### 3.173 $$\int \frac{19 x}{(-1+x)^3 (3+5 x+4 x^2)^2} \, dx$$

Optimal. Leaf size=97 $\frac{19 (44 x+39)}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{209 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{1843}{4416 (1-x)}-\frac{399}{736 (1-x)^2}+\frac{209 \log (1-x)}{2304}+\frac{114437 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}}$

[Out]

-399/(736*(1 - x)^2) - 1843/(4416*(1 - x)) + (19*(39 + 44*x))/(276*(1 - x)^2*(3 + 5*x + 4*x^2)) + (114437*ArcT
an[(5 + 8*x)/Sqrt[23]])/(52992*Sqrt[23]) + (209*Log[1 - x])/2304 - (209*Log[3 + 5*x + 4*x^2])/4608

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Rubi [A]  time = 0.0768091, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.35, Rules used = {12, 822, 800, 634, 618, 204, 628} $\frac{19 (44 x+39)}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{209 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{1843}{4416 (1-x)}-\frac{399}{736 (1-x)^2}+\frac{209 \log (1-x)}{2304}+\frac{114437 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(19*x)/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

-399/(736*(1 - x)^2) - 1843/(4416*(1 - x)) + (19*(39 + 44*x))/(276*(1 - x)^2*(3 + 5*x + 4*x^2)) + (114437*ArcT
an[(5 + 8*x)/Sqrt[23]])/(52992*Sqrt[23]) + (209*Log[1 - x])/2304 - (209*Log[3 + 5*x + 4*x^2])/4608

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

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{19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx &=19 \int \frac{x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx\\ &=\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{19}{276} \int \frac{57+132 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )} \, dx\\ &=\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{19}{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{399}{736 (1-x)^2}-\frac{1843}{4416 (1-x)}+\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{209 \log (1-x)}{2304}+\frac{19 \int \frac{2379-1012 x}{3+5 x+4 x^2} \, dx}{52992}\\ &=-\frac{399}{736 (1-x)^2}-\frac{1843}{4416 (1-x)}+\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{209 \log (1-x)}{2304}-\frac{209 \int \frac{5+8 x}{3+5 x+4 x^2} \, dx}{4608}+\frac{114437 \int \frac{1}{3+5 x+4 x^2} \, dx}{105984}\\ &=-\frac{399}{736 (1-x)^2}-\frac{1843}{4416 (1-x)}+\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{209 \log (1-x)}{2304}-\frac{209 \log \left (3+5 x+4 x^2\right )}{4608}-\frac{114437 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,5+8 x\right )}{52992}\\ &=-\frac{399}{736 (1-x)^2}-\frac{1843}{4416 (1-x)}+\frac{19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{114437 \tan ^{-1}\left (\frac{5+8 x}{\sqrt{23}}\right )}{52992 \sqrt{23}}+\frac{209 \log (1-x)}{2304}-\frac{209 \log \left (3+5 x+4 x^2\right )}{4608}\\ \end{align*}

Mathematica [A]  time = 0.0407205, size = 78, normalized size = 0.8 $\frac{19 \left (\frac{184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left (4 x^2+5 x+3\right )+\frac{59248}{x-1}-\frac{25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt{23} \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )\right )}{7312896}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(19*x)/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

(19*(-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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Maple [A]  time = 0.01, size = 68, normalized size = 0.7 \begin{align*} -{\frac{19}{288\, \left ( -1+x \right ) ^{2}}}+{\frac{133}{-864+864\,x}}+{\frac{209\,\ln \left ( -1+x \right ) }{2304}}-{\frac{19}{6912} \left ( -{\frac{2204\,x}{23}}-{\frac{975}{23}} \right ) \left ({x}^{2}+{\frac{5\,x}{4}}+{\frac{3}{4}} \right ) ^{-1}}-{\frac{209\,\ln \left ( 4\,{x}^{2}+5\,x+3 \right ) }{4608}}+{\frac{114437\,\sqrt{23}}{1218816}\arctan \left ({\frac{ \left ( 5+8\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(19*x/(-1+x)^3/(4*x^2+5*x+3)^2,x)

[Out]

-19/288/(-1+x)^2+133/864/(-1+x)+209/2304*ln(-1+x)-19/6912*(-2204/23*x-975/23)/(x^2+5/4*x+3/4)-209/4608*ln(4*x^
2+5*x+3)+114437/1218816*arctan(1/23*(5+8*x)*23^(1/2))*23^(1/2)

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Maxima [A]  time = 1.41221, size = 101, normalized size = 1.04 \begin{align*} \frac{114437}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{19 \,{\left (388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45\right )}}{4416 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac{209}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{209}{2304} \, \log \left (x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(19*x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="maxima")

[Out]

114437/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 19/4416*(388*x^3 - 407*x^2 - 120*x - 45)/(4*x^4 - 3*
x^3 - 3*x^2 - x + 3) - 209/4608*log(4*x^2 + 5*x + 3) + 209/2304*log(x - 1)

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Fricas [A]  time = 1.97032, size = 378, normalized size = 3.9 \begin{align*} \frac{19 \,{\left (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\right )}}{2437632 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(19*x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="fricas")

[Out]

19/2437632*(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)) - 2246
64*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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Sympy [A]  time = 0.205866, size = 88, normalized size = 0.91 \begin{align*} \frac{19 \left (388 x^{3} - 407 x^{2} - 120 x - 45\right )}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac{209 \log{\left (x - 1 \right )}}{2304} - \frac{209 \log{\left (x^{2} + \frac{5 x}{4} + \frac{3}{4} \right )}}{4608} + \frac{114437 \sqrt{23} \operatorname{atan}{\left (\frac{8 \sqrt{23} x}{23} + \frac{5 \sqrt{23}}{23} \right )}}{1218816} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(19*x/(-1+x)**3/(4*x**2+5*x+3)**2,x)

[Out]

19*(388*x**3 - 407*x**2 - 120*x - 45)/(17664*x**4 - 13248*x**3 - 13248*x**2 - 4416*x + 13248) + 209*log(x - 1)
/2304 - 209*log(x**2 + 5*x/4 + 3/4)/4608 + 114437*sqrt(23)*atan(8*sqrt(23)*x/23 + 5*sqrt(23)/23)/1218816

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Giac [A]  time = 1.05541, size = 96, normalized size = 0.99 \begin{align*} \frac{114437}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{19 \,{\left (388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45\right )}}{4416 \,{\left (4 \, x^{2} + 5 \, x + 3\right )}{\left (x - 1\right )}^{2}} - \frac{209}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{209}{2304} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(19*x/(-1+x)^3/(4*x^2+5*x+3)^2,x, algorithm="giac")

[Out]

114437/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 19/4416*(388*x^3 - 407*x^2 - 120*x - 45)/((4*x^2 + 5
*x + 3)*(x - 1)^2) - 209/4608*log(4*x^2 + 5*x + 3) + 209/2304*log(abs(x - 1))