### 3.2231 $$\int \frac{1}{(1+2 x) (2+3 x+5 x^2)^4} \, dx$$

Optimal. Leaf size=110 $\frac{20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac{4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac{4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac{64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{71528191 \sqrt{31}}$

[Out]

(37 + 20*x)/(651*(2 + 3*x + 5*x^2)^3) + (4*(1983 + 1805*x))/(141267*(2 + 3*x + 5*x^2)^2) + (4*(180133 + 203230
*x))/(10218313*(2 + 3*x + 5*x^2)) + (19007376*ArcTan[(3 + 10*x)/Sqrt[31]])/(71528191*Sqrt[31]) + (128*Log[1 +
2*x])/2401 - (64*Log[2 + 3*x + 5*x^2])/2401

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Rubi [A]  time = 0.0992217, antiderivative size = 110, 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} $\frac{20 x+37}{651 \left (5 x^2+3 x+2\right )^3}+\frac{4 (203230 x+180133)}{10218313 \left (5 x^2+3 x+2\right )}+\frac{4 (1805 x+1983)}{141267 \left (5 x^2+3 x+2\right )^2}-\frac{64 \log \left (5 x^2+3 x+2\right )}{2401}+\frac{128 \log (2 x+1)}{2401}+\frac{19007376 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{71528191 \sqrt{31}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(37 + 20*x)/(651*(2 + 3*x + 5*x^2)^3) + (4*(1983 + 1805*x))/(141267*(2 + 3*x + 5*x^2)^2) + (4*(180133 + 203230
*x))/(10218313*(2 + 3*x + 5*x^2)) + (19007376*ArcTan[(3 + 10*x)/Sqrt[31]])/(71528191*Sqrt[31]) + (128*Log[1 +
2*x])/2401 - (64*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) \left (2+3 x+5 x^2\right )^4} \, dx &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{1}{651} \int \frac{472+200 x}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{\int \frac{135576+86640 x}{(1+2 x) \left (2+3 x+5 x^2\right )^2} \, dx}{282534}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{16317264+9755040 x}{(1+2 x) \left (2+3 x+5 x^2\right )} \, dx}{61309878}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{\int \left (\frac{45758976}{7 (1+2 x)}-\frac{48 (-472977+2383280 x)}{7 \left (2+3 x+5 x^2\right )}\right ) \, dx}{61309878}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{128 \log (1+2 x)}{2401}-\frac{8 \int \frac{-472977+2383280 x}{2+3 x+5 x^2} \, dx}{71528191}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{128 \log (1+2 x)}{2401}-\frac{64 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{2401}+\frac{9503688 \int \frac{1}{2+3 x+5 x^2} \, dx}{71528191}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{128 \log (1+2 x)}{2401}-\frac{64 \log \left (2+3 x+5 x^2\right )}{2401}-\frac{19007376 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{71528191}\\ &=\frac{37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac{4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac{4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac{19007376 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{71528191 \sqrt{31}}+\frac{128 \log (1+2 x)}{2401}-\frac{64 \log \left (2+3 x+5 x^2\right )}{2401}\\ \end{align*}

Mathematica [A]  time = 0.112337, size = 88, normalized size = 0.8 $\frac{16 \left (\frac{217 \left (60969000 x^5+127202700 x^4+143405620 x^3+105257844 x^2+44933184 x+13831165\right )}{16 \left (5 x^2+3 x+2\right )^3}-11082252 \log \left (4 \left (5 x^2+3 x+2\right )\right )+22164504 \log (2 x+1)+3563883 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )\right )}{6652121763}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(16*((217*(13831165 + 44933184*x + 105257844*x^2 + 143405620*x^3 + 127202700*x^4 + 60969000*x^5))/(16*(2 + 3*x
+ 5*x^2)^3) + 3563883*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 22164504*Log[1 + 2*x] - 11082252*Log[4*(2 + 3*x
+ 5*x^2)]))/6652121763

