\(\int \frac {-1+x^3}{(1+x^3) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [2316]

Optimal result

Integrand size = 37, antiderivative size = 179 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {4 \arctan \left (\frac {\sqrt {a-b-c} x}{\sqrt {a}-\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a-b-c}}-\frac {2 \sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 (2 a-2 b+c)} \]

[Out]

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)-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)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {2}{\left (1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (2 \int \left (\frac {1}{3 (1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {2-x}{3 \left (1-x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (\frac {2}{3} \int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {2}{3} \int \frac {2-x}{\left (1-x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (\frac {2}{3} \int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {2}{3} \int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (\frac {2}{3} \int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2}{3} \left (\frac {\arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}+\frac {2 \arctan \left (\frac {\sqrt {a-b-c} x}{\sqrt {a} \left (1-x+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {a-b-c}}\right ) \]

[In]

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

[Out]

(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

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {-a +b +c}+2 \ln \left (\frac {2 \sqrt {-a +b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (2 a +b \right ) x^{2}+\left (-4 a +b +2 c \right ) x +2 a +b}{x^{2}-x +1}\right ) \sqrt {2 a -2 b +c}}{3 \sqrt {-a +b +c}\, \sqrt {2 a -2 b +c}}\)	\(178\)

[In]

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

[Out]

-1/3*(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)+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)

Fricas [A] (verification not implemented)

none

Time = 1.85 (sec) , antiderivative size = 1508, normalized size of antiderivative = 8.42 \[ \int \frac {-1+x^3}{\left (1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6*(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 - 16*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*(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 + 2*b^2 - (a - b)*c - c^2), -1/6*(4*(2*a - 2*b + c)*sqrt(a - b - c)*arctan(2*sqrt(a*x^4 + b*x^3 + c*
x^2 + b*x + a)*sqrt(a - b - c)/((2*a + b)*x^2 - (4*a - b - 2*c)*x + 2*a + b)) - 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 - 16*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 + 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*(2*a - 2*b + c)*sqrt(a - b - c)*arctan(2
*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*sqrt(a - b - c)/((2*a + b)*x^2 - (4*a - b - 2*c)*x + 2*a + b)))/(2*a^2
- 4*a*b + 2*b^2 - (a - b)*c - c^2)]

Sympy [F]

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

[In]

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

[Out]

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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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