\(\int \frac {77-46 x+5 x^2}{(-23+82 x-23 x^2) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\) [608]

Optimal result

Integrand size = 45, antiderivative size = 48 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\frac {\arctan \left (\frac {2 \sqrt {42} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right )}{\sqrt {42}} \]

[Out]

1/42*arctan(2*42^(1/2)*(x^4-3*x^3-21*x^2+83*x-60)^(1/2)/(19*x^2-86*x+103))*42^(1/2)

Rubi [F]

\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \]

[In]

Int[(77 - 46*x + 5*x^2)/((-23 + 82*x - 23*x^2)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]

[Out]

(-5*Defer[Int][1/Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4], x])/23 - (24*(27 + 10*Sqrt[2])*Defer[Int][1/((82 - 4
8*Sqrt[2] - 46*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]), x])/23 - (24*(27 - 10*Sqrt[2])*Defer[Int][1/((82 +
48*Sqrt[2] - 46*x)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]), x])/23

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {5}{23 \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {72 (23-9 x)}{23 \left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx \\ & = -\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \frac {23-9 x}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \\ & = -\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )+\frac {72}{23} \int \left (\frac {-9-\frac {10 \sqrt {2}}{3}}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}+\frac {-9+\frac {10 \sqrt {2}}{3}}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}\right ) \, dx \\ & = -\left (\frac {5}{23} \int \frac {1}{\sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx\right )-\frac {1}{23} \left (24 \left (27-10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82+48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx-\frac {1}{23} \left (24 \left (27+10 \sqrt {2}\right )\right ) \int \frac {1}{\left (82-48 \sqrt {2}-46 x\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=-\sqrt {\frac {2}{21}} \arctan \left (\frac {\sqrt {\frac {21}{2}} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{-20+x+x^2}\right ) \]

[In]

Integrate[(77 - 46*x + 5*x^2)/((-23 + 82*x - 23*x^2)*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4]),x]

[Out]

-(Sqrt[2/21]*ArcTan[(Sqrt[21/2]*Sqrt[-60 + 83*x - 21*x^2 - 3*x^3 + x^4])/(-20 + x + x^2)])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.56

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+42\right ) \ln \left (-\frac {19 \operatorname {RootOf}\left (\textit {\_Z}^{2}+42\right ) x^{2}-86 \operatorname {RootOf}\left (\textit {\_Z}^{2}+42\right ) x +84 \sqrt {x^{4}-3 x^{3}-21 x^{2}+83 x -60}+103 \operatorname {RootOf}\left (\textit {\_Z}^{2}+42\right )}{23 x^{2}-82 x +23}\right )}{42}\)	\(75\)
default	\(\text {Expression too large to display}\)	\(3042\)
elliptic	\(\text {Expression too large to display}\)	\(3260\)

[In]

int((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/42*RootOf(_Z^2+42)*ln(-(19*RootOf(_Z^2+42)*x^2-86*RootOf(_Z^2+42)*x+84*(x^4-3*x^3-21*x^2+83*x-60)^(1/2)+103*
RootOf(_Z^2+42))/(23*x^2-82*x+23))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\frac {1}{42} \, \sqrt {21} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {21} \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60}}{19 \, x^{2} - 86 \, x + 103}\right ) \]

[In]

integrate((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="fricas")

[Out]

1/42*sqrt(21)*sqrt(2)*arctan(2*sqrt(21)*sqrt(2)*sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)/(19*x^2 - 86*x + 103))

Sympy [F]

\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=- \int \left (- \frac {46 x}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\right )\, dx - \int \frac {5 x^{2}}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx - \int \frac {77}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx \]

[In]

integrate((5*x**2-46*x+77)/(-23*x**2+82*x-23)/(x**4-3*x**3-21*x**2+83*x-60)**(1/2),x)

[Out]

-Integral(-46*x/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x
- 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60)), x) - Integral(5*x**2/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x
**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 6
0)), x) - Integral(77/(23*x**2*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60) - 82*x*sqrt(x**4 - 3*x**3 - 21*x**2 +
83*x - 60) + 23*sqrt(x**4 - 3*x**3 - 21*x**2 + 83*x - 60)), x)

Maxima [F]

\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int { -\frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}} \,d x } \]

[In]

integrate((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="maxima")

[Out]

-integrate((5*x^2 - 46*x + 77)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(23*x^2 - 82*x + 23)), x)

Giac [F]

\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int { -\frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}} \,d x } \]

[In]

integrate((5*x^2-46*x+77)/(-23*x^2+82*x-23)/(x^4-3*x^3-21*x^2+83*x-60)^(1/2),x, algorithm="giac")

[Out]

integrate(-(5*x^2 - 46*x + 77)/(sqrt(x^4 - 3*x^3 - 21*x^2 + 83*x - 60)*(23*x^2 - 82*x + 23)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int -\frac {5\,x^2-46\,x+77}{\left (23\,x^2-82\,x+23\right )\,\sqrt {x^4-3\,x^3-21\,x^2+83\,x-60}} \,d x \]

[In]

int(-(5*x^2 - 46*x + 77)/((23*x^2 - 82*x + 23)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)),x)

[Out]

int(-(5*x^2 - 46*x + 77)/((23*x^2 - 82*x + 23)*(83*x - 21*x^2 - 3*x^3 + x^4 - 60)^(1/2)), x)