\(\int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx\) [1081]

Optimal result

Integrand size = 26, antiderivative size = 81 \[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\arctan \left (\frac {-4+4 x}{1-2 x+x^2-\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right )+\log (-1+x)-\log \left (-5+6 x-x^2+\sqrt {-7+4 x+14 x^2-12 x^3+x^4}\right ) \]

[Out]

arctan((-4+4*x)/(1-2*x+x^2-(x^4-12*x^3+14*x^2+4*x-7)^(1/2)))+ln(-1+x)-ln(-5+6*x-x^2+(x^4-12*x^3+14*x^2+4*x-7)^
(1/2))

Rubi [F]

\[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \]

[In]

Int[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

Defer[Int][1/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4], x] + Defer[Int][x/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}+\frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}}\right ) \, dx \\ & = \int \frac {1}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\frac {(-1+x) \sqrt {-7-10 x+x^2} \left (\arctan \left (\frac {1}{4} \left (1-x+\sqrt {-7-10 x+x^2}\right )\right )-\log \left (5-x+\sqrt {-7-10 x+x^2}\right )\right )}{\sqrt {(-1+x)^2 \left (-7-10 x+x^2\right )}} \]

[In]

Integrate[(1 + x)/Sqrt[-7 + 4*x + 14*x^2 - 12*x^3 + x^4],x]

[Out]

((-1 + x)*Sqrt[-7 - 10*x + x^2]*(ArcTan[(1 - x + Sqrt[-7 - 10*x + x^2])/4] - Log[5 - x + Sqrt[-7 - 10*x + x^2]
]))/Sqrt[(-1 + x)^2*(-7 - 10*x + x^2)]

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\left (x -1\right ) \sqrt {x^{2}-10 x -7}\, \left (2 \ln \left (-5+x +\sqrt {x^{2}-10 x -7}\right )-\arctan \left (\frac {3+x}{\sqrt {x^{2}-10 x -7}}\right )\right )}{2 \sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}\)	\(70\)
trager	\(\ln \left (-\frac {x^{2}+\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}-6 x +5}{x -1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{4}-12 x^{3}+14 x^{2}+4 x -7}}{\left (x -1\right )^{2}}\right )}{2}\)	\(101\)

[In]

int((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(x-1)*(x^2-10*x-7)^(1/2)*(2*ln(-5+x+(x^2-10*x-7)^(1/2))-arctan((3+x)/(x^2-10*x-7)^(1/2)))/(x^4-12*x^3+14*x
^2+4*x-7)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\arctan \left (-\frac {x^{2} - 2 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 1}{4 \, {\left (x - 1\right )}}\right ) - \log \left (-\frac {x^{2} - 6 \, x - \sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7} + 5}{x - 1}\right ) \]

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="fricas")

[Out]

arctan(-1/4*(x^2 - 2*x - sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7) + 1)/(x - 1)) - log(-(x^2 - 6*x - sqrt(x^4 - 12
*x^3 + 14*x^2 + 4*x - 7) + 5)/(x - 1))

Sympy [F]

\[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\int \frac {x + 1}{\sqrt {\left (x - 1\right )^{2} \left (x^{2} - 10 x - 7\right )}}\, dx \]

[In]

integrate((1+x)/(x**4-12*x**3+14*x**2+4*x-7)**(1/2),x)

[Out]

Integral((x + 1)/sqrt((x - 1)**2*(x**2 - 10*x - 7)), x)

Maxima [F]

\[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\int { \frac {x + 1}{\sqrt {x^{4} - 12 \, x^{3} + 14 \, x^{2} + 4 \, x - 7}} \,d x } \]

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/sqrt(x^4 - 12*x^3 + 14*x^2 + 4*x - 7), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

integrate((1+x)/(x^4-12*x^3+14*x^2+4*x-7)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: (-atan(i)+ln(4*sqrt(2)))*sign(sageVARx-1
)+2*(1/2*atan(1/4*(-sageVARx+sqrt(sageVARx^2-10*sageVARx-7)+1))/sign(sageVARx-1)-1/2*ln(abs(-sageVARx+sqrt(sag
eVARx^2-10*sageVARx-7

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x}{\sqrt {-7+4 x+14 x^2-12 x^3+x^4}} \, dx=\int \frac {x+1}{\sqrt {x^4-12\,x^3+14\,x^2+4\,x-7}} \,d x \]

[In]

int((x + 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2),x)

[Out]

int((x + 1)/(4*x + 14*x^2 - 12*x^3 + x^4 - 7)^(1/2), x)