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

Optimal. Leaf size=160 \[ \frac {-252815 x-2328909}{174616552 \left (5 x^2+3 x+2\right )}+\frac {9665-1446 x}{512072 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}+\frac {181 \log \left (2 x^2-x+3\right )}{468512}-\frac {181 \log \left (5 x^2+3 x+2\right )}{468512}+\frac {2038497 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{7261936 \sqrt {31}} \]

[Out]

1/174616552*(-2328909-252815*x)/(5*x^2+3*x+2)+1/1012*(13-6*x)/(2*x^2-x+3)^2/(5*x^2+3*x+2)+1/512072*(9665-1446*
x)/(2*x^2-x+3)/(5*x^2+3*x+2)+181/468512*ln(2*x^2-x+3)-181/468512*ln(5*x^2+3*x+2)+2038497/2850192752*arctan(1/2
3*(1-4*x)*23^(1/2))*23^(1/2)+246757/225120016*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \[ \frac {9665-1446 x}{512072 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}-\frac {252815 x+2328909}{174616552 \left (5 x^2+3 x+2\right )}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}+\frac {181 \log \left (2 x^2-x+3\right )}{468512}-\frac {181 \log \left (5 x^2+3 x+2\right )}{468512}+\frac {2038497 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{7261936 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2328909 + 252815*x)/(174616552*(2 + 3*x + 5*x^2)) + (13 - 6*x)/(1012*(3 - x + 2*x^2)^2*(2 + 3*x + 5*x^2)) +
(9665 - 1446*x)/(512072*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)) + (2038497*ArcTan[(1 - 4*x)/Sqrt[23]])/(123921424*S
qrt[23]) + (246757*ArcTan[(3 + 10*x)/Sqrt[31]])/(7261936*Sqrt[31]) + (181*Log[3 - x + 2*x^2])/468512 - (181*Lo
g[2 + 3*x + 5*x^2])/468512

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 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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-4081-3168 x+1650 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2} \, dx}{11132}\\ &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-10650299-21902089 x+2624490 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{61960712}\\ &=-\frac {2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-14070251228+23317637266 x+1345987060 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{464829261424}\\ &=-\frac {2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}-\frac {\int \frac {968672366266-173830777516 x}{3-x+2 x^2} \, dx}{112488681264608}-\frac {\int \frac {-1780781843236+434576943790 x}{2+3 x+5 x^2} \, dx}{112488681264608}\\ &=-\frac {2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {181 \int \frac {-1+4 x}{3-x+2 x^2} \, dx}{468512}-\frac {181 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{468512}-\frac {2038497 \int \frac {1}{3-x+2 x^2} \, dx}{247842848}+\frac {246757 \int \frac {1}{2+3 x+5 x^2} \, dx}{14523872}\\ &=-\frac {2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {181 \log \left (3-x+2 x^2\right )}{468512}-\frac {181 \log \left (2+3 x+5 x^2\right )}{468512}+\frac {2038497 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{123921424}-\frac {246757 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{7261936}\\ &=-\frac {2328909+252815 x}{174616552 \left (2+3 x+5 x^2\right )}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )}+\frac {9665-1446 x}{512072 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )}+\frac {2038497 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{7261936 \sqrt {31}}+\frac {181 \log \left (3-x+2 x^2\right )}{468512}-\frac {181 \log \left (2+3 x+5 x^2\right )}{468512}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 136, normalized size = 0.85 \[ \frac {-2923 x-1782}{1408198 \left (2 x^2-x+3\right )}+\frac {1235 x-1474}{330088 \left (5 x^2+3 x+2\right )}+\frac {-14 x-31}{22264 \left (2 x^2-x+3\right )^2}+\frac {181 \log \left (2 x^2-x+3\right )}{468512}-\frac {181 \log \left (5 x^2+3 x+2\right )}{468512}-\frac {2038497 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{123921424 \sqrt {23}}+\frac {246757 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{7261936 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-31 - 14*x)/(22264*(3 - x + 2*x^2)^2) + (-1782 - 2923*x)/(1408198*(3 - x + 2*x^2)) + (-1474 + 1235*x)/(330088
*(2 + 3*x + 5*x^2)) - (2038497*ArcTan[(-1 + 4*x)/Sqrt[23]])/(123921424*Sqrt[23]) + (246757*ArcTan[(3 + 10*x)/S
qrt[31]])/(7261936*Sqrt[31]) + (181*Log[3 - x + 2*x^2])/468512 - (181*Log[2 + 3*x + 5*x^2])/468512

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fricas [A]  time = 0.87, size = 227, normalized size = 1.42 \[ -\frac {31725248720 \, x^{5} + 260524883872 \, x^{4} - 158204886268 \, x^{3} - 6004584838 \, \sqrt {31} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 3917991234 \, \sqrt {23} {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 679966484692 \, x^{2} + 2116340147 \, {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 2116340147 \, {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )} \log \left (2 \, x^{2} - x + 3\right ) - 184712689040 \, x + 277008109136}{5478070469344 \, {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

-1/5478070469344*(31725248720*x^5 + 260524883872*x^4 - 158204886268*x^3 - 6004584838*sqrt(31)*(20*x^6 - 8*x^5
+ 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*arctan(1/31*sqrt(31)*(10*x + 3)) + 3917991234*sqrt(23)*(20*x^6 - 8*x^5 +
61*x^4 + x^3 + 53*x^2 + 15*x + 18)*arctan(1/23*sqrt(23)*(4*x - 1)) + 679966484692*x^2 + 2116340147*(20*x^6 - 8
*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*log(5*x^2 + 3*x + 2) - 2116340147*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 5
3*x^2 + 15*x + 18)*log(2*x^2 - x + 3) - 184712689040*x + 277008109136)/(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2
+ 15*x + 18)

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giac [A]  time = 0.22, size = 110, normalized size = 0.69 \[ \frac {246757}{225120016} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2038497}{2850192752} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1011260 \, x^{5} + 8304376 \, x^{4} - 5042869 \, x^{3} + 21674311 \, x^{2} - 5887820 \, x + 8829788}{174616552 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x^{2} - x + 3\right )}^{2}} - \frac {181}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {181}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \]

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)^2,x, algorithm="giac")

[Out]

246757/225120016*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 2038497/2850192752*sqrt(23)*arctan(1/23*sqrt(23)*
(4*x - 1)) - 1/174616552*(1011260*x^5 + 8304376*x^4 - 5042869*x^3 + 21674311*x^2 - 5887820*x + 8829788)/((5*x^
2 + 3*x + 2)*(2*x^2 - x + 3)^2) - 181/468512*log(5*x^2 + 3*x + 2) + 181/468512*log(2*x^2 - x + 3)

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maple [A]  time = 0.01, size = 106, normalized size = 0.66 \[ \frac {246757 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{225120016}-\frac {2038497 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{2850192752}+\frac {181 \ln \left (2 x^{2}-x +3\right )}{468512}-\frac {181 \ln \left (5 x^{2}+3 x +2\right )}{468512}-\frac {-\frac {5434 x}{31}+\frac {32428}{155}}{234256 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}+\frac {-\frac {128612}{529} x^{3}-\frac {14102}{529} x^{2}-\frac {173195}{529} x -\frac {321497}{1058}}{58564 \left (2 x^{2}-x +3\right )^{2}} \]

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)^2,x)

[Out]

-1/234256*(-5434/31*x+32428/155)/(x^2+3/5*x+2/5)-181/468512*ln(5*x^2+3*x+2)+246757/225120016*31^(1/2)*arctan(1
/31*(10*x+3)*31^(1/2))+1/58564*(-128612/529*x^3-14102/529*x^2-173195/529*x-321497/1058)/(2*x^2-x+3)^2+181/4685
12*ln(2*x^2-x+3)-2038497/2850192752*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))

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maxima [A]  time = 0.97, size = 116, normalized size = 0.72 \[ \frac {246757}{225120016} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2038497}{2850192752} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1011260 \, x^{5} + 8304376 \, x^{4} - 5042869 \, x^{3} + 21674311 \, x^{2} - 5887820 \, x + 8829788}{174616552 \, {\left (20 \, x^{6} - 8 \, x^{5} + 61 \, x^{4} + x^{3} + 53 \, x^{2} + 15 \, x + 18\right )}} - \frac {181}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {181}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \]

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)^2,x, algorithm="maxima")

[Out]

246757/225120016*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 2038497/2850192752*sqrt(23)*arctan(1/23*sqrt(23)*
(4*x - 1)) - 1/174616552*(1011260*x^5 + 8304376*x^4 - 5042869*x^3 + 21674311*x^2 - 5887820*x + 8829788)/(20*x^
6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18) - 181/468512*log(5*x^2 + 3*x + 2) + 181/468512*log(2*x^2 - x +
3)

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mupad [B]  time = 3.60, size = 136, normalized size = 0.85 \[ -\frac {\frac {50563\,x^5}{174616552}+\frac {1038047\,x^4}{436541380}-\frac {5042869\,x^3}{3492331040}+\frac {21674311\,x^2}{3492331040}-\frac {294391\,x}{174616552}+\frac {200677}{79371160}}{x^6-\frac {2\,x^5}{5}+\frac {61\,x^4}{20}+\frac {x^3}{20}+\frac {53\,x^2}{20}+\frac {3\,x}{4}+\frac {9}{10}}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {181}{468512}+\frac {\sqrt {31}\,246757{}\mathrm {i}}{450240032}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {181}{468512}+\frac {\sqrt {31}\,246757{}\mathrm {i}}{450240032}\right )+\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {181}{468512}+\frac {\sqrt {23}\,2038497{}\mathrm {i}}{5700385504}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {181}{468512}+\frac {\sqrt {23}\,2038497{}\mathrm {i}}{5700385504}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*246757i)/450240032 - 181/468512) - log(x - (31^(1/2)*1i)/10 + 3/10
)*((31^(1/2)*246757i)/450240032 + 181/468512) - ((21674311*x^2)/3492331040 - (294391*x)/174616552 - (5042869*x
^3)/3492331040 + (1038047*x^4)/436541380 + (50563*x^5)/174616552 + 200677/79371160)/((3*x)/4 + (53*x^2)/20 + x
^3/20 + (61*x^4)/20 - (2*x^5)/5 + x^6 + 9/10) + log(x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*2038497i)/5700385504
+ 181/468512) - log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*2038497i)/5700385504 - 181/468512)

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sympy [A]  time = 0.43, size = 143, normalized size = 0.89 \[ \frac {- 1011260 x^{5} - 8304376 x^{4} + 5042869 x^{3} - 21674311 x^{2} + 5887820 x - 8829788}{3492331040 x^{6} - 1396932416 x^{5} + 10651609672 x^{4} + 174616552 x^{3} + 9254677256 x^{2} + 2619248280 x + 3143097936} + \frac {181 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{468512} - \frac {181 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{468512} - \frac {2038497 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{2850192752} + \frac {246757 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{225120016} \]

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)**2,x)

[Out]

(-1011260*x**5 - 8304376*x**4 + 5042869*x**3 - 21674311*x**2 + 5887820*x - 8829788)/(3492331040*x**6 - 1396932
416*x**5 + 10651609672*x**4 + 174616552*x**3 + 9254677256*x**2 + 2619248280*x + 3143097936) + 181*log(x**2 - x
/2 + 3/2)/468512 - 181*log(x**2 + 3*x/5 + 2/5)/468512 - 2038497*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/2
850192752 + 246757*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/225120016

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