\(\int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx\) [847]

Optimal result

Integrand size = 28, antiderivative size = 64 \[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=-\log \left (-16-16 x-24 x^2-8 x^3-3 x^4-x^5+\left (8+4 x+2 x^2+x^3\right ) \sqrt {-12+20 x+5 x^2+2 x^3+x^4}\right ) \]

[Out]

-ln(-16-16*x-24*x^2-8*x^3-3*x^4-x^5+(x^3+2*x^2+4*x+8)*(x^4+2*x^3+5*x^2+20*x-12)^(1/2))

Rubi [F]

\[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx \]

[In]

Int[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]

[Out]

2*Defer[Int][1/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4], x] + 5*Defer[Int][x/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x
^4], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}}+\frac {5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx+5 \int \frac {x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.61 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=-\log \left (-16-16 x-24 x^2-8 x^3-3 x^4-x^5+\left (8+4 x+2 x^2+x^3\right ) \sqrt {-12+20 x+5 x^2+2 x^3+x^4}\right ) \]

[In]

Integrate[(2 + 5*x)/Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4],x]

[Out]

-Log[-16 - 16*x - 24*x^2 - 8*x^3 - 3*x^4 - x^5 + (8 + 4*x + 2*x^2 + x^3)*Sqrt[-12 + 20*x + 5*x^2 + 2*x^3 + x^4
]]

Maple [A] (verified)

Time = 5.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92

method	result	size
trager	\(-\ln \left (-x^{5}+\sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}\, x^{3}-3 x^{4}+2 \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}\, x^{2}-8 x^{3}+4 x \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}-24 x^{2}+8 \sqrt {x^{4}+2 x^{3}+5 x^{2}+20 x -12}-16 x -16\right )\)	\(123\)
default	\(\text {Expression too large to display}\)	\(2769\)
elliptic	\(\text {Expression too large to display}\)	\(2769\)

[In]

int((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-ln(-x^5+(x^4+2*x^3+5*x^2+20*x-12)^(1/2)*x^3-3*x^4+2*(x^4+2*x^3+5*x^2+20*x-12)^(1/2)*x^2-8*x^3+4*x*(x^4+2*x^3+
5*x^2+20*x-12)^(1/2)-24*x^2+8*(x^4+2*x^3+5*x^2+20*x-12)^(1/2)-16*x-16)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\log \left (x^{5} + 3 \, x^{4} + 8 \, x^{3} + 24 \, x^{2} + \sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12} {\left (x^{3} + 2 \, x^{2} + 4 \, x + 8\right )} + 16 \, x + 16\right ) \]

[In]

integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="fricas")

[Out]

log(x^5 + 3*x^4 + 8*x^3 + 24*x^2 + sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12)*(x^3 + 2*x^2 + 4*x + 8) + 16*x + 16)

Sympy [F]

\[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\int \frac {5 x + 2}{\sqrt {\left (x + 3\right ) \left (x^{3} - x^{2} + 8 x - 4\right )}}\, dx \]

[In]

integrate((2+5*x)/(x**4+2*x**3+5*x**2+20*x-12)**(1/2),x)

[Out]

Integral((5*x + 2)/sqrt((x + 3)*(x**3 - x**2 + 8*x - 4)), x)

Maxima [F]

\[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\int { \frac {5 \, x + 2}{\sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12}} \,d x } \]

[In]

integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 2)/sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12), x)

Giac [F]

\[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\int { \frac {5 \, x + 2}{\sqrt {x^{4} + 2 \, x^{3} + 5 \, x^{2} + 20 \, x - 12}} \,d x } \]

[In]

integrate((2+5*x)/(x^4+2*x^3+5*x^2+20*x-12)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 2)/sqrt(x^4 + 2*x^3 + 5*x^2 + 20*x - 12), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {2+5 x}{\sqrt {-12+20 x+5 x^2+2 x^3+x^4}} \, dx=\int \frac {5\,x+2}{\sqrt {x^4+2\,x^3+5\,x^2+20\,x-12}} \,d x \]

[In]

int((5*x + 2)/(20*x + 5*x^2 + 2*x^3 + x^4 - 12)^(1/2),x)

[Out]

int((5*x + 2)/(20*x + 5*x^2 + 2*x^3 + x^4 - 12)^(1/2), x)