a - b)*c)*x)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) - sqrt(2*a + 2*b + c)*(a + b - c)*log(((24*a^2 + 16*a*b + b^2 + 4*a*c) *x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4* a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 1 6*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)))/(2*a^2 + 4*a*b + 2*b^2 - (a + b)*c - c^2), 1/3* ((a + b - c)*sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c )/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)) - (2*a + 2*b + c)*sqrt(-a - b + c)*log(-((8*a*b - b^2 - 4*a*c)*x^4 - 2*(8*a^2 - 4*a*b - 3 *b^2 - 4*(a - b)*c)*x^3 - (24*a^2 + 3*b^2 - 4*(5*a + 2*b)*c + 8*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((2*a - b)*x^2 + (4*a + b - 2*c)* x + 2*a - b)*sqrt(-a - b + c) + 8*a*b - b^2 - 4*a*c - 2*(8*a^2 - 4*a*b - 3 *b^2 - 4*(a - b)*c)*x)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)))/(2*a^2 + 4*a*b...
\[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{\left (x - 1\right ) \left (x^{2} + x + 1\right ) \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
Integral((x + 1)*(x**2 - x + 1)/((x - 1)*(x**2 + x + 1)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x)
\[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x^{3} + 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{3} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x^3+1}{\left (x^3-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]