Integrand size = 45, antiderivative size = 48 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\frac {\arctan \left (\frac {2 \sqrt {42} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{103-86 x+19 x^2}\right )}{\sqrt {42}} \]
Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=-\sqrt {\frac {2}{21}} \arctan \left (\frac {\sqrt {\frac {21}{2}} \sqrt {-60+83 x-21 x^2-3 x^3+x^4}}{-20+x+x^2}\right ) \]
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {5 x^2-46 x+77}{\left (-23 x^2+82 x-23\right ) \sqrt {x^4-3 x^3-21 x^2+83 x-60}} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {72 (23-9 x)}{23 \left (-23 x^2+82 x-23\right ) \sqrt {x^4-3 x^3-21 x^2+83 x-60}}-\frac {5}{23 \sqrt {x^4-3 x^3-21 x^2+83 x-60}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{23} \int \frac {1}{\sqrt {x^4-3 x^3-21 x^2+83 x-60}}dx-\frac {24}{23} \left (27+10 \sqrt {2}\right ) \int \frac {1}{\left (-46 x-48 \sqrt {2}+82\right ) \sqrt {x^4-3 x^3-21 x^2+83 x-60}}dx-\frac {24}{23} \left (27-10 \sqrt {2}\right ) \int \frac {1}{\left (-46 x+48 \sqrt {2}+82\right ) \sqrt {x^4-3 x^3-21 x^2+83 x-60}}dx\) |
3.7.8.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
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 | \(\text {Expression too large to display}\) | \(3260\) |
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,m ethod=_RETURNVERBOSE)
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))
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\frac {1}{42} \, \sqrt {21} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {21} \sqrt {2} \sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60}}{19 \, x^{2} - 86 \, x + 103}\right ) \]
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")
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))
\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=- \int \left (- \frac {46 x}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\right )\, dx - \int \frac {5 x^{2}}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx - \int \frac {77}{23 x^{2} \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} - 82 x \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60} + 23 \sqrt {x^{4} - 3 x^{3} - 21 x^{2} + 83 x - 60}}\, dx \]
-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 - 60)), 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)
\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int { -\frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}} \,d x } \]
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")
-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)
\[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int { -\frac {5 \, x^{2} - 46 \, x + 77}{\sqrt {x^{4} - 3 \, x^{3} - 21 \, x^{2} + 83 \, x - 60} {\left (23 \, x^{2} - 82 \, x + 23\right )}} \,d x } \]
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")
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)
Timed out. \[ \int \frac {77-46 x+5 x^2}{\left (-23+82 x-23 x^2\right ) \sqrt {-60+83 x-21 x^2-3 x^3+x^4}} \, dx=\int -\frac {5\,x^2-46\,x+77}{\left (23\,x^2-82\,x+23\right )\,\sqrt {x^4-3\,x^3-21\,x^2+83\,x-60}} \,d x \]