3.55 \(\int \frac{1}{(3-x+2 x^2)^3 (2+3 x+5 x^2)} \, dx\)

Optimal. Leaf size=115 \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]

[Out]

(13 - 6*x)/(1012*(3 - x + 2*x^2)^2) + (3625 - 746*x)/(256036*(3 - x + 2*x^2)) - (53403*ArcTan[(1 - 4*x)/Sqrt[2
3]])/(5632792*Sqrt[23]) + (247*ArcTan[(3 + 10*x)/Sqrt[31]])/(10648*Sqrt[31]) - (119*Log[3 - x + 2*x^2])/21296
+ (119*Log[2 + 3*x + 5*x^2])/21296

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Rubi [A]  time = 0.1235, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \[ \frac{3625-746 x}{256036 \left (2 x^2-x+3\right )}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac{119 \log \left (2 x^2-x+3\right )}{21296}+\frac{119 \log \left (5 x^2+3 x+2\right )}{21296}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10648 \sqrt{31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13 - 6*x)/(1012*(3 - x + 2*x^2)^2) + (3625 - 746*x)/(256036*(3 - x + 2*x^2)) - (53403*ArcTan[(1 - 4*x)/Sqrt[2
3]])/(5632792*Sqrt[23]) + (247*ArcTan[(3 + 10*x)/Sqrt[31]])/(10648*Sqrt[31]) - (119*Log[3 - x + 2*x^2])/21296
+ (119*Log[2 + 3*x + 5*x^2])/21296

Rule 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
 1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
 q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
 f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
 - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
 (b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1072

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)
), x_Symbol] :> With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, In
t[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e +
a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A
*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2),
 x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]

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}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}-\frac{\int \frac{-3652-1936 x+990 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )} \, dx}{11132}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{\int \frac{-6551908-7779574 x+902660 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{61960712}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{\int \frac{-154867174+335151124 x}{3-x+2 x^2} \, dx}{14994492304}-\frac{\int \frac{-425275796-837877810 x}{2+3 x+5 x^2} \, dx}{14994492304}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}+\frac{53403 \int \frac{1}{3-x+2 x^2} \, dx}{11265584}-\frac{119 \int \frac{-1+4 x}{3-x+2 x^2} \, dx}{21296}+\frac{119 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{21296}+\frac{247 \int \frac{1}{2+3 x+5 x^2} \, dx}{21296}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{119 \log \left (3-x+2 x^2\right )}{21296}+\frac{119 \log \left (2+3 x+5 x^2\right )}{21296}-\frac{53403 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{5632792}-\frac{247 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{10648}\\ &=\frac{13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac{3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac{53403 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5632792 \sqrt{23}}+\frac{247 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{10648 \sqrt{31}}-\frac{119 \log \left (3-x+2 x^2\right )}{21296}+\frac{119 \log \left (2+3 x+5 x^2\right )}{21296}\\ \end{align*}

Mathematica [A]  time = 0.154692, size = 99, normalized size = 0.86 \[ \frac{713 \left (-\frac{44 \left (1492 x^3-7996 x^2+7381 x-14164\right )}{\left (-2 x^2+x-3\right )^2}-62951 \log \left (2 x^2-x+3\right )+62951 \log \left (5 x^2+3 x+2\right )\right )+3310986 \sqrt{23} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+6010498 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{8032361392} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3310986*Sqrt[23]*ArcTan[(-1 + 4*x)/Sqrt[23]] + 6010498*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 713*((-44*(-141
64 + 7381*x - 7996*x^2 + 1492*x^3))/(-3 + x - 2*x^2)^2 - 62951*Log[3 - x + 2*x^2] + 62951*Log[2 + 3*x + 5*x^2]
))/8032361392

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Maple [A]  time = 0.053, size = 89, normalized size = 0.8 \begin{align*}{\frac{119\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{21296}}+{\frac{247\,\sqrt{31}}{330088}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }-{\frac{1}{2662\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ({\frac{8206\,{x}^{3}}{529}}-{\frac{43978\,{x}^{2}}{529}}+{\frac{81191\,x}{1058}}-{\frac{77902}{529}} \right ) }-{\frac{119\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{21296}}+{\frac{53403\,\sqrt{23}}{129554216}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

119/21296*ln(5*x^2+3*x+2)+247/330088*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)-1/2662*(8206/529*x^3-43978/529*x^
2+81191/1058*x-77902/529)/(2*x^2-x+3)^2-119/21296*ln(2*x^2-x+3)+53403/129554216*23^(1/2)*arctan(1/23*(-1+4*x)*
23^(1/2))

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Maxima [A]  time = 1.43151, size = 132, normalized size = 1.15 \begin{align*} \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac{119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) - 1/256036*(1492*x^3 - 7996*x^2 + 7381*x - 14164)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) + 119/21296*log(5*x^2 +
 3*x + 2) - 119/21296*log(2*x^2 - x + 3)

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Fricas [A]  time = 1.02491, size = 556, normalized size = 4.83 \begin{align*} -\frac{46807024 \, x^{3} - 6010498 \, \sqrt{31}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 3310986 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 250850512 \, x^{2} - 44884063 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 44884063 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 231556732 \, x - 444353008}{8032361392 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8032361392*(46807024*x^3 - 6010498*sqrt(31)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/31*sqrt(31)*(10*x +
 3)) - 3310986*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) - 250850512*x^2 - 4
4884063*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(5*x^2 + 3*x + 2) + 44884063*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*
log(2*x^2 - x + 3) + 231556732*x - 444353008)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [A]  time = 0.396684, size = 122, normalized size = 1.06 \begin{align*} - \frac{1492 x^{3} - 7996 x^{2} + 7381 x - 14164}{1024144 x^{4} - 1024144 x^{3} + 3328468 x^{2} - 1536216 x + 2304324} - \frac{119 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{21296} + \frac{119 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{21296} + \frac{53403 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{129554216} + \frac{247 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{330088} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1492*x**3 - 7996*x**2 + 7381*x - 14164)/(1024144*x**4 - 1024144*x**3 + 3328468*x**2 - 1536216*x + 2304324) -
 119*log(x**2 - x/2 + 3/2)/21296 + 119*log(x**2 + 3*x/5 + 2/5)/21296 + 53403*sqrt(23)*atan(4*sqrt(23)*x/23 - s
qrt(23)/23)/129554216 + 247*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/330088

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Giac [A]  time = 1.15984, size = 119, normalized size = 1.03 \begin{align*} \frac{247}{330088} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{53403}{129554216} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac{119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) - 1/256036*(1492*x^3 - 7996*x^2 + 7381*x - 14164)/(2*x^2 - x + 3)^2 + 119/21296*log(5*x^2 + 3*x + 2) - 119/2
1296*log(2*x^2 - x + 3)