### 3.2232 $$\int \frac{1}{(1+2 x)^2 (2+3 x+5 x^2)^4} \, dx$$

Optimal. Leaf size=142 $\frac{20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac{2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac{2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac{1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac{6802312}{71528191 (2 x+1)}+\frac{2048 \log (2 x+1)}{16807}-\frac{116056984 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{500697337 \sqrt{31}}$

[Out]

-6802312/(71528191*(1 + 2*x)) + (37 + 20*x)/(651*(1 + 2*x)*(2 + 3*x + 5*x^2)^3) + (3047 + 2820*x)/(47089*(1 +
2*x)*(2 + 3*x + 5*x^2)^2) + (2*(504757 + 603620*x))/(10218313*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (116056984*ArcTan
[(3 + 10*x)/Sqrt[31]])/(500697337*Sqrt[31]) + (2048*Log[1 + 2*x])/16807 - (1024*Log[2 + 3*x + 5*x^2])/16807

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Rubi [A]  time = 0.108502, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 9, 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} $\frac{20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}+\frac{2 (603620 x+504757)}{10218313 (2 x+1) \left (5 x^2+3 x+2\right )}+\frac{2820 x+3047}{47089 (2 x+1) \left (5 x^2+3 x+2\right )^2}-\frac{1024 \log \left (5 x^2+3 x+2\right )}{16807}-\frac{6802312}{71528191 (2 x+1)}+\frac{2048 \log (2 x+1)}{16807}-\frac{116056984 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{500697337 \sqrt{31}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-6802312/(71528191*(1 + 2*x)) + (37 + 20*x)/(651*(1 + 2*x)*(2 + 3*x + 5*x^2)^3) + (3047 + 2820*x)/(47089*(1 +
2*x)*(2 + 3*x + 5*x^2)^2) + (2*(504757 + 603620*x))/(10218313*(1 + 2*x)*(2 + 3*x + 5*x^2)) - (116056984*ArcTan
[(3 + 10*x)/Sqrt[31]])/(500697337*Sqrt[31]) + (2048*Log[1 + 2*x])/16807 - (1024*Log[2 + 3*x + 5*x^2])/16807

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 )^4} \, dx &=\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{1}{651} \int \frac{546+240 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^3} \, dx\\ &=\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{\int \frac{192972+135360 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^2} \, dx}{282534}\\ &=\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \frac{34893816+28973760 x}{(1+2 x)^2 \left (2+3 x+5 x^2\right )} \, dx}{61309878}\\ &=\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{\int \left (\frac{81627744}{7 (1+2 x)^2}+\frac{732143616}{49 (1+2 x)}-\frac{24 (37386611+76264960 x)}{49 \left (2+3 x+5 x^2\right )}\right ) \, dx}{61309878}\\ &=-\frac{6802312}{71528191 (1+2 x)}+\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{2048 \log (1+2 x)}{16807}-\frac{4 \int \frac{37386611+76264960 x}{2+3 x+5 x^2} \, dx}{500697337}\\ &=-\frac{6802312}{71528191 (1+2 x)}+\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{2048 \log (1+2 x)}{16807}-\frac{1024 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{16807}-\frac{58028492 \int \frac{1}{2+3 x+5 x^2} \, dx}{500697337}\\ &=-\frac{6802312}{71528191 (1+2 x)}+\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}+\frac{2048 \log (1+2 x)}{16807}-\frac{1024 \log \left (2+3 x+5 x^2\right )}{16807}+\frac{116056984 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{500697337}\\ &=-\frac{6802312}{71528191 (1+2 x)}+\frac{37+20 x}{651 (1+2 x) \left (2+3 x+5 x^2\right )^3}+\frac{3047+2820 x}{47089 (1+2 x) \left (2+3 x+5 x^2\right )^2}+\frac{2 (504757+603620 x)}{10218313 (1+2 x) \left (2+3 x+5 x^2\right )}-\frac{116056984 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{500697337 \sqrt{31}}+\frac{2048 \log (1+2 x)}{16807}-\frac{1024 \log \left (2+3 x+5 x^2\right )}{16807}\\ \end{align*}

Mathematica [A]  time = 0.0901292, size = 119, normalized size = 0.84 $\frac{8 \left (-\frac{10218313 (270 x-43)}{8 \left (5 x^2+3 x+2\right )^3}-\frac{651 (3736330 x-1739037)}{4 \left (5 x^2+3 x+2\right )}-\frac{141267 (27530 x-7117)}{8 \left (5 x^2+3 x+2\right )^2}-354632064 \log \left (4 \left (5 x^2+3 x+2\right )\right )-\frac{310303056}{2 x+1}+709264128 \log (2 x+1)-43521369 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{46564852341}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(8*(-310303056/(1 + 2*x) - (10218313*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^3) - (141267*(-7117 + 27530*x))/(8*(2
+ 3*x + 5*x^2)^2) - (651*(-1739037 + 3736330*x))/(4*(2 + 3*x + 5*x^2)) - 43521369*Sqrt[31]*ArcTan[(3 + 10*x)/
Sqrt[31]] + 709264128*Log[1 + 2*x] - 354632064*Log[4*(2 + 3*x + 5*x^2)]))/46564852341

