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

Optimal. Leaf size=176 \[ \frac{13-6 x}{253 \sqrt{2 x^2-x+3}}+\frac{1}{22} \sqrt{\frac{1}{682} \left (247+500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (247+500 \sqrt{2}\right )}} \left (\left (69+65 \sqrt{2}\right ) x+4 \sqrt{2}+61\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{22} \sqrt{\frac{1}{682} \left (500 \sqrt{2}-247\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (500 \sqrt{2}-247\right )}} \left (\left (69-65 \sqrt{2}\right ) x-4 \sqrt{2}+61\right )}{\sqrt{2 x^2-x+3}}\right ) \]

[Out]

(13 - 6*x)/(253*Sqrt[3 - x + 2*x^2]) + (Sqrt[(247 + 500*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(247 + 500*Sqrt[2]))
]*(61 + 4*Sqrt[2] + (69 + 65*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/22 - (Sqrt[(-247 + 500*Sqrt[2])/682]*ArcTanh[(
Sqrt[11/(31*(-247 + 500*Sqrt[2]))]*(61 - 4*Sqrt[2] + (69 - 65*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/22

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Rubi [A]  time = 0.407876, antiderivative size = 176, 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} \[ \frac{13-6 x}{253 \sqrt{2 x^2-x+3}}+\frac{1}{22} \sqrt{\frac{1}{682} \left (247+500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (247+500 \sqrt{2}\right )}} \left (\left (69+65 \sqrt{2}\right ) x+4 \sqrt{2}+61\right )}{\sqrt{2 x^2-x+3}}\right )-\frac{1}{22} \sqrt{\frac{1}{682} \left (500 \sqrt{2}-247\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (500 \sqrt{2}-247\right )}} \left (\left (69-65 \sqrt{2}\right ) x-4 \sqrt{2}+61\right )}{\sqrt{2 x^2-x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13 - 6*x)/(253*Sqrt[3 - x + 2*x^2]) + (Sqrt[(247 + 500*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(247 + 500*Sqrt[2]))
]*(61 + 4*Sqrt[2] + (69 + 65*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/22 - (Sqrt[(-247 + 500*Sqrt[2])/682]*ArcTanh[(
Sqrt[11/(31*(-247 + 500*Sqrt[2]))]*(61 - 4*Sqrt[2] + (69 - 65*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/22

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}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx &=\frac{13-6 x}{253 \sqrt{3-x+2 x^2}}-\frac{\int \frac{-1012-\frac{1265 x}{2}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2783}\\ &=\frac{13-6 x}{253 \sqrt{3-x+2 x^2}}+\frac{\int \frac{\frac{2783}{2} \left (3+8 \sqrt{2}\right )-\frac{2783}{2} \left (13-5 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{61226 \sqrt{2}}-\frac{\int \frac{\frac{2783}{2} \left (3-8 \sqrt{2}\right )-\frac{2783}{2} \left (13+5 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{61226 \sqrt{2}}\\ &=\frac{13-6 x}{253 \sqrt{3-x+2 x^2}}-\frac{1}{8} \left (253 \left (1000-247 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{240097759}{4} \left (247-500 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{2783}{2} \left (61-4 \sqrt{2}\right )+\frac{2783}{2} \left (69-65 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )-\frac{1}{8} \left (253 \left (1000+247 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{240097759}{4} \left (247+500 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{2783}{2} \left (61+4 \sqrt{2}\right )+\frac{2783}{2} \left (69+65 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\\ &=\frac{13-6 x}{253 \sqrt{3-x+2 x^2}}+\frac{1}{22} \sqrt{\frac{1}{682} \left (247+500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (247+500 \sqrt{2}\right )}} \left (61+4 \sqrt{2}+\left (69+65 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )-\frac{1}{22} \sqrt{\frac{1}{682} \left (-247+500 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-247+500 \sqrt{2}\right )}} \left (61-4 \sqrt{2}+\left (69-65 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 1.36352, size = 202, normalized size = 1.15 \[ \frac{-\frac{27280 (6 x-13)}{\sqrt{2 x^2-x+3}}-23 \sqrt{682 \left (13+i \sqrt{31}\right )} \left (13 \sqrt{31}+69 i\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )-23 \sqrt{682 \left (13-i \sqrt{31}\right )} \left (13 \sqrt{31}-69 i\right ) \tanh ^{-1}\left (\frac{\left (-22+4 i \sqrt{31}\right ) x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{6901840} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-27280*(-13 + 6*x))/Sqrt[3 - x + 2*x^2] - 23*Sqrt[682*(13 + I*Sqrt[31])]*(69*I + 13*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])] - 23*Sqrt[682*(13
- I*Sqrt[31])]*(-69*I + 13*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])])/6901840

