3.4.33 \(\int \frac {x}{(-1+x)^3 (3+5 x+4 x^2)^2} \, dx\)

Optimal. Leaf size=97 \[ \frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac {11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac {97}{4416 (1-x)}-\frac {21}{736 (1-x)^2}+\frac {11 \log (1-x)}{2304}+\frac {6023 \tan ^{-1}\left (\frac {8 x+5}{\sqrt {23}}\right )}{52992 \sqrt {23}} \]

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Rubi [A]  time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {822, 800, 634, 618, 204, 628} \begin {gather*} \frac {44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac {11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac {97}{4416 (1-x)}-\frac {21}{736 (1-x)^2}+\frac {11 \log (1-x)}{2304}+\frac {6023 \tan ^{-1}\left (\frac {8 x+5}{\sqrt {23}}\right )}{52992 \sqrt {23}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

-21/(736*(1 - x)^2) - 97/(4416*(1 - x)) + (39 + 44*x)/(276*(1 - x)^2*(3 + 5*x + 4*x^2)) + (6023*ArcTan[(5 + 8*
x)/Sqrt[23]])/(52992*Sqrt[23]) + (11*Log[1 - x])/2304 - (11*Log[3 + 5*x + 4*x^2])/4608

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 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 822

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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx &=\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {1}{276} \int \frac {57+132 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )} \, dx\\ &=\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {1}{276} \int \left (\frac {63}{4 (-1+x)^3}-\frac {97}{16 (-1+x)^2}+\frac {253}{192 (-1+x)}+\frac {2379-1012 x}{192 \left (3+5 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {21}{736 (1-x)^2}-\frac {97}{4416 (1-x)}+\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {11 \log (1-x)}{2304}+\frac {\int \frac {2379-1012 x}{3+5 x+4 x^2} \, dx}{52992}\\ &=-\frac {21}{736 (1-x)^2}-\frac {97}{4416 (1-x)}+\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {11 \log (1-x)}{2304}-\frac {11 \int \frac {5+8 x}{3+5 x+4 x^2} \, dx}{4608}+\frac {6023 \int \frac {1}{3+5 x+4 x^2} \, dx}{105984}\\ &=-\frac {21}{736 (1-x)^2}-\frac {97}{4416 (1-x)}+\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {11 \log (1-x)}{2304}-\frac {11 \log \left (3+5 x+4 x^2\right )}{4608}-\frac {6023 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,5+8 x\right )}{52992}\\ &=-\frac {21}{736 (1-x)^2}-\frac {97}{4416 (1-x)}+\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {6023 \tan ^{-1}\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}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 78, normalized size = 0.80 \begin {gather*} \frac {\frac {184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left (4 x^2+5 x+3\right )+\frac {59248}{x-1}-\frac {25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt {23} \tan ^{-1}\left (\frac {8 x+5}{\sqrt {23}}\right )}{7312896} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

(-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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

IntegrateAlgebraic[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2), x]

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fricas [A]  time = 0.39, size = 134, normalized size = 1.38 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="fricas")

[Out]

1/2437632*(214176*x^3 + 12046*sqrt(23)*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*arctan(1/23*sqrt(23)*(8*x + 5)) - 22466
4*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)

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giac [A]  time = 0.18, size = 71, normalized size = 0.73 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="giac")

[Out]

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))

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maple [A]  time = 0.01, size = 68, normalized size = 0.70 \begin {gather*} \frac {6023 \sqrt {23}\, \arctan \left (\frac {\left (8 x +5\right ) \sqrt {23}}{23}\right )}{1218816}+\frac {11 \ln \left (x -1\right )}{2304}-\frac {11 \ln \left (4 x^{2}+5 x +3\right )}{4608}-\frac {-\frac {2204 x}{23}-\frac {975}{23}}{6912 \left (x^{2}+\frac {5}{4} x +\frac {3}{4}\right )}-\frac {1}{288 \left (x -1\right )^{2}}+\frac {7}{864 \left (x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-1+x)^3/(4*x^2+5*x+3)^2,x)

[Out]

-1/6912*(-2204/23*x-975/23)/(x^2+5/4*x+3/4)-11/4608*ln(4*x^2+5*x+3)+6023/1218816*arctan(1/23*(5+8*x)*23^(1/2))
*23^(1/2)-1/288/(-1+x)^2+7/864/(-1+x)+11/2304*ln(-1+x)

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maxima [A]  time = 0.96, size = 75, normalized size = 0.77 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="maxima")

[Out]

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)

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mupad [B]  time = 0.13, size = 84, normalized size = 0.87 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x - 1)^3*(5*x + 4*x^2 + 3)^2),x)

[Out]

(11*log(x - 1))/2304 + ((5*x)/736 + (407*x^2)/17664 - (97*x^3)/4416 + 15/5888)/(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)

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sympy [A]  time = 0.24, size = 88, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-1+x)**3/(4*x**2+5*x+3)**2,x)

[Out]

(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)/230
4 - 11*log(x**2 + 5*x/4 + 3/4)/4608 + 6023*sqrt(23)*atan(8*sqrt(23)*x/23 + 5*sqrt(23)/23)/1218816

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