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

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

[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

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Rubi [A]  time = 0.161458, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{9446-5765 x}{690184 \left (5 x^2+3 x+2\right )^2}+\frac{3996965 x+1765599}{235352744 \left (5 x^2+3 x+2\right )}+\frac{13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac{97 \log \left (2 x^2-x+3\right )}{468512}-\frac{97 \log \left (5 x^2+3 x+2\right )}{468512}-\frac{25557 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{5387888 \sqrt{23}}+\frac{4464079 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{225120016 \sqrt{31}} \]

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 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 )^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*}

Mathematica [A]  time = 0.0642083, 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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Maple [A]  time = 0.053, size = 106, normalized size = 0.7 \begin{align*} -{\frac{25}{234256\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ( -{\frac{723272\,{x}^{3}}{961}}-{\frac{3656422\,{x}^{2}}{4805}}-{\frac{14280728\,x}{24025}}-{\frac{2238016}{24025}} \right ) }-{\frac{97\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{468512}}+{\frac{4464079\,\sqrt{31}}{6978720496}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }+{\frac{1}{234256} \left ({\frac{990\,x}{23}}-{\frac{121}{23}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}+{\frac{97\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{468512}}+{\frac{25557\,\sqrt{23}}{123921424}\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)^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*arctan(1/31*(3+10*x)*31^(1/2))*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*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.49092, size = 159, normalized size = 1.07 \begin{align*} \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 ) \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)^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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Fricas [A]  time = 1.04071, size = 802, normalized size = 5.42 \begin{align*} \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 )}} \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)^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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Sympy [A]  time = 0.466039, size = 143, normalized size = 0.97 \begin{align*} \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} \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)**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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Giac [A]  time = 1.13189, size = 149, normalized size = 1.01 \begin{align*} \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 ) \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)^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)