3.203 \(\int \frac {1}{x^3 (7-6 x+2 x^2)^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}-\frac {1}{490 x^2}-\frac {40 \log \left (2 x^2-6 x+7\right )}{2401}-\frac {69}{1715 x}+\frac {80 \log (x)}{2401}-\frac {234 \tan ^{-1}\left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}} \]

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Rubi [A]  time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {740, 800, 634, 618, 204, 628} \[ -\frac {2-3 x}{35 x^2 \left (2 x^2-6 x+7\right )}-\frac {1}{490 x^2}-\frac {40 \log \left (2 x^2-6 x+7\right )}{2401}-\frac {69}{1715 x}+\frac {80 \log (x)}{2401}-\frac {234 \tan ^{-1}\left (\frac {3-2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(7 - 6*x + 2*x^2)^2),x]

[Out]

-1/(490*x^2) - 69/(1715*x) - (2 - 3*x)/(35*x^2*(7 - 6*x + 2*x^2)) - (234*ArcTan[(3 - 2*x)/Sqrt[5]])/(12005*Sqr
t[5]) + (80*Log[x])/2401 - (40*Log[7 - 6*x + 2*x^2])/2401

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (7-6 x+2 x^2\right )^2} \, dx &=-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {1}{140} \int \frac {4+36 x}{x^3 \left (7-6 x+2 x^2\right )} \, dx\\ &=-\frac {2-3 x}{35 x^2 \left (7-6 x+2 x^2\right )}+\frac {1}{140} \int \left (\frac {4}{7 x^3}+\frac {276}{49 x^2}+\frac {1600}{343 x}-\frac {8 (-717+400 x)}{343 \left (7-6 x+2 x^2\right )}\right ) \, 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 {80 \log (x)}{2401}-\frac {2 \int \frac {-717+400 x}{7-6 x+2 x^2} \, dx}{12005}\\ &=-\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 {80 \log (x)}{2401}-\frac {40 \int \frac {-6+4 x}{7-6 x+2 x^2} \, dx}{2401}+\frac {234 \int \frac {1}{7-6 x+2 x^2} \, dx}{12005}\\ &=-\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 {80 \log (x)}{2401}-\frac {40 \log \left (7-6 x+2 x^2\right )}{2401}-\frac {468 \operatorname {Subst}\left (\int \frac {1}{-20-x^2} \, dx,x,-6+4 x\right )}{12005}\\ &=-\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 \tan ^{-1}\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}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 70, normalized size = 0.86 \[ \frac {-\frac {140 (9 x-41)}{2 x^2-6 x+7}-\frac {1225}{x^2}-2000 \log \left (2 x^2-6 x+7\right )-\frac {4200}{x}+4000 \log (x)+468 \sqrt {5} \tan ^{-1}\left (\frac {2 x-3}{\sqrt {5}}\right )}{120050} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(7 - 6*x + 2*x^2)^2),x]

[Out]

(-1225/x^2 - 4200/x - (140*(-41 + 9*x))/(7 - 6*x + 2*x^2) + 468*Sqrt[5]*ArcTan[(-3 + 2*x)/Sqrt[5]] + 4000*Log[
x] - 2000*Log[7 - 6*x + 2*x^2])/120050

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IntegrateAlgebraic [A]  time = 0.05, size = 82, normalized size = 1.01 \[ -\frac {40 \log \left (2 x^2-6 x+7\right )}{2401}+\frac {-276 x^3+814 x^2-630 x-245}{3430 x^2 \left (2 x^2-6 x+7\right )}+\frac {80 \log (x)}{2401}-\frac {234 \tan ^{-1}\left (\frac {3}{\sqrt {5}}-\frac {2 x}{\sqrt {5}}\right )}{12005 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(7 - 6*x + 2*x^2)^2),x]

[Out]

