### 3.2230 $$\int \frac{1}{(1+2 x)^2 (2+3 x+5 x^2)^3} \, dx$$

Optimal. Leaf size=114 $\frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}}$

[Out]

-51516/(329623*(1 + 2*x)) + (37 + 20*x)/(434*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (6427 + 5820*x)/(47089*(1 + 2*x)
*(2 + 3*x + 5*x^2)) - (1065012*ArcTan[(3 + 10*x)/Sqrt[31]])/(2307361*Sqrt[31]) + (384*Log[1 + 2*x])/2401 - (19
2*Log[2 + 3*x + 5*x^2])/2401

________________________________________________________________________________________

Rubi [A]  time = 0.0882789, antiderivative size = 114, 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 = {740, 822, 800, 634, 618, 204, 628} $\frac{20 x+37}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}+\frac{5820 x+6427}{47089 (2 x+1) \left (5 x^2+3 x+2\right )}-\frac{192 \log \left (5 x^2+3 x+2\right )}{2401}-\frac{51516}{329623 (2 x+1)}+\frac{384 \log (2 x+1)}{2401}-\frac{1065012 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2307361 \sqrt{31}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-51516/(329623*(1 + 2*x)) + (37 + 20*x)/(434*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + (6427 + 5820*x)/(47089*(1 + 2*x)
*(2 + 3*x + 5*x^2)) - (1065012*ArcTan[(3 + 10*x)/Sqrt[31]])/(2307361*Sqrt[31]) + (384*Log[1 + 2*x])/2401 - (19
2*Log[2 + 3*x + 5*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 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{1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^3} \, dx &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{1}{434} \int \frac{382+160 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \frac{74796+46560 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )} \, dx}{94178}\\ &=\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \left (\frac{206064}{7 (1+2 x)^2}+\frac{1476096}{49 (1+2 x)}-\frac{12 (181007+307520 x)}{49 \left (2+3 x+5 x^2\right )}\right ) \, dx}{94178}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{6 \int \frac{181007+307520 x}{2+3 x+5 x^2} \, dx}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{2401}-\frac{532506 \int \frac{1}{2+3 x+5 x^2} \, dx}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \log \left (2+3 x+5 x^2\right )}{2401}+\frac{1065012 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{2307361}\\ &=-\frac{51516}{329623 (1+2 x)}+\frac{37+20 x}{434 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{6427+5820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )}-\frac{1065012 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{2307361 \sqrt{31}}+\frac{384 \log (1+2 x)}{2401}-\frac{192 \log \left (2+3 x+5 x^2\right )}{2401}\\ \end{align*}

Mathematica [A]  time = 0.0665952, size = 98, normalized size = 0.86 $\frac{4 \left (-\frac{47089 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^2}-\frac{217 (51910 x-15179)}{4 \left (5 x^2+3 x+2\right )}-1429968 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac{1668296}{2 x+1}+2859936 \log (2 x+1)-266253 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{71528191}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*(-1668296/(1 + 2*x) - (47089*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^2) - (217*(-15179 + 51910*x))/(4*(2 + 3*x
+ 5*x^2)) - 266253*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 2859936*Log[1 + 2*x] - 1429968*Log[4*(2 + 3*x + 5*x^
2)]))/71528191

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 77, normalized size = 0.7 \begin{align*} -{\frac{32}{343+686\,x}}+{\frac{384\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{25}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{72674\,{x}^{3}}{961}}+{\frac{111769\,{x}^{2}}{4805}}+{\frac{613046\,x}{24025}}-{\frac{490329}{48050}} \right ) }-{\frac{192\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{2401}}-{\frac{1065012\,\sqrt{31}}{71528191}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/343/(1+2*x)+384/2401*ln(1+2*x)-25/2401*(72674/961*x^3+111769/4805*x^2+613046/24025*x-490329/48050)/(5*x^2+
3*x+2)^2-192/2401*ln(5*x^2+3*x+2)-1065012/71528191*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.51454, size = 117, normalized size = 1.03 \begin{align*} -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2575800 \, x^{4} + 2683560 \, x^{3} + 2293598 \, x^{2} + 773110 \, x + 175969}{659246 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} - \frac{192}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{384}{2401} \, \log \left (2 \, x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1065012/71528191*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 1/659246*(2575800*x^4 + 2683560*x^3 + 2293598*x^
2 + 773110*x + 175969)/(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4) - 192/2401*log(5*x^2 + 3*x + 2) + 384/24
01*log(2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.97443, size = 517, normalized size = 4.54 \begin{align*} -\frac{558948600 \, x^{4} + 582332520 \, x^{3} + 2130024 \, \sqrt{31}{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 497710766 \, x^{2} + 11439744 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 22879488 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \log \left (2 \, x + 1\right ) + 167764870 \, x + 38185273}{143056382 \,{\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/143056382*(558948600*x^4 + 582332520*x^3 + 2130024*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*
arctan(1/31*sqrt(31)*(10*x + 3)) + 497710766*x^2 + 11439744*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*log
(5*x^2 + 3*x + 2) - 22879488*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*log(2*x + 1) + 167764870*x + 38185
273)/(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)

________________________________________________________________________________________

Sympy [A]  time = 0.250377, size = 100, normalized size = 0.88 \begin{align*} - \frac{2575800 x^{4} + 2683560 x^{3} + 2293598 x^{2} + 773110 x + 175969}{32962300 x^{5} + 56035910 x^{4} + 58013648 x^{3} + 34940038 x^{2} + 13184920 x + 2636984} + \frac{384 \log{\left (x + \frac{1}{2} \right )}}{2401} - \frac{192 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{2401} - \frac{1065012 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{71528191} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2575800*x**4 + 2683560*x**3 + 2293598*x**2 + 773110*x + 175969)/(32962300*x**5 + 56035910*x**4 + 58013648*x*
*3 + 34940038*x**2 + 13184920*x + 2636984) + 384*log(x + 1/2)/2401 - 192*log(x**2 + 3*x/5 + 2/5)/2401 - 106501
2*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/71528191

________________________________________________________________________________________

Giac [A]  time = 1.10521, size = 146, normalized size = 1.28 \begin{align*} -\frac{1065012}{71528191} \, \sqrt{31} \arctan \left (-\frac{1}{31} \, \sqrt{31}{\left (\frac{7}{2 \, x + 1} - 2\right )}\right ) - \frac{32}{343 \,{\left (2 \, x + 1\right )}} + \frac{4 \,{\left (\frac{1178375}{2 \, x + 1} - \frac{2320190}{{\left (2 \, x + 1\right )}^{2}} + \frac{87843}{{\left (2 \, x + 1\right )}^{3}} - 1304250\right )}}{2307361 \,{\left (\frac{4}{2 \, x + 1} - \frac{7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{2}} - \frac{192}{2401} \, \log \left (-\frac{4}{2 \, x + 1} + \frac{7}{{\left (2 \, x + 1\right )}^{2}} + 5\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1065012/71528191*sqrt(31)*arctan(-1/31*sqrt(31)*(7/(2*x + 1) - 2)) - 32/343/(2*x + 1) + 4/2307361*(1178375/(2
*x + 1) - 2320190/(2*x + 1)^2 + 87843/(2*x + 1)^3 - 1304250)/(4/(2*x + 1) - 7/(2*x + 1)^2 - 5)^2 - 192/2401*lo
g(-4/(2*x + 1) + 7/(2*x + 1)^2 + 5)