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

Optimal. Leaf size=148 \[ \frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {3996965 x+1765599}{235352744 \left (5 x^2+3 x+2\right )}+\frac {5765 x-9446}{690184 \left (5 x^2+3 x+2\right )^2}+\frac {97 \log \left (2 x^2-x+3\right )}{468512}-\frac {97 \log \left (5 x^2+3 x+2\right )}{468512}-\frac {25557 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{225120016 \sqrt {31}} \]

[Out]

1/690184*(-9446+5765*x)/(5*x^2+3*x+2)^2+1/506*(13-6*x)/(2*x^2-x+3)/(5*x^2+3*x+2)^2+1/235352744*(1765599+399696
5*x)/(5*x^2+3*x+2)+97/468512*ln(2*x^2-x+3)-97/468512*ln(5*x^2+3*x+2)-25557/123921424*arctan(1/23*(1-4*x)*23^(1
/2))*23^(1/2)+4464079/6978720496*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 148, 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} \[ -\frac {9446-5765 x}{690184 \left (5 x^2+3 x+2\right )^2}+\frac {3996965 x+1765599}{235352744 \left (5 x^2+3 x+2\right )}+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {97 \log \left (2 x^2-x+3\right )}{468512}-\frac {97 \log \left (5 x^2+3 x+2\right )}{468512}-\frac {25557 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{225120016 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(9446 - 5765*x)/(690184*(2 + 3*x + 5*x^2)^2) + (13 - 6*x)/(506*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^2) + (176559
9 + 3996965*x)/(235352744*(2 + 3*x + 5*x^2)) - (25557*ArcTan[(1 - 4*x)/Sqrt[23]])/(5387888*Sqrt[23]) + (446407
9*ArcTan[(3 + 10*x)/Sqrt[31]])/(225120016*Sqrt[31]) + (97*Log[3 - x + 2*x^2])/468512 - (97*Log[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 )^2 \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-2750-3531 x+1650 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx}{5566}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-8251111+12910579 x-4185390 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{83512264}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-20180265292+4607727674 x-21279841660 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{626509004528}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-328196843326-125560688924 x}{3-x+2 x^2} \, dx}{151615179095776}-\frac {\int \frac {-1409076838004+313901722310 x}{2+3 x+5 x^2} \, dx}{151615179095776}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}+\frac {97 \int \frac {-1+4 x}{3-x+2 x^2} \, dx}{468512}-\frac {97 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{468512}+\frac {25557 \int \frac {1}{3-x+2 x^2} \, dx}{10775776}+\frac {4464079 \int \frac {1}{2+3 x+5 x^2} \, dx}{450240032}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}+\frac {97 \log \left (3-x+2 x^2\right )}{468512}-\frac {97 \log \left (2+3 x+5 x^2\right )}{468512}-\frac {25557 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{5387888}-\frac {4464079 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{225120016}\\ &=-\frac {9446-5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}-\frac {25557 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{225120016 \sqrt {31}}+\frac {97 \log \left (3-x+2 x^2\right )}{468512}-\frac {97 \log \left (2+3 x+5 x^2\right )}{468512}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 136, normalized size = 0.92 \[ \frac {90 x-11}{244904 \left (2 x^2-x+3\right )}+\frac {164380 x+67573}{10232728 \left (5 x^2+3 x+2\right )}+\frac {345 x-98}{30008 \left (5 x^2+3 x+2\right )^2}+\frac {97 \log \left (2 x^2-x+3\right )}{468512}-\frac {97 \log \left (5 x^2+3 x+2\right )}{468512}+\frac {25557 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{225120016 \sqrt {31}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11 + 90*x)/(244904*(3 - x + 2*x^2)) + (-98 + 345*x)/(30008*(2 + 3*x + 5*x^2)^2) + (67573 + 164380*x)/(102327
28*(2 + 3*x + 5*x^2)) + (25557*ArcTan[(-1 + 4*x)/Sqrt[23]])/(5387888*Sqrt[23]) + (4464079*ArcTan[(3 + 10*x)/Sq
rt[31]])/(225120016*Sqrt[31]) + (97*Log[3 - x + 2*x^2])/468512 - (97*Log[2 + 3*x + 5*x^2])/468512

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fricas [A]  time = 1.07, size = 237, normalized size = 1.60 \[ \frac {1253927859800 \, x^{5} + 679296504260 \, x^{4} + 2185021181068 \, x^{3} + 4722995582 \, \sqrt {31} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 1522737174 \, \sqrt {23} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 1500218514344 \, x^{2} - 1528665583 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1528665583 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \log \left (2 \, x^{2} - x + 3\right ) + 1338609358240 \, x + 218880812656}{7383486284768 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/7383486284768*(1253927859800*x^5 + 679296504260*x^4 + 2185021181068*x^3 + 4722995582*sqrt(31)*(50*x^6 + 35*x
^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1522737174*sqrt(23)*(50*x^6 + 3
5*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1500218514344*x^2 - 152866558
3*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*log(5*x^2 + 3*x + 2) + 1528665583*(50*x^6 + 35*x^5
+ 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*log(2*x^2 - x + 3) + 1338609358240*x + 218880812656)/(50*x^6 + 35*x^
5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)

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giac [A]  time = 0.20, size = 110, normalized size = 0.74 \[ \frac {4464079}{6978720496} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {25557}{123921424} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {39969650 \, x^{5} + 21652955 \, x^{4} + 69648769 \, x^{3} + 47820302 \, x^{2} + 42668920 \, x + 6976948}{235352744 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x^{2} - x + 3\right )}} - \frac {97}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {97}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4464079/6978720496*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 25557/123921424*sqrt(23)*arctan(1/23*sqrt(23)*(
4*x - 1)) + 1/235352744*(39969650*x^5 + 21652955*x^4 + 69648769*x^3 + 47820302*x^2 + 42668920*x + 6976948)/((5
*x^2 + 3*x + 2)^2*(2*x^2 - x + 3)) - 97/468512*log(5*x^2 + 3*x + 2) + 97/468512*log(2*x^2 - x + 3)

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maple [A]  time = 0.01, size = 106, normalized size = 0.72 \[ \frac {4464079 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{6978720496}+\frac {25557 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{123921424}+\frac {97 \ln \left (2 x^{2}-x +3\right )}{468512}-\frac {97 \ln \left (5 x^{2}+3 x +2\right )}{468512}-\frac {25 \left (-\frac {723272}{961} x^{3}-\frac {3656422}{4805} x^{2}-\frac {14280728}{24025} x -\frac {2238016}{24025}\right )}{234256 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {\frac {990 x}{23}-\frac {121}{23}}{234256 x^{2}-117128 x +351384} \]

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)^3,x)

[Out]

-25/234256*(-723272/961*x^3-3656422/4805*x^2-14280728/24025*x-2238016/24025)/(5*x^2+3*x+2)^2-97/468512*ln(5*x^
2+3*x+2)+4464079/6978720496*31^(1/2)*arctan(1/31*(10*x+3)*31^(1/2))+1/234256*(990/23*x-121/23)/(x^2-1/2*x+3/2)
+97/468512*ln(2*x^2-x+3)+25557/123921424*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))

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maxima [A]  time = 0.97, size = 118, normalized size = 0.80 \[ \frac {4464079}{6978720496} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {25557}{123921424} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {39969650 \, x^{5} + 21652955 \, x^{4} + 69648769 \, x^{3} + 47820302 \, x^{2} + 42668920 \, x + 6976948}{235352744 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )}} - \frac {97}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {97}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4464079/6978720496*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 25557/123921424*sqrt(23)*arctan(1/23*sqrt(23)*(
4*x - 1)) + 1/235352744*(39969650*x^5 + 21652955*x^4 + 69648769*x^3 + 47820302*x^2 + 42668920*x + 6976948)/(50
*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12) - 97/468512*log(5*x^2 + 3*x + 2) + 97/468512*log(2*x^2
- x + 3)

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mupad [B]  time = 3.59, size = 135, normalized size = 0.91 \[ \frac {\frac {799393\,x^5}{235352744}+\frac {4330591\,x^4}{2353527440}+\frac {69648769\,x^3}{11767637200}+\frac {23910151\,x^2}{5883818600}+\frac {1066723\,x}{294190930}+\frac {158567}{267446300}}{x^6+\frac {7\,x^5}{10}+\frac {103\,x^4}{50}+\frac {17\,x^3}{10}+\frac {83\,x^2}{50}+\frac {16\,x}{25}+\frac {6}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {97}{468512}+\frac {\sqrt {23}\,25557{}\mathrm {i}}{247842848}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {97}{468512}+\frac {\sqrt {23}\,25557{}\mathrm {i}}{247842848}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {97}{468512}+\frac {\sqrt {31}\,4464079{}\mathrm {i}}{13957440992}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {97}{468512}+\frac {\sqrt {31}\,4464079{}\mathrm {i}}{13957440992}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*25557i)/247842848 + 97/468512) - log(x - (23^(1/2)*1i)/4 - 1/4)*((23
^(1/2)*25557i)/247842848 - 97/468512) + ((1066723*x)/294190930 + (23910151*x^2)/5883818600 + (69648769*x^3)/11
767637200 + (4330591*x^4)/2353527440 + (799393*x^5)/235352744 + 158567/267446300)/((16*x)/25 + (83*x^2)/50 + (
17*x^3)/10 + (103*x^4)/50 + (7*x^5)/10 + x^6 + 6/25) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*4464079i)/1
3957440992 + 97/468512) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*4464079i)/13957440992 - 97/468512)

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sympy [A]  time = 0.40, size = 143, normalized size = 0.97 \[ \frac {39969650 x^{5} + 21652955 x^{4} + 69648769 x^{3} + 47820302 x^{2} + 42668920 x + 6976948}{11767637200 x^{6} + 8237346040 x^{5} + 24241332632 x^{4} + 20004983240 x^{3} + 19534277752 x^{2} + 7531287808 x + 2824232928} + \frac {97 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{468512} - \frac {97 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{468512} + \frac {25557 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{123921424} + \frac {4464079 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{6978720496} \]

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)**3,x)

[Out]

(39969650*x**5 + 21652955*x**4 + 69648769*x**3 + 47820302*x**2 + 42668920*x + 6976948)/(11767637200*x**6 + 823
7346040*x**5 + 24241332632*x**4 + 20004983240*x**3 + 19534277752*x**2 + 7531287808*x + 2824232928) + 97*log(x*
*2 - x/2 + 3/2)/468512 - 97*log(x**2 + 3*x/5 + 2/5)/468512 + 25557*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23
)/123921424 + 4464079*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/6978720496

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