\(\int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [1424]

Optimal result

Integrand size = 35, antiderivative size = 101 \[ \int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {\sqrt {a+b x+c x^2+b x^3+a x^4}}{a x}+\frac {b \log (x)}{2 a^{3/2}}-\frac {b \log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{2 a^{3/2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {\sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}{a x}+\frac {b \text {arctanh}\left (\frac {2 \sqrt {a} \sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}{b x+2 a \left (1+x^2\right )}\right )}{2 a^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.68 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}{a x}+\frac {b \left (-\ln \left (2\right )+\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )\right )}{2 a^{\frac {3}{2}}}\)	\(91\)
default	\(-\frac {-\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right ) b x +\ln \left (2\right ) b x +2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}{2 a^{\frac {3}{2}} x}\)	\(94\)
pseudoelliptic	\(\frac {\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right ) b x -\ln \left (2\right ) b x -2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}{2 a^{\frac {3}{2}} x}\)	\(94\)
elliptic	\(\text {Expression too large to display}\)	\(3402\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.82 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.14 \[ \int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\sqrt {a} b x \log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{4 \, a^{2} x}, -\frac {\sqrt {-a} b x \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-a}}{2 \, a x^{2} + b x + 2 \, a}\right ) + 2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{2 \, a^{2} x}\right ] \]

[In]

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

[Out]

[1/4*(sqrt(a)*b*x*log((8*a^2*x^4 + 8*a*b*x^3 + 8*a*b*x + (8*a^2 + b^2 + 4*a*c)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*
x^2 + b*x + a)*(2*a*x^2 + b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*a)/(a^2*x
), -1/2*(sqrt(-a)*b*x*arctan(2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*sqrt(-a)/(2*a*x^2 + b*x + 2*a)) + 2*sqrt(
a*x^4 + b*x^3 + c*x^2 + b*x + a)*a)/(a^2*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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