\(\int \frac {1+x^3}{(-1+x^3) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [2287]

Optimal result

Integrand size = 37, antiderivative size = 174 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {2 \sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 (2 a+2 b+c)}+\frac {4 \arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a}+\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a+b-c}} \]

[Out]

-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)

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 \\ & = 2 \int \frac {1}{\left (-1+x^3\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 \\ & = 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+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = \frac {2}{3} \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\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 \\ & = \frac {2}{3} \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\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 \\ & = \frac {2}{3} \int \frac {1}{(-1+x) \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-\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 = 1.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2}{3} \left (\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {2 \arctan \left (\frac {\sqrt {a+b-c} x}{\sqrt {a} \left (1+x+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {a+b-c}}\right ) \]

[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])])/Sqr
t[a + b - c]))/3

Maple [A] (verified)

Time = 1.91 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {2 \ln \left (\frac {2 \sqrt {-a -b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (-2 a +b \right ) x^{2}+\left (-4 a -b +2 c \right ) x -2 a +b}{x^{2}+x +1}\right ) \sqrt {2 a +2 b +c}+\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {-a -b +c}}{3 \sqrt {2 a +2 b +c}\, \sqrt {-a -b +c}}\)	\(184\)

[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*(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)

Fricas [A] (verification not implemented)

none

Time = 1.84 (sec) , antiderivative size = 1497, normalized size of antiderivative = 8.60 \[ \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]

[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*(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 + 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), -1/6*(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*(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)