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

Optimal. Leaf size=269 \[ -\frac{1134826571-1504660754 x}{476353953856 \sqrt{2 x^2-x+3}}+\frac{86885 x+46386}{1860496 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}-\frac{12280939-19536786 x}{2824232928 \left (2 x^2-x+3\right )^{3/2}}+\frac{65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}+\frac{35 \sqrt{\frac{1}{682} \left (2243059557247+2011748500000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2243059557247+2011748500000 \sqrt{2}\right )}} \left (\left (6290431+3861685 \sqrt{2}\right ) x+2428746 \sqrt{2}+1432939\right )}{\sqrt{2 x^2-x+3}}\right )}{1800960128}-\frac{35 \sqrt{\frac{1}{682} \left (2011748500000 \sqrt{2}-2243059557247\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2011748500000 \sqrt{2}-2243059557247\right )}} \left (\left (6290431-3861685 \sqrt{2}\right ) x-2428746 \sqrt{2}+1432939\right )}{\sqrt{2 x^2-x+3}}\right )}{1800960128} \]

[Out]

-(12280939 - 19536786*x)/(2824232928*(3 - x + 2*x^2)^(3/2)) - (1134826571 - 1504660754*x)/(476353953856*Sqrt[3
- x + 2*x^2]) + (4 + 65*x)/(1364*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2) + (46386 + 86885*x)/(1860496*(3 -
x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (35*Sqrt[(2243059557247 + 2011748500000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(
31*(2243059557247 + 2011748500000*Sqrt[2]))]*(1432939 + 2428746*Sqrt[2] + (6290431 + 3861685*Sqrt[2])*x))/Sqrt
[3 - x + 2*x^2]])/1800960128 - (35*Sqrt[(-2243059557247 + 2011748500000*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-2
243059557247 + 2011748500000*Sqrt[2]))]*(1432939 - 2428746*Sqrt[2] + (6290431 - 3861685*Sqrt[2])*x))/Sqrt[3 -
x + 2*x^2]])/1800960128

________________________________________________________________________________________

Rubi [A]  time = 0.589435, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {974, 1060, 1035, 1029, 206, 204} \[ -\frac{1134826571-1504660754 x}{476353953856 \sqrt{2 x^2-x+3}}+\frac{86885 x+46386}{1860496 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}-\frac{12280939-19536786 x}{2824232928 \left (2 x^2-x+3\right )^{3/2}}+\frac{65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}+\frac{35 \sqrt{\frac{1}{682} \left (2243059557247+2011748500000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2243059557247+2011748500000 \sqrt{2}\right )}} \left (\left (6290431+3861685 \sqrt{2}\right ) x+2428746 \sqrt{2}+1432939\right )}{\sqrt{2 x^2-x+3}}\right )}{1800960128}-\frac{35 \sqrt{\frac{1}{682} \left (2011748500000 \sqrt{2}-2243059557247\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2011748500000 \sqrt{2}-2243059557247\right )}} \left (\left (6290431-3861685 \sqrt{2}\right ) x-2428746 \sqrt{2}+1432939\right )}{\sqrt{2 x^2-x+3}}\right )}{1800960128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(12280939 - 19536786*x)/(2824232928*(3 - x + 2*x^2)^(3/2)) - (1134826571 - 1504660754*x)/(476353953856*Sqrt[3
- x + 2*x^2]) + (4 + 65*x)/(1364*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2) + (46386 + 86885*x)/(1860496*(3 -
x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (35*Sqrt[(2243059557247 + 2011748500000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(
31*(2243059557247 + 2011748500000*Sqrt[2]))]*(1432939 + 2428746*Sqrt[2] + (6290431 + 3861685*Sqrt[2])*x))/Sqrt
[3 - x + 2*x^2]])/1800960128 - (35*Sqrt[(-2243059557247 + 2011748500000*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-2
243059557247 + 2011748500000*Sqrt[2]))]*(1432939 - 2428746*Sqrt[2] + (6290431 - 3861685*Sqrt[2])*x))/Sqrt[3 -
x + 2*x^2]])/1800960128