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Maple [A]  time = 0.05, size = 87, normalized size = 0.6 \begin{align*} -{\frac{128}{2401+4802\,x}}+{\frac{2048\,\ln \left ( 1+2\,x \right ) }{16807}}-{\frac{125}{16807\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ({\frac{10461724\,{x}^{5}}{29791}}+{\frac{38423826\,{x}^{4}}{148955}}+{\frac{199128958\,{x}^{3}}{744775}}-{\frac{6944987\,{x}^{2}}{3723875}}-{\frac{410739\,x}{744775}}-{\frac{371196343}{11171625}} \right ) }-{\frac{1024\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{16807}}-{\frac{116056984\,\sqrt{31}}{15521617447}\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)^4,x)

[Out]

-128/2401/(1+2*x)+2048/16807*ln(1+2*x)-125/16807*(10461724/29791*x^5+38423826/148955*x^4+199128958/744775*x^3-
6944987/3723875*x^2-410739/744775*x-371196343/11171625)/(5*x^2+3*x+2)^3-1024/16807*ln(5*x^2+3*x+2)-116056984/1
5521617447*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Maxima [A]  time = 1.56445, size = 144, normalized size = 1.01 \begin{align*} -\frac{116056984}{15521617447} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2550867000 \, x^{6} + 3957759600 \, x^{5} + 4525420710 \, x^{4} + 2788779072 \, x^{3} + 1299394083 \, x^{2} + 304894531 \, x + 38489903}{214584573 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} - \frac{1024}{16807} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{2048}{16807} \, \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)^4,x, algorithm="maxima")

[Out]

-116056984/15521617447*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 1/214584573*(2550867000*x^6 + 3957759600*x^
5 + 4525420710*x^4 + 2788779072*x^3 + 1299394083*x^2 + 304894531*x + 38489903)/(250*x^7 + 575*x^6 + 795*x^5 +
699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8) - 1024/16807*log(5*x^2 + 3*x + 2) + 2048/16807*log(2*x + 1)

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Fricas [A]  time = 2.43371, size = 726, normalized size = 5.11 \begin{align*} -\frac{553538139000 \, x^{6} + 858833833200 \, x^{5} + 982016294070 \, x^{4} + 605165058624 \, x^{3} + 348170952 \, \sqrt{31}{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 281968516011 \, x^{2} + 2837056512 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 5674113024 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 66162113227 \, x + 8352308951}{46564852341 \,{\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/46564852341*(553538139000*x^6 + 858833833200*x^5 + 982016294070*x^4 + 605165058624*x^3 + 348170952*sqrt(31)
*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)*arctan(1/31*sqrt(31)*(10*x + 3)) + 281
968516011*x^2 + 2837056512*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)*log(5*x^2 +
3*x + 2) - 5674113024*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)*log(2*x + 1) + 66
162113227*x + 8352308951)/(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)

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Sympy [A]  time = 0.299741, size = 121, normalized size = 0.85 \begin{align*} - \frac{2550867000 x^{6} + 3957759600 x^{5} + 4525420710 x^{4} + 2788779072 x^{3} + 1299394083 x^{2} + 304894531 x + 38489903}{53646143250 x^{7} + 123386129475 x^{6} + 170594735535 x^{5} + 149994616527 x^{4} + 93344289255 x^{3} + 39912730578 x^{2} + 11158397796 x + 1716676584} + \frac{2048 \log{\left (x + \frac{1}{2} \right )}}{16807} - \frac{1024 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{16807} - \frac{116056984 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{15521617447} \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)**4,x)

[Out]

-(2550867000*x**6 + 3957759600*x**5 + 4525420710*x**4 + 2788779072*x**3 + 1299394083*x**2 + 304894531*x + 3848
9903)/(53646143250*x**7 + 123386129475*x**6 + 170594735535*x**5 + 149994616527*x**4 + 93344289255*x**3 + 39912
730578*x**2 + 11158397796*x + 1716676584) + 2048*log(x + 1/2)/16807 - 1024*log(x**2 + 3*x/5 + 2/5)/16807 - 116
056984*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/15521617447

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Giac [A]  time = 1.09174, size = 170, normalized size = 1.2 \begin{align*} -\frac{116056984}{15521617447} \, \sqrt{31} \arctan \left (-\frac{1}{31} \, \sqrt{31}{\left (\frac{7}{2 \, x + 1} - 2\right )}\right ) - \frac{128}{2401 \,{\left (2 \, x + 1\right )}} - \frac{8 \,{\left (\frac{3841449975}{2 \, x + 1} - \frac{8833663680}{{\left (2 \, x + 1\right )}^{2}} + \frac{7499779568}{{\left (2 \, x + 1\right )}^{3}} - \frac{7050406230}{{\left (2 \, x + 1\right )}^{4}} + \frac{1291725897}{{\left (2 \, x + 1\right )}^{5}} - 2009265250\right )}}{1502092011 \,{\left (\frac{4}{2 \, x + 1} - \frac{7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{3}} - \frac{1024}{16807} \, \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)^4,x, algorithm="giac")

[Out]

-116056984/15521617447*sqrt(31)*arctan(-1/31*sqrt(31)*(7/(2*x + 1) - 2)) - 128/2401/(2*x + 1) - 8/1502092011*(
3841449975/(2*x + 1) - 8833663680/(2*x + 1)^2 + 7499779568/(2*x + 1)^3 - 7050406230/(2*x + 1)^4 + 1291725897/(
2*x + 1)^5 - 2009265250)/(4/(2*x + 1) - 7/(2*x + 1)^2 - 5)^3 - 1024/16807*log(-4/(2*x + 1) + 7/(2*x + 1)^2 + 5
)