\(\int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx\) [725]

Optimal result

Integrand size = 28, antiderivative size = 56 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=-\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \]

[Out]

-1/2*ln(-90-76*x+89*x^2-18*x^3+x^4+(-x^2+13*x-38)*(x^4-10*x^3-3*x^2+4*x-8)^(1/2))

Rubi [F]

\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx \]

[In]

Int[(-1 + 2*x)/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4],x]

[Out]

-Defer[Int][1/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4], x] + 2*Defer[Int][x/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]
, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}+\frac {2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}}\right ) \, dx \\ & = 2 \int \frac {x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx-\int \frac {1}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.95 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=-\frac {1}{2} \log \left (-90-76 x+89 x^2-18 x^3+x^4+\left (-38+13 x-x^2\right ) \sqrt {-8+4 x-3 x^2-10 x^3+x^4}\right ) \]

[In]

Integrate[(-1 + 2*x)/Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4],x]

[Out]

-1/2*Log[-90 - 76*x + 89*x^2 - 18*x^3 + x^4 + (-38 + 13*x - x^2)*Sqrt[-8 + 4*x - 3*x^2 - 10*x^3 + x^4]]

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.62

method	result	size
trager	\(\frac {\ln \left (x^{4}+\sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}\, x^{2}-18 x^{3}-13 x \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}+89 x^{2}+38 \sqrt {x^{4}-10 x^{3}-3 x^{2}+4 x -8}-76 x -90\right )}{2}\)	\(91\)
default	\(\text {Expression too large to display}\)	\(2700\)
elliptic	\(\text {Expression too large to display}\)	\(2700\)

[In]

int((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(x^4+(x^4-10*x^3-3*x^2+4*x-8)^(1/2)*x^2-18*x^3-13*x*(x^4-10*x^3-3*x^2+4*x-8)^(1/2)+89*x^2+38*(x^4-10*x^3
-3*x^2+4*x-8)^(1/2)-76*x-90)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\frac {1}{2} \, \log \left (x^{4} - 18 \, x^{3} + 89 \, x^{2} + \sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8} {\left (x^{2} - 13 \, x + 38\right )} - 76 \, x - 90\right ) \]

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(x^4 - 18*x^3 + 89*x^2 + sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8)*(x^2 - 13*x + 38) - 76*x - 90)

Sympy [F]

\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {2 x - 1}{\sqrt {x^{4} - 10 x^{3} - 3 x^{2} + 4 x - 8}}\, dx \]

[In]

integrate((-1+2*x)/(x**4-10*x**3-3*x**2+4*x-8)**(1/2),x)

[Out]

Integral((2*x - 1)/sqrt(x**4 - 10*x**3 - 3*x**2 + 4*x - 8), x)

Maxima [F]

\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8}} \,d x } \]

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x - 1)/sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8), x)

Giac [F]

\[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 10 \, x^{3} - 3 \, x^{2} + 4 \, x - 8}} \,d x } \]

[In]

integrate((-1+2*x)/(x^4-10*x^3-3*x^2+4*x-8)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x - 1)/sqrt(x^4 - 10*x^3 - 3*x^2 + 4*x - 8), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+2 x}{\sqrt {-8+4 x-3 x^2-10 x^3+x^4}} \, dx=\int \frac {2\,x-1}{\sqrt {x^4-10\,x^3-3\,x^2+4\,x-8}} \,d x \]

[In]

int((2*x - 1)/(4*x - 3*x^2 - 10*x^3 + x^4 - 8)^(1/2),x)

[Out]

int((2*x - 1)/(4*x - 3*x^2 - 10*x^3 + x^4 - 8)^(1/2), x)