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 1035

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}-\frac{\int \frac{-5687+\frac{8635 x}{2}-8580 x^2}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{27962737}{2}-\frac{34457291 x}{4}-42052340 x^2}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=-\frac{12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}+\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{119979599619}{4}-\frac{16689617967 x}{8}-65008655415 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{939763506792}\\ &=-\frac{12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac{1134826571-1504660754 x}{476353953856 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{107038717723245}{8}+\frac{333226839035475 x}{16}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2615361839402136}\\ &=-\frac{12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac{1134826571-1504660754 x}{476353953856 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\frac{8945577795}{16} \left (672997+263242 \sqrt{2}\right )+\frac{8945577795}{16} \left (146513-409755 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{57537960466846992 \sqrt{2}}-\frac{\int \frac{\frac{8945577795}{16} \left (672997-263242 \sqrt{2}\right )+\frac{8945577795}{16} \left (146513+409755 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{57537960466846992 \sqrt{2}}\\ &=-\frac{12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac{1134826571-1504660754 x}{476353953856 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\left (21384825 \left (4023497000000-2243059557247 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2480724224678308922775}{256} \left (2243059557247-2011748500000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{8945577795}{16} \left (1432939-2428746 \sqrt{2}\right )+\frac{8945577795}{16} \left (6290431-3861685 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3936256}-\frac{\left (21384825 \left (4023497000000+2243059557247 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2480724224678308922775}{256} \left (2243059557247+2011748500000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{8945577795}{16} \left (1432939+2428746 \sqrt{2}\right )+\frac{8945577795}{16} \left (6290431+3861685 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3936256}\\ &=-\frac{12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac{1134826571-1504660754 x}{476353953856 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac{46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac{35 \sqrt{\frac{1}{682} \left (2243059557247+2011748500000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2243059557247+2011748500000 \sqrt{2}\right )}} \left (1432939+2428746 \sqrt{2}+\left (6290431+3861685 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1800960128}-\frac{35 \sqrt{\frac{1}{682} \left (-2243059557247+2011748500000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-2243059557247+2011748500000 \sqrt{2}\right )}} \left (1432939-2428746 \sqrt{2}+\left (6290431-3861685 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1800960128}\\ \end{align*}

Mathematica [C]  time = 1.97068, size = 242, normalized size = 0.9 \[ \frac{\frac{5456 \left (225699113100 x^7-12234606480 x^6+592923725931 x^5+174241614961 x^4+519223213785 x^3+178650961091 x^2+218659985088 x+9739335532\right )}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}+11109 \sqrt{286+22 i \sqrt{31}} \left (4541903-6290431 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )-11109 i \sqrt{286-22 i \sqrt{31}} \left (6290431 \sqrt{31}-4541903 i\right ) \tanh ^{-1}\left (\frac{\left (22-4 i \sqrt{31}\right ) x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{7796961516715008} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5456*(9739335532 + 218659985088*x + 178650961091*x^2 + 519223213785*x^3 + 174241614961*x^4 + 592923725931*x^
5 - 12234606480*x^6 + 225699113100*x^7))/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2) + 11109*Sqrt[286 + (22*I)
*Sqrt[31]]*(4541903 - (6290431*I)*Sqrt[31])*ArcTanh[(63 + I*Sqrt[31] + (-22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 +
(22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] - (11109*I)*Sqrt[286 - (22*I)*Sqrt[31]]*(-4541903*I + 6290431*Sqrt[31]
)*ArcTanh[(-63 + I*Sqrt[31] + (22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/7
796961516715008

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Maple [B]  time = 0.277, size = 19014, normalized size = 70.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 5.31818, size = 11142, normalized size = 41.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/611377875290135815296770157063555072*(2164988593398757980*129508224872072^(1/4)*sqrt(4023497)*sqrt(341)*sqrt
(2)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*sqrt(2243059557247*sqrt(2
) + 4023497000000)*arctan(1/452534011574628261925237033857859439*(11475013444*sqrt(4023497)*(11*12950822487207
2^(3/4)*sqrt(341)*(2673027292*x^7 - 11768684222*x^6 + 24008796626*x^5 - 42687622824*x^4 + 22428040912*x^3 - 12
956821056*x^2 - sqrt(2)*(2612082154*x^7 - 9010050347*x^6 + 19426337114*x^5 - 28170626609*x^4 + 13394761640*x^3
- 4698131400*x^2 - 17594323200*x + 10110341376) - 20220682752*x + 17594323200) + 124728407*129508224872072^(1
/4)*sqrt(341)*(214583731*x^7 - 3372306249*x^6 + 18434388344*x^5 - 43845503580*x^4 + 57631717152*x^3 - 41786349
984*x^2 - sqrt(2)*(190078101*x^7 - 2862100476*x^6 + 14688003420*x^5 - 32231022496*x^4 + 40927641120*x^3 - 2195
9568000*x^2 - 31156503552*x + 19060075008) - 38120150016*x + 31156503552))*sqrt(2*x^2 - x + 3)*sqrt(2243059557
247*sqrt(2) + 4023497000000) + 1284612678018299582239382547725536472*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7
+ 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^
6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(804
6994/10139750351)*(sqrt(4023497)*(11*129508224872072^(3/4)*sqrt(341)*(8140972152*x^7 - 11907581308*x^6 + 39777
303828*x^5 - 24395365568*x^4 + 37103094432*x^3 - 1836165888*x^2 - sqrt(2)*(10387383478*x^7 - 14753211883*x^6 +
46462095753*x^5 - 11926110640*x^4 + 8224291080*x^3 + 34793549568*x^2 - 34793549568*x) + 1836165888*x) + 12472
8407*129508224872072^(1/4)*sqrt(341)*(692762453*x^7 - 8972954292*x^6 + 34803726780*x^5 - 46915651008*x^4 + 674
21983392*x^3 + 10625375232*x^2 - 2*sqrt(2)*(367903387*x^7 - 4754813452*x^6 + 18261523780*x^5 - 22991417280*x^4
+ 27054001440*x^3 + 26759248128*x^2 - 26759248128*x) - 10625375232*x))*sqrt(2*x^2 - x + 3)*sqrt(2243059557247
*sqrt(2) + 4023497000000) + 111948686098489209076292438*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^
6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 70737
4*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) + 5088576640840418594376929*sqrt(31)
*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 1548
8*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt
(-(129508224872072^(1/4)*sqrt(4023497)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(643213*x + 2195288) -
2838501*x + 1552075)*sqrt(2243059557247*sqrt(2) + 4023497000000) - 1921101946251381781783*x^2 - 17250711354094
04048948*sqrt(2)*(2*x^2 - x + 3) + 5920130487427727531617*x - 7841232433679109313400)/x^2) + 14597871341117040
707265710769608369*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300
096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 108
0*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 135
62944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 2164988593398757980*129508224872072^(1/4)*
sqrt(4023497)*sqrt(341)*sqrt(2)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 3
6)*sqrt(2243059557247*sqrt(2) + 4023497000000)*arctan(1/452534011574628261925237033857859439*(11475013444*sqrt
(4023497)*(11*129508224872072^(3/4)*sqrt(341)*(2673027292*x^7 - 11768684222*x^6 + 24008796626*x^5 - 4268762282
4*x^4 + 22428040912*x^3 - 12956821056*x^2 - sqrt(2)*(2612082154*x^7 - 9010050347*x^6 + 19426337114*x^5 - 28170
626609*x^4 + 13394761640*x^3 - 4698131400*x^2 - 17594323200*x + 10110341376) - 20220682752*x + 17594323200) +
124728407*129508224872072^(1/4)*sqrt(341)*(214583731*x^7 - 3372306249*x^6 + 18434388344*x^5 - 43845503580*x^4
+ 57631717152*x^3 - 41786349984*x^2 - sqrt(2)*(190078101*x^7 - 2862100476*x^6 + 14688003420*x^5 - 32231022496*
x^4 + 40927641120*x^3 - 21959568000*x^2 - 31156503552*x + 19060075008) - 38120150016*x + 31156503552))*sqrt(2*
x^2 - x + 3)*sqrt(2243059557247*sqrt(2) + 4023497000000) - 1284612678018299582239382547725536472*sqrt(31)*sqrt
(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*
x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154
304*x - 456192) - 2*sqrt(8046994/10139750351)*(sqrt(4023497)*(11*129508224872072^(3/4)*sqrt(341)*(8140972152*x
^7 - 11907581308*x^6 + 39777303828*x^5 - 24395365568*x^4 + 37103094432*x^3 - 1836165888*x^2 - sqrt(2)*(1038738
3478*x^7 - 14753211883*x^6 + 46462095753*x^5 - 11926110640*x^4 + 8224291080*x^3 + 34793549568*x^2 - 3479354956
8*x) + 1836165888*x) + 124728407*129508224872072^(1/4)*sqrt(341)*(692762453*x^7 - 8972954292*x^6 + 34803726780
*x^5 - 46915651008*x^4 + 67421983392*x^3 + 10625375232*x^2 - 2*sqrt(2)*(367903387*x^7 - 4754813452*x^6 + 18261
523780*x^5 - 22991417280*x^4 + 27054001440*x^3 + 26759248128*x^2 - 26759248128*x) - 10625375232*x))*sqrt(2*x^2
- x + 3)*sqrt(2243059557247*sqrt(2) + 4023497000000) - 111948686098489209076292438*sqrt(31)*sqrt(2)*(123408*x
^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118
051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 508857
6640840418594376929*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 7421932
8*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 -
1944*x) + 144820224*x))*sqrt((129508224872072^(1/4)*sqrt(4023497)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt
(2)*(643213*x + 2195288) - 2838501*x + 1552075)*sqrt(2243059557247*sqrt(2) + 4023497000000) + 1921101946251381
781783*x^2 + 1725071135409404048948*sqrt(2)*(2*x^2 - x + 3) - 5920130487427727531617*x + 784123243367910931340
0)/x^2) - 14597871341117040707265710769608369*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x
^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 +
15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 1419
1920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 55545*129508224
872072^(1/4)*sqrt(4023497)*sqrt(341)*sqrt(31)*(402349700000000*x^8 + 80469940000000*x^7 + 1291542537000000*x^6
+ 692041484000000*x^5 + 1569163830000000*x^4 + 949545292000000*x^3 + 969662777000000*x^2 - 2243059557247*sqrt
(2)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36) + 337973748000000*x + 144
845892000000)*sqrt(2243059557247*sqrt(2) + 4023497000000)*log(2464391912500000000000/10139750351*(129508224872
072^(1/4)*sqrt(4023497)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(643213*x + 2195288) - 2838501*x + 155
2075)*sqrt(2243059557247*sqrt(2) + 4023497000000) + 1921101946251381781783*x^2 + 1725071135409404048948*sqrt(2
)*(2*x^2 - x + 3) - 5920130487427727531617*x + 7841232433679109313400)/x^2) - 55545*129508224872072^(1/4)*sqrt
(4023497)*sqrt(341)*sqrt(31)*(402349700000000*x^8 + 80469940000000*x^7 + 1291542537000000*x^6 + 69204148400000
0*x^5 + 1569163830000000*x^4 + 949545292000000*x^3 + 969662777000000*x^2 - 2243059557247*sqrt(2)*(100*x^8 + 20
*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36) + 337973748000000*x + 144845892000000)*sqr
t(2243059557247*sqrt(2) + 4023497000000)*log(-2464391912500000000000/10139750351*(129508224872072^(1/4)*sqrt(4
023497)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(643213*x + 2195288) - 2838501*x + 1552075)*sqrt(22430
59557247*sqrt(2) + 4023497000000) - 1921101946251381781783*x^2 - 1725071135409404048948*sqrt(2)*(2*x^2 - x + 3
) + 5920130487427727531617*x - 7841232433679109313400)/x^2) + 427817641581532204139104*(225699113100*x^7 - 122
34606480*x^6 + 592923725931*x^5 + 174241614961*x^4 + 519223213785*x^3 + 178650961091*x^2 + 218659985088*x + 97
39335532)*sqrt(2*x^2 - x + 3))/(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError