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

Optimal. Leaf size=188 \[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (2343727+1678700 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2343727+1678700 \sqrt{2}\right )}} \left (\left (5751+3935 \sqrt{2}\right ) x+1816 \sqrt{2}+2119\right )}{\sqrt{2 x^2-x+3}}\right )}{1364}-\frac{\sqrt{\frac{1}{682} \left (1678700 \sqrt{2}-2343727\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (1678700 \sqrt{2}-2343727\right )}} \left (\left (5751-3935 \sqrt{2}\right ) x-1816 \sqrt{2}+2119\right )}{\sqrt{2 x^2-x+3}}\right )}{1364} \]

[Out]

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(2343727 + 1678700*Sqrt[2])/682]*ArcTan[(Sqrt
[11/(31*(2343727 + 1678700*Sqrt[2]))]*(2119 + 1816*Sqrt[2] + (5751 + 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1
364 - (Sqrt[(-2343727 + 1678700*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-2343727 + 1678700*Sqrt[2]))]*(2119 - 1816
*Sqrt[2] + (5751 - 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1364

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Rubi [A]  time = 0.428841, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {974, 1035, 1029, 206, 204} \[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (2343727+1678700 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2343727+1678700 \sqrt{2}\right )}} \left (\left (5751+3935 \sqrt{2}\right ) x+1816 \sqrt{2}+2119\right )}{\sqrt{2 x^2-x+3}}\right )}{1364}-\frac{\sqrt{\frac{1}{682} \left (1678700 \sqrt{2}-2343727\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (1678700 \sqrt{2}-2343727\right )}} \left (\left (5751-3935 \sqrt{2}\right ) x-1816 \sqrt{2}+2119\right )}{\sqrt{2 x^2-x+3}}\right )}{1364} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(2343727 + 1678700*Sqrt[2])/682]*ArcTan[(Sqrt
[11/(31*(2343727 + 1678700*Sqrt[2]))]*(2119 + 1816*Sqrt[2] + (5751 + 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1
364 - (Sqrt[(-2343727 + 1678700*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-2343727 + 1678700*Sqrt[2]))]*(2119 - 1816
*Sqrt[2] + (5751 - 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1364

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 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}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-1826+\frac{2255 x}{2}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{7502}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{\frac{121}{2} \left (537-332 \sqrt{2}\right )-\frac{121}{2} \left (127-205 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{165044 \sqrt{2}}+\frac{\int \frac{\frac{121}{2} \left (537+332 \sqrt{2}\right )-\frac{121}{2} \left (127+205 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{165044 \sqrt{2}}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac{1}{496} \left (11 \left (3357400-2343727 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{453871}{4} \left (2343727-1678700 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{121}{2} \left (2119-1816 \sqrt{2}\right )+\frac{121}{2} \left (5751-3935 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )-\frac{1}{496} \left (11 \left (3357400+2343727 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{453871}{4} \left (2343727+1678700 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{121}{2} \left (2119+1816 \sqrt{2}\right )+\frac{121}{2} \left (5751+3935 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac{\sqrt{\frac{1}{682} \left (2343727+1678700 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (2343727+1678700 \sqrt{2}\right )}} \left (2119+1816 \sqrt{2}+\left (5751+3935 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1364}-\frac{\sqrt{\frac{1}{682} \left (-2343727+1678700 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-2343727+1678700 \sqrt{2}\right )}} \left (2119-1816 \sqrt{2}+\left (5751-3935 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1364}\\ \end{align*}

Mathematica [C]  time = 1.0074, size = 287, normalized size = 1.53 \[ \frac{25 \left (\frac{i \sqrt{286+22 i \sqrt{31}} \left (224 \sqrt{31}+1023 i\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\left (\sqrt{31}-13 i\right )^2}+\frac{10 i \left (1364 \left (\sqrt{31}+13 i\right ) (65 x+4) \sqrt{2 x^2-x+3}-5 \sqrt{286-22 i \sqrt{31}} \left (787 \sqrt{31}-1271 i\right ) \left (5 x^2+3 x+2\right ) \tanh ^{-1}\left (\frac{\left (-22+4 i \sqrt{31}\right ) x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )\right )}{\left (\sqrt{31}+13 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right ) \left (5 \left (\sqrt{31}-13 i\right ) x+8 \sqrt{31}-4 i\right )}\right )}{116281} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25*((I*Sqrt[286 + (22*I)*Sqrt[31]]*(1023*I + 224*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])])/(-13*I + Sqrt[31])^2 + ((10*I)*(1364*(13*I + Sqrt[31]
)*(4 + 65*x)*Sqrt[3 - x + 2*x^2] - 5*Sqrt[286 - (22*I)*Sqrt[31]]*(-1271*I + 787*Sqrt[31])*(2 + 3*x + 5*x^2)*Ar
cTanh[(63 - I*Sqrt[31] + (-22 + (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/((13
*I + Sqrt[31])^2*(3*I + Sqrt[31] + (10*I)*x)*(-4*I + 8*Sqrt[31] + 5*(-13*I + Sqrt[31])*x))))/116281

________________________________________________________________________________________

Maple [B]  time = 0.153, size = 5225, normalized size = 27.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 5.14088, size = 8625, normalized size = 45.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/263043507934399808*(8422204*563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(2343727*sqr
t(2) + 3357400)*arctan(1/7101900221517254683789*(47876524*sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(2950932*x
^7 - 11691762*x^6 + 24397746*x^5 - 40053004*x^4 + 20309552*x^3 - 10145376*x^2 - sqrt(2)*(2248634*x^7 - 8421787
*x^6 + 17801494*x^5 - 27869789*x^4 + 13808040*x^3 - 6172200*x^2 - 15724800*x + 10596096) - 21192192*x + 157248
00) + 520397*563606738^(1/4)*sqrt(341)*(226651*x^7 - 3496629*x^6 + 18614024*x^5 - 42860780*x^4 + 55586592*x^3
- 36274464*x^2 - sqrt(2)*(168871*x^7 - 2579646*x^6 + 13533020*x^5 - 30582616*x^4 + 39345120*x^3 - 23947200*x^2
- 28449792*x + 19450368) - 38900736*x + 28449792))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) + 2016
0232886887690715272*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 + 39614
4*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(33574/2191)*(sqrt(33574)*(22*563606738^(3/4)*sqrt(34
1)*(10257392*x^7 - 14773368*x^6 + 47877288*x^5 - 20710528*x^4 + 26321472*x^3 + 17079552*x^2 - sqrt(2)*(8292238
*x^7 - 11867543*x^6 + 37968813*x^5 - 13449840*x^4 + 14570280*x^3 + 20176128*x^2 - 20176128*x) - 17079552*x) +
520397*563606738^(1/4)*sqrt(341)*(795513*x^7 - 10292932*x^6 + 39734380*x^5 - 51864768*x^4 + 68281632*x^3 + 342
55872*x^2 - 8*sqrt(2)*(77213*x^7 - 998548*x^6 + 3846220*x^5 - 4943520*x^4 + 6215760*x^3 + 4318272*x^2 - 431827
2*x) - 34255872*x))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) + 421088065768678*sqrt(31)*sqrt(2)*(12
3408*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 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) +
19140366625849*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*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(-(563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1123*x
+ 898) - 2021*x - 225)*sqrt(2343727*sqrt(2) + 3357400) - 1731948347213*x^2 - 1555218924028*sqrt(2)*(2*x^2 - x
+ 3) + 5337228580187*x - 7069176927400)/x^2) + 229093555532814667219*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 533
85560*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 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579
456)) + 8422204*563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(2343727*sqrt(2) + 3357400
)*arctan(1/7101900221517254683789*(47876524*sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(2950932*x^7 - 11691762*
x^6 + 24397746*x^5 - 40053004*x^4 + 20309552*x^3 - 10145376*x^2 - sqrt(2)*(2248634*x^7 - 8421787*x^6 + 1780149
4*x^5 - 27869789*x^4 + 13808040*x^3 - 6172200*x^2 - 15724800*x + 10596096) - 21192192*x + 15724800) + 520397*5
63606738^(1/4)*sqrt(341)*(226651*x^7 - 3496629*x^6 + 18614024*x^5 - 42860780*x^4 + 55586592*x^3 - 36274464*x^2
- sqrt(2)*(168871*x^7 - 2579646*x^6 + 13533020*x^5 - 30582616*x^4 + 39345120*x^3 - 23947200*x^2 - 28449792*x
+ 19450368) - 38900736*x + 28449792))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) - 201602328868876907
15272*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(33574/2191)*(sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(10257392*x
^7 - 14773368*x^6 + 47877288*x^5 - 20710528*x^4 + 26321472*x^3 + 17079552*x^2 - sqrt(2)*(8292238*x^7 - 1186754
3*x^6 + 37968813*x^5 - 13449840*x^4 + 14570280*x^3 + 20176128*x^2 - 20176128*x) - 17079552*x) + 520397*5636067
38^(1/4)*sqrt(341)*(795513*x^7 - 10292932*x^6 + 39734380*x^5 - 51864768*x^4 + 68281632*x^3 + 34255872*x^2 - 8*
sqrt(2)*(77213*x^7 - 998548*x^6 + 3846220*x^5 - 4943520*x^4 + 6215760*x^3 + 4318272*x^2 - 4318272*x) - 3425587
2*x))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) - 421088065768678*sqrt(31)*sqrt(2)*(123408*x^8 - 914
152*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 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 19140366625849
*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*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((563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1123*x + 898) - 2021*x
- 225)*sqrt(2343727*sqrt(2) + 3357400) + 1731948347213*x^2 + 1555218924028*sqrt(2)*(2*x^2 - x + 3) - 53372285
80187*x + 7069176927400)/x^2) - 229093555532814667219*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142
835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 100
70*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^
7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 5636067
38^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*(16787000*x^2 - 2343727*sqrt(2)*(5*x^2 + 3*x + 2) + 10072200*x + 67148
00)*sqrt(2343727*sqrt(2) + 3357400)*log(335740000/2191*(563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*
x^2 - x + 3)*(sqrt(2)*(1123*x + 898) - 2021*x - 225)*sqrt(2343727*sqrt(2) + 3357400) + 1731948347213*x^2 + 155
5218924028*sqrt(2)*(2*x^2 - x + 3) - 5337228580187*x + 7069176927400)/x^2) - 563606738^(1/4)*sqrt(33574)*sqrt(
341)*sqrt(31)*(16787000*x^2 - 2343727*sqrt(2)*(5*x^2 + 3*x + 2) + 10072200*x + 6714800)*sqrt(2343727*sqrt(2) +
3357400)*log(-335740000/2191*(563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(11
23*x + 898) - 2021*x - 225)*sqrt(2343727*sqrt(2) + 3357400) - 1731948347213*x^2 - 1555218924028*sqrt(2)*(2*x^2
- x + 3) + 5337228580187*x - 7069176927400)/x^2) + 385694293158944*sqrt(2*x^2 - x + 3)*(65*x + 4))/(5*x^2 + 3
*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError