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

Optimal. Leaf size=127 \[ -\frac{25 (117-137 x)}{172546 \left (5 x^2+3 x+2\right )}+\frac{13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}+\frac{19 \log \left (2 x^2-x+3\right )}{10648}-\frac{19 \log \left (5 x^2+3 x+2\right )}{10648}+\frac{2769 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{122452 \sqrt{23}}+\frac{12643 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{165044 \sqrt{31}} \]

[Out]

(-25*(117 - 137*x))/(172546*(2 + 3*x + 5*x^2)) + (13 - 6*x)/(506*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)) + (2769*Ar
cTan[(1 - 4*x)/Sqrt[23]])/(122452*Sqrt[23]) + (12643*ArcTan[(3 + 10*x)/Sqrt[31]])/(165044*Sqrt[31]) + (19*Log[
3 - x + 2*x^2])/10648 - (19*Log[2 + 3*x + 5*x^2])/10648

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Rubi [A]  time = 0.122755, antiderivative size = 127, 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} \[ -\frac{25 (117-137 x)}{172546 \left (5 x^2+3 x+2\right )}+\frac{13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}+\frac{19 \log \left (2 x^2-x+3\right )}{10648}-\frac{19 \log \left (5 x^2+3 x+2\right )}{10648}+\frac{2769 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{122452 \sqrt{23}}+\frac{12643 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{165044 \sqrt{31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25*(117 - 137*x))/(172546*(2 + 3*x + 5*x^2)) + (13 - 6*x)/(506*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)) + (2769*Ar
cTan[(1 - 4*x)/Sqrt[23]])/(122452*Sqrt[23]) + (12643*ArcTan[(3 + 10*x)/Sqrt[31]])/(165044*Sqrt[31]) + (19*Log[
3 - x + 2*x^2])/10648 - (19*Log[2 + 3*x + 5*x^2])/10648

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 )^2 \left (2+3 x+5 x^2\right )^2} \, dx &=\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-2321-2299 x+990 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{5566}\\ &=-\frac{25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-3034196+4654870 x-1657700 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{41756132}\\ &=-\frac{25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac{\int \frac{132282766-72124228 x}{3-x+2 x^2} \, dx}{10104983944}-\frac{\int \frac{-332946988+180310570 x}{2+3 x+5 x^2} \, dx}{10104983944}\\ &=-\frac{25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{19 \int \frac{-1+4 x}{3-x+2 x^2} \, dx}{10648}-\frac{19 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{10648}-\frac{2769 \int \frac{1}{3-x+2 x^2} \, dx}{244904}+\frac{12643 \int \frac{1}{2+3 x+5 x^2} \, dx}{330088}\\ &=-\frac{25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{19 \log \left (3-x+2 x^2\right )}{10648}-\frac{19 \log \left (2+3 x+5 x^2\right )}{10648}+\frac{2769 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{122452}-\frac{12643 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{165044}\\ &=-\frac{25 (117-137 x)}{172546 \left (2+3 x+5 x^2\right )}+\frac{13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac{2769 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{122452 \sqrt{23}}+\frac{12643 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{165044 \sqrt{31}}+\frac{19 \log \left (3-x+2 x^2\right )}{10648}-\frac{19 \log \left (2+3 x+5 x^2\right )}{10648}\\ \end{align*}

Mathematica [A]  time = 0.057922, size = 106, normalized size = 0.83 \[ \frac{\frac{31372 \left (6850 x^3-9275 x^2+11154 x-4342\right )}{10 x^4+x^3+16 x^2+7 x+6}+9659011 \log \left (2 x^2-x+3\right )-9659011 \log \left (5 x^2+3 x+2\right )-5322018 \sqrt{23} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )[phone] \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{5413113112} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((31372*(-4342 + 11154*x - 9275*x^2 + 6850*x^3))/(6 + 7*x + 16*x^2 + x^3 + 10*x^4) - 5322018*Sqrt[23]*ArcTan[(
-1 + 4*x)/Sqrt[23]] + 13376294*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 9659011*Log[3 - x + 2*x^2] - 9659011*Log
[2 + 3*x + 5*x^2])/5413113112

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Maple [A]  time = 0.052, size = 94, normalized size = 0.7 \begin{align*} -{\frac{1}{5324} \left ( -{\frac{759\,x}{31}}+{\frac{1078}{155}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}-{\frac{19\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{10648}}+{\frac{12643\,\sqrt{31}}{5116364}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }+{\frac{1}{5324} \left ( -{\frac{77\,x}{23}}-{\frac{341}{46}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{19\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{10648}}-{\frac{2769\,\sqrt{23}}{2816396}\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)^2/(5*x^2+3*x+2)^2,x)

[Out]

-1/5324*(-759/31*x+1078/155)/(x^2+3/5*x+2/5)-19/10648*ln(5*x^2+3*x+2)+12643/5116364*arctan(1/31*(3+10*x)*31^(1
/2))*31^(1/2)+1/5324*(-77/23*x-341/46)/(x^2-1/2*x+3/2)+19/10648*ln(2*x^2-x+3)-2769/2816396*23^(1/2)*arctan(1/2
3*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.44659, size = 130, normalized size = 1.02 \begin{align*} \frac{12643}{5116364} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2769}{2816396} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{6850 \, x^{3} - 9275 \, x^{2} + 11154 \, x - 4342}{172546 \,{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} - \frac{19}{10648} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{19}{10648} \, \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)^2/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

12643/5116364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 2769/2816396*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) + 1/172546*(6850*x^3 - 9275*x^2 + 11154*x - 4342)/(10*x^4 + x^3 + 16*x^2 + 7*x + 6) - 19/10648*log(5*x^2 + 3
*x + 2) + 19/10648*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.991225, size = 548, normalized size = 4.31 \begin{align*} \frac{214898200 \, x^{3} + 13376294 \, \sqrt{31}{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 5322018 \, \sqrt{23}{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - 290975300 \, x^{2} - 9659011 \,{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 9659011 \,{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )} \log \left (2 \, x^{2} - x + 3\right ) + 349923288 \, x - 136217224}{5413113112 \,{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5413113112*(214898200*x^3 + 13376294*sqrt(31)*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*arctan(1/31*sqrt(31)*(10*x +
3)) - 5322018*sqrt(23)*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*arctan(1/23*sqrt(23)*(4*x - 1)) - 290975300*x^2 - 96
59011*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*log(5*x^2 + 3*x + 2) + 9659011*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*log(2
*x^2 - x + 3) + 349923288*x - 136217224)/(10*x^4 + x^3 + 16*x^2 + 7*x + 6)

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Sympy [A]  time = 0.449151, size = 122, normalized size = 0.96 \begin{align*} \frac{6850 x^{3} - 9275 x^{2} + 11154 x - 4342}{1725460 x^{4} + 172546 x^{3} + 2760736 x^{2} + 1207822 x + 1035276} + \frac{19 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{10648} - \frac{19 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{10648} - \frac{2769 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{2816396} + \frac{12643 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{5116364} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6850*x**3 - 9275*x**2 + 11154*x - 4342)/(1725460*x**4 + 172546*x**3 + 2760736*x**2 + 1207822*x + 1035276) + 1
9*log(x**2 - x/2 + 3/2)/10648 - 19*log(x**2 + 3*x/5 + 2/5)/10648 - 2769*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(2
3)/23)/2816396 + 12643*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/5116364

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Giac [A]  time = 1.1847, size = 130, normalized size = 1.02 \begin{align*} \frac{12643}{5116364} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{2769}{2816396} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{6850 \, x^{3} - 9275 \, x^{2} + 11154 \, x - 4342}{172546 \,{\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}} - \frac{19}{10648} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac{19}{10648} \, \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)^2/(5*x^2+3*x+2)^2,x, algorithm="giac")

[Out]

12643/5116364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 2769/2816396*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) + 1/172546*(6850*x^3 - 9275*x^2 + 11154*x - 4342)/(10*x^4 + x^3 + 16*x^2 + 7*x + 6) - 19/10648*log(5*x^2 + 3
*x + 2) + 19/10648*log(2*x^2 - x + 3)