\(\int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [818]

Optimal result

Integrand size = 35, antiderivative size = 62 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {\log (x)}{\sqrt {a}}+\frac {\log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \]

[Out]

-ln(x)/a^(1/2)+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^(1/2)

Rubi [F]

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

[In]

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

[Out]

Defer[Int][1/(x*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x] - Defer[Int][x/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 \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {x}{\sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = \int \frac {1}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\int \frac {x}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {-\log (x)+\log \left (-b x-2 a \left (1+x^2\right )+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\ln \left (2\right )-\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\)	\(58\)
default	\(-\frac {-\ln \left (2\right )+\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\)	\(59\)
elliptic	\(\text {Expression too large to display}\)	\(3365\)

[In]

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

[Out]

(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)))/a^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.82 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.39 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-a}}{2 \, a x^{2} + b x + 2 \, a}\right )}{a}\right ] \]

[In]

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

[Out]

[1/2*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)/sqrt(a), sqrt(-a)*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))/a]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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