3.1.55 \(\int \frac {1}{(3-x+2 x^2)^3 (2+3 x+5 x^2)} \, dx\) [55]

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

[Out]

1/1012*(13-6*x)/(2*x^2-x+3)^2+1/256036*(3625-746*x)/(2*x^2-x+3)-119/21296*ln(2*x^2-x+3)+119/21296*ln(5*x^2+3*x
+2)-53403/129554216*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+247/330088*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {988, 1074, 1086, 648, 632, 210, 642} \begin {gather*} -\frac {53403 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{5632792 \sqrt {23}}+\frac {247 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{10648 \sqrt {31}}+\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} \end {gather*}

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 988

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 1074

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)/((b^2 - 4*a*c)*((c*d - a*f)^
2 - (b*d - a*e)*(c*e - b*f))*(p + 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), 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 1086

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 )} \, 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 \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{5632792}-\frac {247 \text {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*}

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Mathematica [A]
time = 0.12, size = 99, normalized size = 0.86 \begin {gather*} \frac {3310986 \sqrt {23} \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )+6010498 \sqrt {31} \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )+713 \left (-\frac {44 \left (-14164+7381 x-7996 x^2+1492 x^3\right )}{\left (-3+x-2 x^2\right )^2}-62951 \log \left (3-x+2 x^2\right )+62951 \log \left (2+3 x+5 x^2\right )\right )}{8032361392} \end {gather*}

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 = 0.16, size = 89, normalized size = 0.77

method	result	size
default	\(-\frac {\frac {8206}{529} x^{3}-\frac {43978}{529} x^{2}+\frac {81191}{1058} x -\frac {77902}{529}}{2662 \left (2 x^{2}-x +3\right )^{2}}-\frac {119 \ln \left (2 x^{2}-x +3\right )}{21296}+\frac {53403 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{129554216}+\frac {119 \ln \left (5 x^{2}+3 x +2\right )}{21296}+\frac {247 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{330088}\)	\(89\)
risch	\(\frac {-\frac {373}{64009} x^{3}+\frac {1999}{64009} x^{2}-\frac {61}{2116} x +\frac {3541}{64009}}{\left (2 x^{2}-x +3\right )^{2}}-\frac {119 \ln \left (16 x^{2}-8 x +24\right )}{21296}+\frac {53403 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{129554216}+\frac {247 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{330088}+\frac {119 \ln \left (100 x^{2}+60 x +40\right )}{21296}\)	\(89\)

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,method=_RETURNVERBOSE)

[Out]

-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)+53403/129554
216*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))+119/21296*ln(5*x^2+3*x+2)+247/330088*arctan(1/31*(3+10*x)*31^(1/2))
*31^(1/2)

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Maxima [A]
time = 0.52, size = 98, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 2.03, size = 177, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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 = 0.15, size = 122, normalized size = 1.06 \begin {gather*} \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 {gather*}

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 = 1.97, size = 88, normalized size = 0.77 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 3.58, size = 116, normalized size = 1.01 \begin {gather*} -\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {119}{21296}+\frac {\sqrt {31}\,247{}\mathrm {i}}{660176}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {119}{21296}+\frac {\sqrt {31}\,247{}\mathrm {i}}{660176}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {119}{21296}+\frac {\sqrt {23}\,53403{}\mathrm {i}}{259108432}\right )+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {119}{21296}+\frac {\sqrt {23}\,53403{}\mathrm {i}}{259108432}\right )-\frac {\frac {373\,x^3}{256036}-\frac {1999\,x^2}{256036}+\frac {61\,x}{8464}-\frac {3541}{256036}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \end {gather*}

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)),x)

[Out]

log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*247i)/660176 + 119/21296) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^
(1/2)*247i)/660176 - 119/21296) - log(x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*53403i)/259108432 + 119/21296) + l
og(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*53403i)/259108432 - 119/21296) - ((61*x)/8464 - (1999*x^2)/256036 + (
373*x^3)/256036 - 3541/256036)/((13*x^2)/4 - (3*x)/2 - x^3 + x^4 + 9/4)

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