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Maple [A]  time = 0.051, size = 78, normalized size = 0.7 \begin{align*}{\frac{128\,\ln \left ( 1+2\,x \right ) }{2401}}-{\frac{125}{2401\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{3}} \left ( -{\frac{1138088\,{x}^{5}}{29791}}-{\frac{11872252\,{x}^{4}}{148955}}-{\frac{200767868\,{x}^{3}}{2234325}}-{\frac{245601636\,{x}^{2}}{3723875}}-{\frac{104844096\,x}{3723875}}-{\frac{19363631}{2234325}} \right ) }-{\frac{64\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{2401}}+{\frac{19007376\,\sqrt{31}}{2217373921}\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)/(5*x^2+3*x+2)^4,x)

[Out]

128/2401*ln(1+2*x)-125/2401*(-1138088/29791*x^5-11872252/148955*x^4-200767868/2234325*x^3-245601636/3723875*x^
2-104844096/3723875*x-19363631/2234325)/(5*x^2+3*x+2)^3-64/2401*ln(5*x^2+3*x+2)+19007376/2217373921*arctan(1/3
1*(3+10*x)*31^(1/2))*31^(1/2)

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Maxima [A]  time = 1.51665, size = 131, normalized size = 1.19 \begin{align*} \frac{19007376}{2217373921} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} - \frac{64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{128}{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)/(5*x^2+3*x+2)^4,x, algorithm="maxima")

[Out]

19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939*(60969000*x^5 + 127202700*x^4 + 143
405620*x^3 + 105257844*x^2 + 44933184*x + 13831165)/(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x +
8) - 64/2401*log(5*x^2 + 3*x + 2) + 128/2401*log(2*x + 1)

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Fricas [A]  time = 2.38046, size = 633, normalized size = 5.75 \begin{align*} \frac{13230273000 \, x^{5} + 27602985900 \, x^{4} + 31119019540 \, x^{3} + 57022128 \, \sqrt{31}{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 22840952148 \, x^{2} - 177316032 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 354632064 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 9750500928 \, x + 3001362805}{6652121763 \,{\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6652121763*(13230273000*x^5 + 27602985900*x^4 + 31119019540*x^3 + 57022128*sqrt(31)*(125*x^6 + 225*x^5 + 285
*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*arctan(1/31*sqrt(31)*(10*x + 3)) + 22840952148*x^2 - 177316032*(125*x^6 +
225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*log(5*x^2 + 3*x + 2) + 354632064*(125*x^6 + 225*x^5 + 285*x
^4 + 207*x^3 + 114*x^2 + 36*x + 8)*log(2*x + 1) + 9750500928*x + 3001362805)/(125*x^6 + 225*x^5 + 285*x^4 + 20
7*x^3 + 114*x^2 + 36*x + 8)

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Sympy [A]  time = 0.274918, size = 110, normalized size = 1. \begin{align*} \frac{60969000 x^{5} + 127202700 x^{4} + 143405620 x^{3} + 105257844 x^{2} + 44933184 x + 13831165}{3831867375 x^{6} + 6897361275 x^{5} + 8736657615 x^{4} + 6345572373 x^{3} + 3494663046 x^{2} + 1103577804 x + 245239512} + \frac{128 \log{\left (x + \frac{1}{2} \right )}}{2401} - \frac{64 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{2401} + \frac{19007376 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{2217373921} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(60969000*x**5 + 127202700*x**4 + 143405620*x**3 + 105257844*x**2 + 44933184*x + 13831165)/(3831867375*x**6 +
6897361275*x**5 + 8736657615*x**4 + 6345572373*x**3 + 3494663046*x**2 + 1103577804*x + 245239512) + 128*log(x
+ 1/2)/2401 - 64*log(x**2 + 3*x/5 + 2/5)/2401 + 19007376*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/22173
73921

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Giac [A]  time = 1.07708, size = 105, normalized size = 0.95 \begin{align*} \frac{19007376}{2217373921} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} - \frac{64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{128}{2401} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939*(60969000*x^5 + 127202700*x^4 + 143
405620*x^3 + 105257844*x^2 + 44933184*x + 13831165)/(5*x^2 + 3*x + 2)^3 - 64/2401*log(5*x^2 + 3*x + 2) + 128/2
401*log(abs(2*x + 1))