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

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

[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

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Rubi [A]  time = 0.0561464, antiderivative size = 81, normalized size of antiderivative = 1., 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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 0.0348387, 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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Maple [A]  time = 0.013, size = 62, normalized size = 0.8 \begin{align*} -{\frac{1}{98\,{x}^{2}}}-{\frac{12}{343\,x}}+{\frac{80\,\ln \left ( x \right ) }{2401}}-{\frac{4}{2401} \left ({\frac{63\,x}{20}}-{\frac{287}{20}} \right ) \left ({x}^{2}-3\,x+{\frac{7}{2}} \right ) ^{-1}}-{\frac{40\,\ln \left ( 2\,{x}^{2}-6\,x+7 \right ) }{2401}}+{\frac{234\,\sqrt{5}}{60025}\arctan \left ({\frac{ \left ( 4\,x-6 \right ) \sqrt{5}}{10}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Maxima [A]  time = 1.41005, size = 93, normalized size = 1.15 \begin{align*} \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 ) \end{align*}

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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Fricas [A]  time = 1.7611, size = 315, normalized size = 3.89 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 0.202163, size = 80, normalized size = 0.99 \begin{align*} \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}} \end{align*}

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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Giac [A]  time = 1.07603, size = 90, normalized size = 1.11 \begin{align*} \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 ) \end{align*}

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))