\(\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx\) [709]

Optimal result

Integrand size = 24, antiderivative size = 55 \[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=-\frac {1}{4} \log \left (-11-68 x^2-24 x^3-2 x^4+\left (34+18 x+2 x^2\right ) \sqrt {-17+18 x-11 x^2+6 x^3+x^4}\right ) \]

[Out]

-1/4*ln(-11-68*x^2-24*x^3-2*x^4+(2*x^2+18*x+34)*(x^4+6*x^3-11*x^2+18*x-17)^(1/2))

Rubi [F]

\[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx \]

[In]

Int[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]

[Out]

Defer[Int][x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.84 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=-\frac {1}{4} \log \left (-11-68 x^2-24 x^3-2 x^4+\left (34+18 x+2 x^2\right ) \sqrt {-17+18 x-11 x^2+6 x^3+x^4}\right ) \]

[In]

Integrate[x/Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4],x]

[Out]

-1/4*Log[-11 - 68*x^2 - 24*x^3 - 2*x^4 + (34 + 18*x + 2*x^2)*Sqrt[-17 + 18*x - 11*x^2 + 6*x^3 + x^4]]

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\ln \left (2 x^{4}+2 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}\, x^{2}+24 x^{3}+18 x \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}+68 x^{2}+34 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}+11\right )}{4}\)	\(91\)
trager	\(\frac {\ln \left (2 x^{4}+2 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}\, x^{2}+24 x^{3}+18 x \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}+68 x^{2}+34 \sqrt {x^{4}+6 x^{3}-11 x^{2}+18 x -17}+11\right )}{4}\)	\(91\)
elliptic	\(\text {Expression too large to display}\)	\(1609\)

[In]

int(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*ln(2*x^4+2*(x^4+6*x^3-11*x^2+18*x-17)^(1/2)*x^2+24*x^3+18*x*(x^4+6*x^3-11*x^2+18*x-17)^(1/2)+68*x^2+34*(x^
4+6*x^3-11*x^2+18*x-17)^(1/2)+11)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\frac {1}{4} \, \log \left (2 \, x^{4} + 24 \, x^{3} + 68 \, x^{2} + 2 \, \sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17} {\left (x^{2} + 9 \, x + 17\right )} + 11\right ) \]

[In]

integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="fricas")

[Out]

1/4*log(2*x^4 + 24*x^3 + 68*x^2 + 2*sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17)*(x^2 + 9*x + 17) + 11)

Sympy [F]

\[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\int \frac {x}{\sqrt {x^{4} + 6 x^{3} - 11 x^{2} + 18 x - 17}}\, dx \]

[In]

integrate(x/(x**4+6*x**3-11*x**2+18*x-17)**(1/2),x)

[Out]

Integral(x/sqrt(x**4 + 6*x**3 - 11*x**2 + 18*x - 17), x)

Maxima [F]

\[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\int { \frac {x}{\sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17}} \,d x } \]

[In]

integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17), x)

Giac [F]

\[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\int { \frac {x}{\sqrt {x^{4} + 6 \, x^{3} - 11 \, x^{2} + 18 \, x - 17}} \,d x } \]

[In]

integrate(x/(x^4+6*x^3-11*x^2+18*x-17)^(1/2),x, algorithm="giac")

[Out]

integrate(x/sqrt(x^4 + 6*x^3 - 11*x^2 + 18*x - 17), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {-17+18 x-11 x^2+6 x^3+x^4}} \, dx=\int \frac {x}{\sqrt {x^4+6\,x^3-11\,x^2+18\,x-17}} \,d x \]

[In]

int(x/(18*x - 11*x^2 + 6*x^3 + x^4 - 17)^(1/2),x)

[Out]

int(x/(18*x - 11*x^2 + 6*x^3 + x^4 - 17)^(1/2), x)