\(\int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx\) [497]

Optimal result

Integrand size = 37, antiderivative size = 38 \[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\log \left (2-x^2+x^3+\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right ) \]

[Out]

ln(2-x^2+x^3+(x^6-2*x^5+x^4+4*x^3-4*x^2+5)^(1/2))

Rubi [F]

\[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx \]

[In]

Int[(-2*x + 3*x^2)/Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6],x]

[Out]

-2*Defer[Int][x/Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6], x] + 3*Defer[Int][x^2/Sqrt[5 - 4*x^2 + 4*x^3 + x^
4 - 2*x^5 + x^6], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x (-2+3 x)}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx \\ & = \int \left (-\frac {2 x}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}}+\frac {3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx\right )+3 \int \frac {x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=-\log \left (-2+x^2-x^3+\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}\right ) \]

[In]

Integrate[(-2*x + 3*x^2)/Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6],x]

[Out]

-Log[-2 + x^2 - x^3 + Sqrt[5 - 4*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6]]

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.32

method	result	size
pseudoelliptic	\(\operatorname {arcsinh}\left (x^{3}-x^{2}+2\right )\)	\(12\)
trager	\(\ln \left (2-x^{2}+x^{3}+\sqrt {x^{6}-2 x^{5}+x^{4}+4 x^{3}-4 x^{2}+5}\right )\)	\(37\)

[In]

int((3*x^2-2*x)/(x^6-2*x^5+x^4+4*x^3-4*x^2+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsinh(x^3-x^2+2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\log \left (x^{3} - x^{2} + \sqrt {x^{6} - 2 \, x^{5} + x^{4} + 4 \, x^{3} - 4 \, x^{2} + 5} + 2\right ) \]

[In]

integrate((3*x^2-2*x)/(x^6-2*x^5+x^4+4*x^3-4*x^2+5)^(1/2),x, algorithm="fricas")

[Out]

log(x^3 - x^2 + sqrt(x^6 - 2*x^5 + x^4 + 4*x^3 - 4*x^2 + 5) + 2)

Sympy [F]

\[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\int \frac {x \left (3 x - 2\right )}{\sqrt {x^{6} - 2 x^{5} + x^{4} + 4 x^{3} - 4 x^{2} + 5}}\, dx \]

[In]

integrate((3*x**2-2*x)/(x**6-2*x**5+x**4+4*x**3-4*x**2+5)**(1/2),x)

[Out]

Integral(x*(3*x - 2)/sqrt(x**6 - 2*x**5 + x**4 + 4*x**3 - 4*x**2 + 5), x)

Maxima [F]

\[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\int { \frac {3 \, x^{2} - 2 \, x}{\sqrt {x^{6} - 2 \, x^{5} + x^{4} + 4 \, x^{3} - 4 \, x^{2} + 5}} \,d x } \]

[In]

integrate((3*x^2-2*x)/(x^6-2*x^5+x^4+4*x^3-4*x^2+5)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x^2 - 2*x)/sqrt(x^6 - 2*x^5 + x^4 + 4*x^3 - 4*x^2 + 5), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (36) = 72\).

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\frac {1}{2} \, \sqrt {4 \, x^{3} + {\left (x^{3} - x^{2}\right )}^{2} - 4 \, x^{2} + 5} {\left (x^{3} - x^{2} + 2\right )} - \frac {1}{2} \, \log \left (-x^{3} + x^{2} + \sqrt {4 \, x^{3} + {\left (x^{3} - x^{2}\right )}^{2} - 4 \, x^{2} + 5} - 2\right ) \]

[In]

integrate((3*x^2-2*x)/(x^6-2*x^5+x^4+4*x^3-4*x^2+5)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(4*x^3 + (x^3 - x^2)^2 - 4*x^2 + 5)*(x^3 - x^2 + 2) - 1/2*log(-x^3 + x^2 + sqrt(4*x^3 + (x^3 - x^2)^2
- 4*x^2 + 5) - 2)

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 x+3 x^2}{\sqrt {5-4 x^2+4 x^3+x^4-2 x^5+x^6}} \, dx=\int -\frac {2\,x-3\,x^2}{\sqrt {x^6-2\,x^5+x^4+4\,x^3-4\,x^2+5}} \,d x \]

[In]

int(-(2*x - 3*x^2)/(4*x^3 - 4*x^2 + x^4 - 2*x^5 + x^6 + 5)^(1/2),x)

[Out]

int(-(2*x - 3*x^2)/(4*x^3 - 4*x^2 + x^4 - 2*x^5 + x^6 + 5)^(1/2), x)