\(\int \frac {19 x}{(-1+x)^3 (3+5 x+4 x^2)^2} \, dx\) [173]

Optimal result

Integrand size = 20, antiderivative size = 97 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=-\frac {399}{736 (1-x)^2}-\frac {1843}{4416 (1-x)}+\frac {19 (39+44 x)}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {114437 \arctan \left (\frac {5+8 x}{\sqrt {23}}\right )}{52992 \sqrt {23}}+\frac {209 \log (1-x)}{2304}-\frac {209 \log \left (3+5 x+4 x^2\right )}{4608} \]

[Out]

-399/736/(1-x)^2-1843/4416/(1-x)+19/276*(39+44*x)/(1-x)^2/(4*x^2+5*x+3)+209/2304*ln(1-x)-209/4608*ln(4*x^2+5*x
+3)+114437/1218816*arctan(1/23*(5+8*x)*23^(1/2))*23^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {12, 836, 814, 648, 632, 210, 642} \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {114437 \arctan \left (\frac {8 x+5}{\sqrt {23}}\right )}{52992 \sqrt {23}}+\frac {19 (44 x+39)}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac {209 \log \left (4 x^2+5 x+3\right )}{4608}-\frac {1843}{4416 (1-x)}-\frac {399}{736 (1-x)^2}+\frac {209 \log (1-x)}{2304} \]

[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 210

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

Rule 632

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 642

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 648

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 814

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 836

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*} \text {integral}& = 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 \text {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 \arctan \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] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {19 \left (-\frac {25392}{(-1+x)^2}+\frac {59248}{-1+x}+\frac {184 (975+2204 x)}{3+5 x+4 x^2}+36138 \sqrt {23} \arctan \left (\frac {5+8 x}{\sqrt {23}}\right )+34914 \log (1-x)-17457 \log \left (3+5 x+4 x^2\right )\right )}{7312896} \]

[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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
default	\(-\frac {19}{288 \left (-1+x \right )^{2}}+\frac {133}{864 \left (-1+x \right )}+\frac {209 \ln \left (-1+x \right )}{2304}-\frac {19 \left (-\frac {2204 x}{23}-\frac {975}{23}\right )}{6912 \left (x^{2}+\frac {5}{4} x +\frac {3}{4}\right )}-\frac {209 \ln \left (4 x^{2}+5 x +3\right )}{4608}+\frac {114437 \arctan \left (\frac {\left (5+8 x \right ) \sqrt {23}}{23}\right ) \sqrt {23}}{1218816}\)	\(68\)
risch	\(\frac {\frac {1843}{1104} x^{3}-\frac {7733}{4416} x^{2}-\frac {95}{184} x -\frac {285}{1472}}{\left (-1+x \right )^{2} \left (4 x^{2}+5 x +3\right )}+\frac {209 \ln \left (-1+x \right )}{2304}-\frac {209 \ln \left (580424464 x^{2}[phone] x [phone]\right )}{4608}+\frac {114437 \sqrt {23}\, \arctan \left (\frac {2 \left (24092 x +\frac {30115}{2}\right ) \sqrt {23}}{138529}\right )}{1218816}\)	\(71\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.38 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {19 \, {\left (214176 \, x^{3} + 12046 \, \sqrt {23} {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) - 224664 \, x^{2} - 5819 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (4 \, x^{2} + 5 \, x + 3\right ) + 11638 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (x - 1\right ) - 66240 \, x - 24840\right )}}{2437632 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {19 \cdot \left (388 x^{3} - 407 x^{2} - 120 x - 45\right )}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac {209 \log {\left (x - 1 \right )}}{2304} - \frac {209 \log {\left (x^{2} + \frac {5 x}{4} + \frac {3}{4} \right )}}{4608} + \frac {114437 \sqrt {23} \operatorname {atan}{\left (\frac {8 \sqrt {23} x}{23} + \frac {5 \sqrt {23}}{23} \right )}}{1218816} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {114437}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {19 \, {\left (388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45\right )}}{4416 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac {209}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {209}{2304} \, \log \left (x - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {114437}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {19 \, {\left (388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45\right )}}{4416 \, {\left (4 \, x^{2} + 5 \, x + 3\right )} {\left (x - 1\right )}^{2}} - \frac {209}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {209}{2304} \, \log \left ({\left | x - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {19 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx=\frac {209\,\ln \left (x-1\right )}{2304}+\frac {-\frac {1843\,x^3}{4416}+\frac {7733\,x^2}{17664}+\frac {95\,x}{736}+\frac {285}{5888}}{-x^4+\frac {3\,x^3}{4}+\frac {3\,x^2}{4}+\frac {x}{4}-\frac {3}{4}}-\ln \left (x+\frac {5}{8}-\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {209}{4608}+\frac {\sqrt {23}\,114437{}\mathrm {i}}{2437632}\right )+\ln \left (x+\frac {5}{8}+\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {209}{4608}+\frac {\sqrt {23}\,114437{}\mathrm {i}}{2437632}\right ) \]

[In]

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

[Out]

(209*log(x - 1))/2304 + ((95*x)/736 + (7733*x^2)/17664 - (1843*x^3)/4416 + 285/5888)/(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)*114437i)/2437632 + 209/4608) + log(x + (23^(1/2)*
1i)/8 + 5/8)*((23^(1/2)*114437i)/2437632 - 209/4608)