Integrand size = 16, antiderivative size = 81 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=-\frac {1}{490 x^2}-\frac {69}{1715 x}-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}-\frac {234 \arctan \left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}}+\frac {80 \log (x)}{2401}-\frac {40 \log \left (7-6 x+2 x^2\right )}{2401} \] Output:
-1/490/x^2-69/1715/x+1/35*(-2+3*x)/x^2/(2*x^2-6*x+7)+80/2401*ln(x)-40/2401 *ln(2*x^2-6*x+7)-234/60025*arctan(1/5*(3-2*x)*5^(1/2))*5^(1/2)
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {-\frac {1225}{x^2}-\frac {4200}{x}-\frac {140 (-41+9 x)}{7-6 x+2 x^2}+468 \sqrt {5} \arctan \left (\frac {-3+2 x}{\sqrt {5}}\right )+4000 \log (x)-2000 \log \left (7-6 x+2 x^2\right )}{120050} \] Input:
Integrate[1/(x^3*(7 - 6*x + 2*x^2)^2),x]
Output:
(-1225/x^2 - 4200/x - (140*(-41 + 9*x))/(7 - 6*x + 2*x^2) + 468*Sqrt[5]*Ar cTan[(-3 + 2*x)/Sqrt[5]] + 4000*Log[x] - 2000*Log[7 - 6*x + 2*x^2])/120050
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1165, 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.
\(\displaystyle \int \frac {1}{x^3 \left (2 x^2-6 x+7\right )^2} \, dx\) |
\(\Big \downarrow \) 1165 |
\(\displaystyle \frac {1}{140} \int \frac {4 (9 x+1)}{x^3 \left (2 x^2-6 x+7\right )}dx-\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{35} \int \frac {9 x+1}{x^3 \left (2 x^2-6 x+7\right )}dx-\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{35} \int \left (-\frac {2 (400 x-717)}{343 \left (2 x^2-6 x+7\right )}+\frac {400}{343 x}+\frac {69}{49 x^2}+\frac {1}{7 x^3}\right )dx-\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{35} \left (-\frac {234 \arctan \left (\frac {3-2 x}{\sqrt {5}}\right )}{343 \sqrt {5}}-\frac {1}{14 x^2}-\frac {200}{343} \log \left (2 x^2-6 x+7\right )-\frac {69}{49 x}+\frac {400 \log (x)}{343}\right )-\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}\) |
Input:
Int[1/(x^3*(7 - 6*x + 2*x^2)^2),x]
Output:
-1/35*(2 - 3*x)/(x^2*(7 - 6*x + 2*x^2)) + (-1/14*1/x^2 - 69/(49*x) - (234* ArcTan[(3 - 2*x)/Sqrt[5]])/(343*Sqrt[5]) + (400*Log[x])/343 - (200*Log[7 - 6*x + 2*x^2])/343)/35
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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, 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]
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {1}{98 x^{2}}-\frac {12}{343 x}+\frac {80 \ln \left (x \right )}{2401}-\frac {4 \left (\frac {63 x}{20}-\frac {287}{20}\right )}{2401 \left (x^{2}-3 x +\frac {7}{2}\right )}-\frac {40 \ln \left (2 x^{2}-6 x +7\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (4 x -6\right ) \sqrt {5}}{10}\right )}{60025}\) | \(62\) |
risch | \(\frac {-\frac {138}{1715} x^{3}+\frac {407}{1715} x^{2}-\frac {9}{49} x -\frac {1}{14}}{x^{2} \left (2 x^{2}-6 x +7\right )}-\frac {40 \ln \left (4 x^{2}-12 x +14\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (2 x -3\right ) \sqrt {5}}{5}\right )}{60025}+\frac {80 \ln \left (x \right )}{2401}\) | \(67\) |
Input:
int(1/x^3/(2*x^2-6*x+7)^2,x,method=_RETURNVERBOSE)
Output:
-1/98/x^2-12/343/x+80/2401*ln(x)-4/2401*(63/20*x-287/20)/(x^2-3*x+7/2)-40/ 2401*ln(2*x^2-6*x+7)+234/60025*5^(1/2)*arctan(1/10*(4*x-6)*5^(1/2))
Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=-\frac {9660 \, x^{3} - 468 \, \sqrt {5} {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - 28490 \, x^{2} + 2000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \left (2 \, x^{2} - 6 \, x + 7\right ) - 4000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \left (x\right ) + 22050 \, x + 8575}{120050 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} \] Input:
integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="fricas")
Output:
-1/120050*(9660*x^3 - 468*sqrt(5)*(2*x^4 - 6*x^3 + 7*x^2)*arctan(1/5*sqrt( 5)*(2*x - 3)) - 28490*x^2 + 2000*(2*x^4 - 6*x^3 + 7*x^2)*log(2*x^2 - 6*x + 7) - 4000*(2*x^4 - 6*x^3 + 7*x^2)*log(x) + 22050*x + 8575)/(2*x^4 - 6*x^3 + 7*x^2)
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {80 \log {\left (x \right )}}{2401} - \frac {40 \log {\left (x^{2} - 3 x + \frac {7}{2} \right )}}{2401} + \frac {234 \sqrt {5} \operatorname {atan}{\left (\frac {2 \sqrt {5} x}{5} - \frac {3 \sqrt {5}}{5} \right )}}{60025} + \frac {- 276 x^{3} + 814 x^{2} - 630 x - 245}{6860 x^{4} - 20580 x^{3} + 24010 x^{2}} \] Input:
integrate(1/x**3/(2*x**2-6*x+7)**2,x)
Output:
80*log(x)/2401 - 40*log(x**2 - 3*x + 7/2)/2401 + 234*sqrt(5)*atan(2*sqrt(5 )*x/5 - 3*sqrt(5)/5)/60025 + (-276*x**3 + 814*x**2 - 630*x - 245)/(6860*x* *4 - 20580*x**3 + 24010*x**2)
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \left (x\right ) \] Input:
integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="maxima")
Output:
234/60025*sqrt(5)*arctan(1/5*sqrt(5)*(2*x - 3)) - 1/3430*(276*x^3 - 814*x^ 2 + 630*x + 245)/(2*x^4 - 6*x^3 + 7*x^2) - 40/2401*log(2*x^2 - 6*x + 7) + 80/2401*log(x)
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{2} - 6 \, x + 7\right )} x^{2}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="giac")
Output:
234/60025*sqrt(5)*arctan(1/5*sqrt(5)*(2*x - 3)) - 1/3430*(276*x^3 - 814*x^ 2 + 630*x + 245)/((2*x^2 - 6*x + 7)*x^2) - 40/2401*log(2*x^2 - 6*x + 7) + 80/2401*log(abs(x))
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {80\,\ln \left (x\right )}{2401}-\frac {\frac {69\,x^3}{1715}-\frac {407\,x^2}{3430}+\frac {9\,x}{98}+\frac {1}{28}}{x^4-3\,x^3+\frac {7\,x^2}{2}}-\ln \left (x-\frac {3}{2}-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right )+\ln \left (x-\frac {3}{2}+\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right ) \] Input:
int(1/(x^3*(2*x^2 - 6*x + 7)^2),x)
Output:
(80*log(x))/2401 - ((9*x)/98 - (407*x^2)/3430 + (69*x^3)/1715 + 1/28)/((7* x^2)/2 - 3*x^3 + x^4) - log(x - (5^(1/2)*1i)/2 - 3/2)*((5^(1/2)*117i)/6002 5 + 40/2401) + log(x + (5^(1/2)*1i)/2 - 3/2)*((5^(1/2)*117i)/60025 - 40/24 01)
Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.91 \[ \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx=\frac {936 \sqrt {5}\, \mathit {atan} \left (\frac {2 x -3}{\sqrt {5}}\right ) x^{4}-2808 \sqrt {5}\, \mathit {atan} \left (\frac {2 x -3}{\sqrt {5}}\right ) x^{3}+3276 \sqrt {5}\, \mathit {atan} \left (\frac {2 x -3}{\sqrt {5}}\right ) x^{2}-4000 \,\mathrm {log}\left (2 x^{2}-6 x +7\right ) x^{4}+12000 \,\mathrm {log}\left (2 x^{2}-6 x +7\right ) x^{3}-14000 \,\mathrm {log}\left (2 x^{2}-6 x +7\right ) x^{2}+8000 \,\mathrm {log}\left (x \right ) x^{4}-24000 \,\mathrm {log}\left (x \right ) x^{3}+28000 \,\mathrm {log}\left (x \right ) x^{2}-3220 x^{4}+17220 x^{2}-22050 x -8575}{120050 x^{2} \left (2 x^{2}-6 x +7\right )} \] Input:
int(1/x^3/(2*x^2-6*x+7)^2,x)
Output:
(936*sqrt(5)*atan((2*x - 3)/sqrt(5))*x**4 - 2808*sqrt(5)*atan((2*x - 3)/sq rt(5))*x**3 + 3276*sqrt(5)*atan((2*x - 3)/sqrt(5))*x**2 - 4000*log(2*x**2 - 6*x + 7)*x**4 + 12000*log(2*x**2 - 6*x + 7)*x**3 - 14000*log(2*x**2 - 6* x + 7)*x**2 + 8000*log(x)*x**4 - 24000*log(x)*x**3 + 28000*log(x)*x**2 - 3 220*x**4 + 17220*x**2 - 22050*x - 8575)/(120050*x**2*(2*x**2 - 6*x + 7))