(-245 - 630*x + 814*x^2 - 276*x^3)/(3430*x^2*(7 - 6*x + 2*x^2)) - (234*ArcTan[3/Sqrt[5] - (2*x)/Sqrt[5]])/(120
05*Sqrt[5]) + (80*Log[x])/2401 - (40*Log[7 - 6*x + 2*x^2])/2401

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fricas [A]  time = 0.92, size = 116, normalized size = 1.43 \[ -\frac {9660 \, x^{3} - 468 \, \sqrt {5} {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - 28490 \, x^{2} + 2000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \left (2 \, x^{2} - 6 \, x + 7\right ) - 4000 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )} \log \relax (x) + 22050 \, x + 8575}{120050 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="fricas")

[Out]

-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)

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giac [A]  time = 1.16, size = 67, normalized size = 0.83 \[ \frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{2} - 6 \, x + 7\right )} x^{2}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="giac")

[Out]

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))

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maple [A]  time = 0.43, size = 62, normalized size = 0.77

method	result	size
default	\(-\frac {4 \left (\frac {63 x}{20}-\frac {287}{20}\right )}{2401 \left (x^{2}-3 x +\frac {7}{2}\right )}-\frac {40 \ln \left (2 x^{2}-6 x +7\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (4 x -6\right ) \sqrt {5}}{10}\right )}{60025}-\frac {1}{98 x^{2}}-\frac {12}{343 x}+\frac {80 \ln \relax (x )}{2401}\)	\(62\)
risch	\(\frac {-\frac {138}{1715} x^{3}+\frac {407}{1715} x^{2}-\frac {9}{49} x -\frac {1}{14}}{x^{2} \left (2 x^{2}-6 x +7\right )}-\frac {40 \ln \left (4 x^{2}-12 x +14\right )}{2401}+\frac {234 \sqrt {5}\, \arctan \left (\frac {\left (-3+2 x \right ) \sqrt {5}}{5}\right )}{60025}+\frac {80 \ln \relax (x )}{2401}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(2*x^2-6*x+7)^2,x,method=_RETURNVERBOSE)

[Out]

-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))-
1/98/x^2-12/343/x+80/2401*ln(x)

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maxima [A]  time = 1.31, size = 69, normalized size = 0.85 \[ \frac {234}{60025} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, x - 3\right )}\right ) - \frac {276 \, x^{3} - 814 \, x^{2} + 630 \, x + 245}{3430 \, {\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2}\right )}} - \frac {40}{2401} \, \log \left (2 \, x^{2} - 6 \, x + 7\right ) + \frac {80}{2401} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(2*x^2-6*x+7)^2,x, algorithm="maxima")

[Out]

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)

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mupad [B]  time = 0.11, size = 77, normalized size = 0.95 \[ \frac {80\,\ln \relax (x)}{2401}-\frac {\frac {69\,x^3}{1715}-\frac {407\,x^2}{3430}+\frac {9\,x}{98}+\frac {1}{28}}{x^4-3\,x^3+\frac {7\,x^2}{2}}-\ln \left (x-\frac {3}{2}-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right )+\ln \left (x-\frac {3}{2}+\frac {\sqrt {5}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {40}{2401}+\frac {\sqrt {5}\,117{}\mathrm {i}}{60025}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(2*x^2 - 6*x + 7)^2),x)

[Out]

(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)/60025 + 40/2401) + log(x + (5^(1/2)*1i)/2 - 3/2)*((5^(1/2)*117i)/60025 - 40/
2401)

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sympy [A]  time = 0.23, size = 80, normalized size = 0.99 \[ \frac {80 \log {\relax (x )}}{2401} - \frac {40 \log {\left (x^{2} - 3 x + \frac {7}{2} \right )}}{2401} + \frac {234 \sqrt {5} \operatorname {atan}{\left (\frac {2 \sqrt {5} x}{5} - \frac {3 \sqrt {5}}{5} \right )}}{60025} + \frac {- 276 x^{3} + 814 x^{2} - 630 x - 245}{6860 x^{4} - 20580 x^{3} + 24010 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(2*x**2-6*x+7)**2,x)

[Out]

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)

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