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Maple [B]  time = 0.124, size = 718, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/465124*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)*2^(1/
2)*(2197*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2
^(1/2))^(1/2)*(-23*(8+3*2^(1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x
)^2/(2^(1/2)+1-x)^2*2^(1/2)+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^
(1/2)+1-x)^4+82*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+3070*(-775687+5
49362*2^(1/2))^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-775687+549362*2^(1/2))^(1/2)*(-23*(8+3*2^(
1/2))*(-23*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+24*2^(1/2)-41))^(1/2)*(6485*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)
+10368*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+22379*2^(1/2)+32016)/(23*(2^(1/2)-1+x)^4/(2^(1/2)+1-x)^4+82*(2^(1/2)-1+
x)^2/(2^(1/2)+1-x)^2+23)*(8+3*2^(1/2))*(2^(1/2)-1+x)/(2^(1/2)+1-x))+1712502*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2
^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2))^(1/2)/(-8866+6820*2^(1/2))^(1/2))*2^(1/2)
-6617446*arctanh(31/2*(8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(1/2)+8-3*2^(1/2)
)^(1/2)/(-8866+6820*2^(1/2))^(1/2)))/((8*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2+3*(2^(1/2)-1+x)^2/(2^(1/2)+1-x)^2*2^(
1/2)+8-3*2^(1/2))/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))^2)^(1/2)/(1+(2^(1/2)-1+x)/(2^(1/2)+1-x))/(8+3*2^(1/2))/(-886
6+6820*2^(1/2))^(1/2)+1/22/(2*x^2-x+3)^(1/2)-3/506*(-1+4*x)/(2*x^2-x+3)^(1/2)

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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 )}{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 4.79084, size = 7618, normalized size = 43.28 \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)^(3/2)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

