Integrand size = 19, antiderivative size = 90 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=-\frac {1}{288 (1-x)^2}-\frac {7}{864 (1-x)}+\frac {975+2204 x}{39744 \left (3+5 x+4 x^2\right )}+\frac {6023 \arctan \left (\frac {5+8 x}{\sqrt {23}}\right )}{52992 \sqrt {23}}+\frac {11 \log (1-x)}{2304}-\frac {11 \log \left (3+5 x+4 x^2\right )}{4608} \] Output:
-1/288/(1-x)^2-7/(864-864*x)+(975+2204*x)/(158976*x^2+198720*x+119232)+602 3/1218816*arctan(1/23*(5+8*x)*23^(1/2))*23^(1/2)+11/2304*ln(1-x)-11/4608*l n(4*x^2+5*x+3)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.87 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {-\frac {25392}{(-1+x)^2}+\frac {59248}{-1+x}+\frac {184 (975+2204 x)}{3+5 x+4 x^2}+36138 \sqrt {23} \arctan \left (\frac {5+8 x}{\sqrt {23}}\right )+34914 \log (1-x)-17457 \log \left (3+5 x+4 x^2\right )}{7312896} \] Input:
Integrate[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]
Output:
(-25392/(-1 + x)^2 + 59248/(-1 + x) + (184*(975 + 2204*x))/(3 + 5*x + 4*x^ 2) + 36138*Sqrt[23]*ArcTan[(5 + 8*x)/Sqrt[23]] + 34914*Log[1 - x] - 17457* Log[3 + 5*x + 4*x^2])/7312896
Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1235, 27, 1200, 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.
\(\displaystyle \int \frac {x}{(x-1)^3 \left (4 x^2+5 x+3\right )^2} \, dx\) |
\(\Big \downarrow \) 1235 |
\(\displaystyle \frac {1}{276} \int -\frac {3 (44 x+19)}{(1-x)^3 \left (4 x^2+5 x+3\right )}dx+\frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac {1}{92} \int \frac {44 x+19}{(1-x)^3 \left (4 x^2+5 x+3\right )}dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac {1}{92} \int \left (\frac {1012 x-2379}{576 \left (4 x^2+5 x+3\right )}-\frac {253}{576 (x-1)}+\frac {97}{48 (x-1)^2}-\frac {21}{4 (x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{92} \left (\frac {6023 \arctan \left (\frac {8 x+5}{\sqrt {23}}\right )}{576 \sqrt {23}}-\frac {253 \log \left (4 x^2+5 x+3\right )}{1152}-\frac {97}{48 (1-x)}-\frac {21}{8 (1-x)^2}+\frac {253}{576} \log (1-x)\right )+\frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}\) |
Input:
Int[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]
Output:
(39 + 44*x)/(276*(1 - x)^2*(3 + 5*x + 4*x^2)) + (-21/(8*(1 - x)^2) - 97/(4 8*(1 - x)) + (6023*ArcTan[(5 + 8*x)/Sqrt[23]])/(576*Sqrt[23]) + (253*Log[1 - x])/576 - (253*Log[3 + 5*x + 4*x^2])/1152)/92
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Time = 2.59 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {-\frac {2204 x}{23}-\frac {975}{23}}{6912 \left (x^{2}+\frac {5}{4} x +\frac {3}{4}\right )}-\frac {11 \ln \left (4 x^{2}+5 x +3\right )}{4608}+\frac {6023 \arctan \left (\frac {\left (5+8 x \right ) \sqrt {23}}{23}\right ) \sqrt {23}}{1218816}-\frac {1}{288 \left (x -1\right )^{2}}+\frac {7}{864 \left (x -1\right )}+\frac {11 \ln \left (x -1\right )}{2304}\) | \(68\) |
risch | \(\frac {\frac {97}{1104} x^{3}-\frac {407}{4416} x^{2}-\frac {5}{184} x -\frac {15}{1472}}{\left (x -1\right )^{2} \left (4 x^{2}+5 x +3\right )}-\frac {11 \ln \left (64 x^{2}+80 x +48\right )}{4608}+\frac {6023 \arctan \left (\frac {\left (5+8 x \right ) \sqrt {23}}{23}\right ) \sqrt {23}}{1218816}+\frac {11 \ln \left (x -1\right )}{2304}\) | \(71\) |
Input:
int(x/(x-1)^3/(4*x^2+5*x+3)^2,x,method=_RETURNVERBOSE)
Output:
-1/6912*(-2204/23*x-975/23)/(x^2+5/4*x+3/4)-11/4608*ln(4*x^2+5*x+3)+6023/1 218816*arctan(1/23*(5+8*x)*23^(1/2))*23^(1/2)-1/288/(x-1)^2+7/864/(x-1)+11 /2304*ln(x-1)
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {214176 \, x^{3} + 12046 \, \sqrt {23} {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) - 224664 \, x^{2} - 5819 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (4 \, x^{2} + 5 \, x + 3\right ) + 11638 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (x - 1\right ) - 66240 \, x - 24840}{2437632 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \] Input:
integrate(x/(x-1)^3/(4*x^2+5*x+3)^2,x, algorithm="fricas")
Output:
1/2437632*(214176*x^3 + 12046*sqrt(23)*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*arc tan(1/23*sqrt(23)*(8*x + 5)) - 224664*x^2 - 5819*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*log(4*x^2 + 5*x + 3) + 11638*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*log(x - 1) - 66240*x - 24840)/(4*x^4 - 3*x^3 - 3*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {388 x^{3} - 407 x^{2} - 120 x - 45}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac {11 \log {\left (x - 1 \right )}}{2304} - \frac {11 \log {\left (x^{2} + \frac {5 x}{4} + \frac {3}{4} \right )}}{4608} + \frac {6023 \sqrt {23} \operatorname {atan}{\left (\frac {8 \sqrt {23} x}{23} + \frac {5 \sqrt {23}}{23} \right )}}{1218816} \] Input:
integrate(x/(x-1)**3/(4*x**2+5*x+3)**2,x)
Output:
(388*x**3 - 407*x**2 - 120*x - 45)/(17664*x**4 - 13248*x**3 - 13248*x**2 - 4416*x + 13248) + 11*log(x - 1)/2304 - 11*log(x**2 + 5*x/4 + 3/4)/4608 + 6023*sqrt(23)*atan(8*sqrt(23)*x/23 + 5*sqrt(23)/23)/1218816
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.83 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {6023}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac {11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {11}{2304} \, \log \left (x - 1\right ) \] Input:
integrate(x/(x-1)^3/(4*x^2+5*x+3)^2,x, algorithm="maxima")
Output:
6023/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 1/4416*(388*x^3 - 407*x^2 - 120*x - 45)/(4*x^4 - 3*x^3 - 3*x^2 - x + 3) - 11/4608*log(4*x^2 + 5*x + 3) + 11/2304*log(x - 1)
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.79 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {6023}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \, {\left (4 \, x^{2} + 5 \, x + 3\right )} {\left (x - 1\right )}^{2}} - \frac {11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {11}{2304} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:
integrate(x/(x-1)^3/(4*x^2+5*x+3)^2,x, algorithm="giac")
Output:
6023/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 1/4416*(388*x^3 - 407*x^2 - 120*x - 45)/((4*x^2 + 5*x + 3)*(x - 1)^2) - 11/4608*log(4*x^2 + 5*x + 3) + 11/2304*log(abs(x - 1))
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {11\,\ln \left (x-1\right )}{2304}+\frac {-\frac {97\,x^3}{4416}+\frac {407\,x^2}{17664}+\frac {5\,x}{736}+\frac {15}{5888}}{-x^4+\frac {3\,x^3}{4}+\frac {3\,x^2}{4}+\frac {x}{4}-\frac {3}{4}}-\ln \left (x+\frac {5}{8}-\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {11}{4608}+\frac {\sqrt {23}\,6023{}\mathrm {i}}{2437632}\right )+\ln \left (x+\frac {5}{8}+\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {11}{4608}+\frac {\sqrt {23}\,6023{}\mathrm {i}}{2437632}\right ) \] Input:
int(x/((x - 1)^3*(5*x + 4*x^2 + 3)^2),x)
Output:
(11*log(x - 1))/2304 + ((5*x)/736 + (407*x^2)/17664 - (97*x^3)/4416 + 15/5 888)/(x/4 + (3*x^2)/4 + (3*x^3)/4 - x^4 - 3/4) - log(x - (23^(1/2)*1i)/8 + 5/8)*((23^(1/2)*6023i)/2437632 + 11/4608) + log(x + (23^(1/2)*1i)/8 + 5/8 )*((23^(1/2)*6023i)/2437632 - 11/4608)
Time = 0.30 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.64 \[ \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {48184 \sqrt {23}\, \mathit {atan} \left (\frac {8 x +5}{\sqrt {23}}\right ) x^{4}-36138 \sqrt {23}\, \mathit {atan} \left (\frac {8 x +5}{\sqrt {23}}\right ) x^{3}-36138 \sqrt {23}\, \mathit {atan} \left (\frac {8 x +5}{\sqrt {23}}\right ) x^{2}-12046 \sqrt {23}\, \mathit {atan} \left (\frac {8 x +5}{\sqrt {23}}\right ) x +36138 \sqrt {23}\, \mathit {atan} \left (\frac {8 x +5}{\sqrt {23}}\right )-23276 \,\mathrm {log}\left (4 x^{2}+5 x +3\right ) x^{4}+17457 \,\mathrm {log}\left (4 x^{2}+5 x +3\right ) x^{3}+17457 \,\mathrm {log}\left (4 x^{2}+5 x +3\right ) x^{2}+5819 \,\mathrm {log}\left (4 x^{2}+5 x +3\right ) x -17457 \,\mathrm {log}\left (4 x^{2}+5 x +3\right )+46552 \,\mathrm {log}\left (x -1\right ) x^{4}-34914 \,\mathrm {log}\left (x -1\right ) x^{3}-34914 \,\mathrm {log}\left (x -1\right ) x^{2}-11638 \,\mathrm {log}\left (x -1\right ) x +34914 \,\mathrm {log}\left (x -1\right )+285568 x^{4}-438840 x^{2}-137632 x +189336}{9750528 x^{4}-7312896 x^{3}-7312896 x^{2}-2437632 x +7312896} \] Input:
int(x/(x-1)^3/(4*x^2+5*x+3)^2,x)
Output:
(48184*sqrt(23)*atan((8*x + 5)/sqrt(23))*x**4 - 36138*sqrt(23)*atan((8*x + 5)/sqrt(23))*x**3 - 36138*sqrt(23)*atan((8*x + 5)/sqrt(23))*x**2 - 12046* sqrt(23)*atan((8*x + 5)/sqrt(23))*x + 36138*sqrt(23)*atan((8*x + 5)/sqrt(2 3)) - 23276*log(4*x**2 + 5*x + 3)*x**4 + 17457*log(4*x**2 + 5*x + 3)*x**3 + 17457*log(4*x**2 + 5*x + 3)*x**2 + 5819*log(4*x**2 + 5*x + 3)*x - 17457* log(4*x**2 + 5*x + 3) + 46552*log(x - 1)*x**4 - 34914*log(x - 1)*x**3 - 34 914*log(x - 1)*x**2 - 11638*log(x - 1)*x + 34914*log(x - 1) + 285568*x**4 - 438840*x**2 - 137632*x + 189336)/(2437632*(4*x**4 - 3*x**3 - 3*x**2 - x + 3))