Integrand size = 30, antiderivative size = 70 \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\frac {4}{15 \sqrt {5-\frac {4}{x}}}+\frac {\text {arctanh}\left (\frac {\sqrt {5-\frac {4}{x}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {5-\frac {4}{x}}}{\sqrt {5}}\right )}{5 \sqrt {5}} \] Output:
4/15/(5-4/x)^(1/2)+1/9*arctanh(1/3*(5-4/x)^(1/2)*3^(1/2))*3^(1/2)+2/25*arc tanh(1/5*(5-4/x)^(1/2)*5^(1/2))*5^(1/2)
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03 \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\frac {4 \sqrt {5-\frac {4}{x}} x}{-60+75 x}+\frac {\text {arctanh}\left (\frac {\sqrt {5-\frac {4}{x}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {4}{5 x}}\right )}{5 \sqrt {5}} \] Input:
Integrate[((1 - x)*(4 - x))/((5 - 4/x)^(3/2)*(-2 + x)*x^2),x]
Output:
(4*Sqrt[5 - 4/x]*x)/(-60 + 75*x) + ArcTanh[Sqrt[5 - 4/x]/Sqrt[3]]/(3*Sqrt[ 3]) + (2*ArcTanh[Sqrt[1 - 4/(5*x)]])/(5*Sqrt[5])
Time = 1.52 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7268, 2195, 2009}
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.
| \(\displaystyle \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (x-2) x^2} \, dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle -\int \frac {\left (\frac {4}{x}-4\right ) \left (\frac {4}{x}-1\right )}{\left (\left (5-\frac {4}{x}\right )^2-8 \left (5-\frac {4}{x}\right )+15\right ) \left (5-\frac {4}{x}\right )}d\sqrt {5-\frac {4}{x}}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle -\int \left (-\frac {x}{10}+\frac {1}{3 \left (2-\frac {4}{x}\right )}+\frac {4}{15 \left (5-\frac {4}{x}\right )}\right )d\sqrt {5-\frac {4}{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {5-\frac {4}{x}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {5-\frac {4}{x}}}{\sqrt {5}}\right )}{5 \sqrt {5}}+\frac {4}{15 \sqrt {5-\frac {4}{x}}}\) |
Input:
Int[((1 - x)*(4 - x))/((5 - 4/x)^(3/2)*(-2 + x)*x^2),x]
Output:
4/(15*Sqrt[5 - 4/x]) + ArcTanh[Sqrt[5 - 4/x]/Sqrt[3]]/(3*Sqrt[3]) + (2*Arc Tanh[Sqrt[5 - 4/x]/Sqrt[5]])/(5*Sqrt[5])
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.64
| method | result | size |
| trager | \(\frac {4 x \sqrt {-\frac {-5 x +4}{x}}}{15 \left (5 x -4\right )}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \sqrt {-\frac {-5 x +4}{x}}\, x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{25}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {3 \sqrt {-\frac {-5 x +4}{x}}\, x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x -2}\right )}{18}\) | \(115\) |
| default | \(\frac {\sqrt {\frac {5 x -4}{x}}\, x \left (450 \ln \left (\sqrt {5}\, x -\frac {2 \sqrt {5}}{5}+\sqrt {5 x^{2}-4 x}\right ) \sqrt {5}\, x^{2}+625 \,\operatorname {arctanh}\left (\frac {2 \left (2 x -1\right ) \sqrt {3}}{3 \sqrt {5 x^{2}-4 x}}\right ) \sqrt {3}\, x^{2}-720 \ln \left (\sqrt {5}\, x -\frac {2 \sqrt {5}}{5}+\sqrt {5 x^{2}-4 x}\right ) \sqrt {5}\, x -1000 \,\operatorname {arctanh}\left (\frac {2 \left (2 x -1\right ) \sqrt {3}}{3 \sqrt {5 x^{2}-4 x}}\right ) \sqrt {3}\, x +150 \left (5 x^{2}-4 x \right )^{\frac {3}{2}}-750 \sqrt {5 x^{2}-4 x}\, x^{2}+288 \ln \left (\sqrt {5}\, x -\frac {2 \sqrt {5}}{5}+\sqrt {5 x^{2}-4 x}\right ) \sqrt {5}+400 \sqrt {3}\, \operatorname {arctanh}\left (\frac {2 \left (2 x -1\right ) \sqrt {3}}{3 \sqrt {5 x^{2}-4 x}}\right )+1200 \sqrt {5 x^{2}-4 x}\, x -480 \sqrt {5 x^{2}-4 x}\right )}{450 \sqrt {\left (5 x -4\right ) x}\, \left (5 x -4\right )^{2}}\) | \(261\) |
Input:
int((1-x)*(4-x)/(5-4/x)^(3/2)/(x-2)/x^2,x,method=_RETURNVERBOSE)
Output:
4/15*x/(5*x-4)*(-(-5*x+4)/x)^(1/2)+1/25*RootOf(_Z^2-5)*ln(5*(-(-5*x+4)/x)^ (1/2)*x+5*RootOf(_Z^2-5)*x-2*RootOf(_Z^2-5))+1/18*RootOf(_Z^2-3)*ln(-(3*(- (-5*x+4)/x)^(1/2)*x+4*RootOf(_Z^2-3)*x-2*RootOf(_Z^2-3))/(x-2))
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.36 \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\frac {18 \, \sqrt {5} {\left (5 \, x - 4\right )} \log \left (-\sqrt {5} x \sqrt {\frac {5 \, x - 4}{x}} - 5 \, x + 2\right ) + 25 \, \sqrt {3} {\left (5 \, x - 4\right )} \log \left (\frac {\sqrt {3} x \sqrt {\frac {5 \, x - 4}{x}} + 4 \, x - 2}{x - 2}\right ) + 120 \, x \sqrt {\frac {5 \, x - 4}{x}}}{450 \, {\left (5 \, x - 4\right )}} \] Input:
integrate((1-x)*(4-x)/(5-4/x)^(3/2)/(-2+x)/x^2,x, algorithm="fricas")
Output:
1/450*(18*sqrt(5)*(5*x - 4)*log(-sqrt(5)*x*sqrt((5*x - 4)/x) - 5*x + 2) + 25*sqrt(3)*(5*x - 4)*log((sqrt(3)*x*sqrt((5*x - 4)/x) + 4*x - 2)/(x - 2)) + 120*x*sqrt((5*x - 4)/x))/(5*x - 4)
\[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\int \frac {\left (x - 4\right ) \left (x - 1\right )}{x^{2} \left (5 - \frac {4}{x}\right )^{\frac {3}{2}} \left (x - 2\right )}\, dx \] Input:
integrate((1-x)*(4-x)/(5-4/x)**(3/2)/(-2+x)/x**2,x)
Output:
Integral((x - 4)*(x - 1)/(x**2*(5 - 4/x)**(3/2)*(x - 2)), x)
\[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\int { \frac {{\left (x - 1\right )} {\left (x - 4\right )}}{{\left (x - 2\right )} x^{2} {\left (-\frac {4}{x} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((1-x)*(4-x)/(5-4/x)^(3/2)/(-2+x)/x^2,x, algorithm="maxima")
Output:
integrate((x - 1)*(x - 4)/((x - 2)*x^2*(-4/x + 5)^(3/2)), x)
Exception generated. \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1-x)*(4-x)/(5-4/x)^(3/2)/(-2+x)/x^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{480,[2,1]%%%}+%%%{%%{[768,0]:[1,0,-5]%%},[1,1]%%%}+%%%{153 6,[0,1]%%
Timed out. \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\int \frac {\left (x-1\right )\,\left (x-4\right )}{x^2\,{\left (5-\frac {4}{x}\right )}^{3/2}\,\left (x-2\right )} \,d x \] Input:
int(((x - 1)*(x - 4))/(x^2*(5 - 4/x)^(3/2)*(x - 2)),x)
Output:
int(((x - 1)*(x - 4))/(x^2*(5 - 4/x)^(3/2)*(x - 2)), x)
Time = 0.18 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.13 \[ \int \frac {(1-x) (4-x)}{\left (5-\frac {4}{x}\right )^{3/2} (-2+x) x^2} \, dx=\frac {36 \sqrt {5 x -4}\, \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x -4}}{2}+\frac {\sqrt {x}\, \sqrt {5}}{2}\right )+24 \sqrt {5 x -4}\, \sqrt {5}-25 \sqrt {5 x -4}\, \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {5 x -4}}{2}+\frac {\sqrt {x}\, \sqrt {5}}{2}-\frac {\sqrt {10}}{2}-\frac {\sqrt {6}}{2}\right )-25 \sqrt {5 x -4}\, \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {5 x -4}}{2}+\frac {\sqrt {x}\, \sqrt {5}}{2}+\frac {\sqrt {10}}{2}+\frac {\sqrt {6}}{2}\right )+25 \sqrt {5 x -4}\, \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {x}\, \sqrt {5 x -4}\, \sqrt {5}}{2}+\sqrt {15}+\frac {5 x}{2}-5\right )+120 \sqrt {x}}{450 \sqrt {5 x -4}} \] Input:
int((1-x)*(4-x)/(5-4/x)^(3/2)/(-2+x)/x^2,x)
Output:
(36*sqrt(5*x - 4)*sqrt(5)*log((sqrt(5*x - 4) + sqrt(x)*sqrt(5))/2) + 24*sq rt(5*x - 4)*sqrt(5) - 25*sqrt(5*x - 4)*sqrt(3)*log((sqrt(5*x - 4) + sqrt(x )*sqrt(5) - sqrt(10) - sqrt(6))/2) - 25*sqrt(5*x - 4)*sqrt(3)*log((sqrt(5* x - 4) + sqrt(x)*sqrt(5) + sqrt(10) + sqrt(6))/2) + 25*sqrt(5*x - 4)*sqrt( 3)*log((sqrt(x)*sqrt(5*x - 4)*sqrt(5) + 2*sqrt(15) + 5*x - 10)/2) + 120*sq rt(x))/(450*sqrt(5*x - 4))