-1/50921775520*(339388*sqrt(341)*50^(1/4)*sqrt(10)*sqrt(2)*(2*x^2 - x + 3)*sqrt(247*sqrt(2) + 1000)*arctan(1/3
28782125*(14260*sqrt(341)*sqrt(10)*sqrt(2*x^2 - x + 3)*(22*50^(3/4)*(57708*x^7 - 181278*x^6 + 400374*x^5 - 525
676*x^4 + 235088*x^3 - 46944*x^2 - sqrt(2)*(20846*x^7 - 109153*x^6 + 215386*x^5 - 427391*x^4 + 234360*x^3 - 15
6600*x^2 - 172800*x + 186624) - 373248*x + 172800) + 5*50^(1/4)*(125839*x^7 - 1864281*x^6 + 9323336*x^5 - 1972
5020*x^4 + 24624288*x^3 - 10862496*x^2 - sqrt(2)*(56119*x^7 - 908994*x^6 + 5175980*x^5 - 12895624*x^4 + 172612
80*x^3 - 14184000*x^2 - 10533888*x + 9994752) - 19989504*x + 10533888))*sqrt(247*sqrt(2) + 1000) + 933317000*s
qrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sq
rt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 53
9136) + 1154304*x - 456192) - 2*sqrt(310/119)*(sqrt(341)*sqrt(10)*sqrt(2*x^2 - x + 3)*(22*50^(3/4)*(246848*x^7
- 348192*x^6 + 1080672*x^5 - 178432*x^4 - 18432*x^3 + 1029888*x^2 - sqrt(2)*(46522*x^7 - 71117*x^6 + 257247*x
^5 - 273360*x^4 + 484920*x^3 - 269568*x^2 + 269568*x) - 1029888*x) + 5*50^(1/4)*(516957*x^7 - 6676948*x^6 + 25
569820*x^5 - 31522752*x^4 + 34450848*x^3 + 46199808*x^2 - 4*sqrt(2)*(38689*x^7 - 502244*x^6 + 1967660*x^5 - 28
28160*x^4 + 4711680*x^3 - 1689984*x^2 + 1689984*x) - 46199808*x))*sqrt(247*sqrt(2) + 1000) + 65450*sqrt(31)*sq
rt(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 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276
288*x) + 2975*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(-(sqrt(341)*50^(1/4)*sqrt(31)*sqrt(10)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(37*x - 38) + x - 7
5)*sqrt(247*sqrt(2) + 1000) - 903805*x^2 - 811580*sqrt(2)*(2*x^2 - x + 3) + 2785195*x - 3689000)/x^2) + 106058
75*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 379814
40*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 + 4424
9088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 339388*sqrt(341)*50^(1/4)*sqrt(10)*sqrt(2)*(2*x^2 - x + 3)
*sqrt(247*sqrt(2) + 1000)*arctan(1/328782125*(14260*sqrt(341)*sqrt(10)*sqrt(2*x^2 - x + 3)*(22*50^(3/4)*(57708
*x^7 - 181278*x^6 + 400374*x^5 - 525676*x^4 + 235088*x^3 - 46944*x^2 - sqrt(2)*(20846*x^7 - 109153*x^6 + 21538
6*x^5 - 427391*x^4 + 234360*x^3 - 156600*x^2 - 172800*x + 186624) - 373248*x + 172800) + 5*50^(1/4)*(125839*x^
7 - 1864281*x^6 + 9323336*x^5 - 19725020*x^4 + 24624288*x^3 - 10862496*x^2 - sqrt(2)*(56119*x^7 - 908994*x^6 +
5175980*x^5 - 12895624*x^4 + 17261280*x^3 - 14184000*x^2 - 10533888*x + 9994752) - 19989504*x + 10533888))*sq
rt(247*sqrt(2) + 1000) - 933317000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 15491
44*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 - 752
088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(310/119)*(sqrt(341)*sqrt(10)*sqrt(2*x
^2 - x + 3)*(22*50^(3/4)*(246848*x^7 - 348192*x^6 + 1080672*x^5 - 178432*x^4 - 18432*x^3 + 1029888*x^2 - sqrt(
2)*(46522*x^7 - 71117*x^6 + 257247*x^5 - 273360*x^4 + 484920*x^3 - 269568*x^2 + 269568*x) - 1029888*x) + 5*50^
(1/4)*(516957*x^7 - 6676948*x^6 + 25569820*x^5 - 31522752*x^4 + 34450848*x^3 + 46199808*x^2 - 4*sqrt(2)*(38689
*x^7 - 502244*x^6 + 1967660*x^5 - 2828160*x^4 + 4711680*x^3 - 1689984*x^2 + 1689984*x) - 46199808*x))*sqrt(247
*sqrt(2) + 1000) - 65450*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 - 707374*x^5 + 1053960*x^4 - 1667952*x
^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 2975*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 + 238
5*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((sqrt(341)*50^(1/4)*sqrt(31)*sqrt(10)*sqrt(2*x^2 -
x + 3)*(sqrt(2)*(37*x - 38) + x - 75)*sqrt(247*sqrt(2) + 1000) + 903805*x^2 + 811580*sqrt(2)*(2*x^2 - x + 3) -
2785195*x + 3689000)/x^2) - 10605875*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254
146592*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 + 18579456)) - 23*sqrt(341)*50^(1/4)*s
qrt(31)*sqrt(10)*(2000*x^2 - 247*sqrt(2)*(2*x^2 - x + 3) - 1000*x + 3000)*sqrt(247*sqrt(2) + 1000)*log(3100000
/119*(sqrt(341)*50^(1/4)*sqrt(31)*sqrt(10)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(37*x - 38) + x - 75)*sqrt(247*sqrt(2)
+ 1000) + 903805*x^2 + 811580*sqrt(2)*(2*x^2 - x + 3) - 2785195*x + 3689000)/x^2) + 23*sqrt(341)*50^(1/4)*sqr
t(31)*sqrt(10)*(2000*x^2 - 247*sqrt(2)*(2*x^2 - x + 3) - 1000*x + 3000)*sqrt(247*sqrt(2) + 1000)*log(-3100000/
119*(sqrt(341)*50^(1/4)*sqrt(31)*sqrt(10)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(37*x - 38) + x - 75)*sqrt(247*sqrt(2)
+ 1000) - 903805*x^2 - 811580*sqrt(2)*(2*x^2 - x + 3) + 2785195*x - 3689000)/x^2) + 201271840*sqrt(2*x^2 - x +
3)*(6*x - 13))/(2*x^2 - x + 3)

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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{3}{2}} \left (5 x^{2} + 3 x + 2\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError