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

Optimal. Leaf size=232 \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}+\frac{4}{155} (4-5 x) \sqrt{2 x^2-x+3}+\frac{\sqrt{\frac{11}{31} \left (3169333+2265350 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3169333+2265350 \sqrt{2}\right )}} \left (\left (9440+6477 \sqrt{2}\right ) x+2963 \sqrt{2}+3514\right )}{\sqrt{2 x^2-x+3}}\right )}{1550}-\frac{\sqrt{\frac{11}{31} \left (2265350 \sqrt{2}-3169333\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (2265350 \sqrt{2}-3169333\right )}} \left (\left (9440-6477 \sqrt{2}\right ) x-2963 \sqrt{2}+3514\right )}{\sqrt{2 x^2-x+3}}\right )}{1550}-\frac{2}{25} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

[Out]

(4*(4 - 5*x)*Sqrt[3 - x + 2*x^2])/155 + ((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (2*Sqrt[2]
*ArcSinh[(1 - 4*x)/Sqrt[23]])/25 + (Sqrt[(11*(3169333 + 2265350*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(3169333 + 2
265350*Sqrt[2]))]*(3514 + 2963*Sqrt[2] + (9440 + 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550 - (Sqrt[(11*(-31
69333 + 2265350*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(-3169333 + 2265350*Sqrt[2]))]*(3514 - 2963*Sqrt[2] + (9440
- 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550

________________________________________________________________________________________

Rubi [A]  time = 0.575455, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {971, 1066, 1076, 619, 215, 1035, 1029, 206, 204} \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}+\frac{4}{155} (4-5 x) \sqrt{2 x^2-x+3}+\frac{\sqrt{\frac{11}{31} \left (3169333+2265350 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3169333+2265350 \sqrt{2}\right )}} \left (\left (9440+6477 \sqrt{2}\right ) x+2963 \sqrt{2}+3514\right )}{\sqrt{2 x^2-x+3}}\right )}{1550}-\frac{\sqrt{\frac{11}{31} \left (2265350 \sqrt{2}-3169333\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (2265350 \sqrt{2}-3169333\right )}} \left (\left (9440-6477 \sqrt{2}\right ) x-2963 \sqrt{2}+3514\right )}{\sqrt{2 x^2-x+3}}\right )}{1550}-\frac{2}{25} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(4 - 5*x)*Sqrt[3 - x + 2*x^2])/155 + ((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (2*Sqrt[2]
*ArcSinh[(1 - 4*x)/Sqrt[23]])/25 + (Sqrt[(11*(3169333 + 2265350*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(3169333 + 2
265350*Sqrt[2]))]*(3514 + 2963*Sqrt[2] + (9440 + 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550 - (Sqrt[(11*(-31
69333 + 2265350*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(-3169333 + 2265350*Sqrt[2]))]*(3514 - 2963*Sqrt[2] + (9440
- 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550

Rule 971

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((b +
2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p
+ 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e
*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]

Rule 1066

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*
(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p
+ q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)
*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (
p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 -
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{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] && GtQ[p, 0] && NeQ[p +
q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; 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 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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{\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \int \frac{\sqrt{3-x+2 x^2} \left (-\frac{69}{2}+13 x+40 x^2\right )}{2+3 x+5 x^2} \, dx\\ &=\frac{4}{155} (4-5 x) \sqrt{3-x+2 x^2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{13070-5750 x+2480 x^2}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3100}\\ &=\frac{4}{155} (4-5 x) \sqrt{3-x+2 x^2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{60390-36190 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{15500}+\frac{4}{25} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{4}{155} (4-5 x) \sqrt{3-x+2 x^2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{25} \left (2 \sqrt{\frac{2}{23}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )+\frac{\int \frac{-1210 \left (878-549 \sqrt{2}\right )+1210 \left (220-329 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt{2}}-\frac{\int \frac{-1210 \left (878+549 \sqrt{2}\right )+1210 \left (220+329 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt{2}}\\ &=\frac{4}{155} (4-5 x) \sqrt{3-x+2 x^2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{2}{25} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )+\frac{1}{155} \left (1331 \left (4530700-3169333 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-90774200 \left (3169333-2265350 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{-1210 \left (3514-2963 \sqrt{2}\right )-1210 \left (9440-6477 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )+\frac{1}{155} \left (1331 \left (4530700+3169333 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-90774200 \left (3169333+2265350 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{-1210 \left (3514+2963 \sqrt{2}\right )-1210 \left (9440+6477 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )\\ &=\frac{4}{155} (4-5 x) \sqrt{3-x+2 x^2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{2}{25} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )+\frac{\sqrt{\frac{11}{31} \left (3169333+2265350 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3169333+2265350 \sqrt{2}\right )}} \left (3514+2963 \sqrt{2}+\left (9440+6477 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1550}-\frac{\sqrt{\frac{11}{31} \left (-3169333+2265350 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (-3169333+2265350 \sqrt{2}\right )}} \left (3514-2963 \sqrt{2}+\left (9440-6477 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{1550}\\ \end{align*}

Mathematica [C]  time = 2.58377, size = 530, normalized size = 2.28 \[ \frac{\frac{62000 \sqrt{2 x^2-x+3} x^2}{10 x-i \sqrt{31}+3}+\frac{62000 \sqrt{2 x^2-x+3} x^2}{10 x+i \sqrt{31}+3}-\frac{31000 \sqrt{2 x^2-x+3} x}{10 x-i \sqrt{31}+3}-\frac{31000 \sqrt{2 x^2-x+3} x}{10 x+i \sqrt{31}+3}-12400 \sqrt{2 x^2-x+3} x+\frac{93000 \sqrt{2 x^2-x+3}}{10 x-i \sqrt{31}+3}+\frac{93000 \sqrt{2 x^2-x+3}}{10 x+i \sqrt{31}+3}+9920 \sqrt{2 x^2-x+3}-\frac{\sqrt{286+22 i \sqrt{31}} \left (6477 \sqrt{31}+10199 i\right ) \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x-22 x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\sqrt{31}-13 i}+\frac{10199 i \sqrt{286-22 i \sqrt{31}} \tanh ^{-1}\left (\frac{4 i \sqrt{31} x-22 x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\sqrt{31}+13 i}-\frac{6477 \sqrt{682 \left (13-i \sqrt{31}\right )} \tanh ^{-1}\left (\frac{4 i \sqrt{31} x-22 x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\sqrt{31}+13 i}-7688 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{96100} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9920*Sqrt[3 - x + 2*x^2] - 12400*x*Sqrt[3 - x + 2*x^2] + (93000*Sqrt[3 - x + 2*x^2])/(3 - I*Sqrt[31] + 10*x)
- (31000*x*Sqrt[3 - x + 2*x^2])/(3 - I*Sqrt[31] + 10*x) + (62000*x^2*Sqrt[3 - x + 2*x^2])/(3 - I*Sqrt[31] + 10
*x) + (93000*Sqrt[3 - x + 2*x^2])/(3 + I*Sqrt[31] + 10*x) - (31000*x*Sqrt[3 - x + 2*x^2])/(3 + I*Sqrt[31] + 10
*x) + (62000*x^2*Sqrt[3 - x + 2*x^2])/(3 + I*Sqrt[31] + 10*x) - 7688*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] - (Sq
rt[286 + (22*I)*Sqrt[31]]*(10199*I + 6477*Sqrt[31])*ArcTanh[(63 + I*Sqrt[31] - 22*x - (4*I)*Sqrt[31]*x)/(2*Sqr
t[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(-13*I + Sqrt[31]) - (6477*Sqrt[682*(13 - I*Sqrt[31])]*ArcTanh
[(63 - I*Sqrt[31] - 22*x + (4*I)*Sqrt[31]*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(13*I + Sqr
t[31]) + ((10199*I)*Sqrt[286 - (22*I)*Sqrt[31]]*ArcTanh[(63 - I*Sqrt[31] - 22*x + (4*I)*Sqrt[31]*x)/(2*Sqrt[28
6 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(13*I + Sqrt[31]))/96100

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Maple [B]  time = 0.355, size = 28185, normalized size = 121.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.10758, size = 8892, normalized size = 38.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90746855745853600*(10421084*1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(3169333*
sqrt(2) + 4530700)*arctan(1/172758074198807633719789*(64607782*sqrt(45307)*(2*1987037073032^(3/4)*sqrt(62)*(24
33118*x^7 - 9616349*x^6 + 20077988*x^5 - 32895253*x^4 + 16664280*x^3 - 8289000*x^2 - sqrt(2)*(1842432*x^7 - 69
16062*x^6 + 14611071*x^5 - 22920229*x^4 + 11367152*x^3 - 5107176*x^2 - 12897792*x + 8726400) - 17452800*x + 12
897792) + 1404517*1987037073032^(1/4)*sqrt(62)*(373384*x^7 - 5757834*x^6 + 30631880*x^5 - 70476664*x^4 + 91370
880*x^3 - 59457600*x^2 - sqrt(2)*(276977*x^7 - 4232733*x^6 + 22218448*x^5 - 50249260*x^4 + 64668384*x^3 - 3947
9328*x^2 - 46697472*x + 32016384) - 64032768*x + 46697472))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700
) + 490410017080486186043272*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) - sqrt(45307/2711)*(sqrt(45307)*(2*1987037073032^(3/
4)*sqrt(62)*(8480726*x^7 - 12210811*x^6 + 39548601*x^5 - 16962480*x^4 + 21434760*x^3 + 14432256*x^2 - sqrt(2)*
(6779042*x^7 - 9704193*x^6 + 31062363*x^5 - 11094928*x^4 + 12114072*x^3 + 16301952*x^2 - 16301952*x) - 1443225
6*x) + 1404517*1987037073032^(1/4)*sqrt(62)*(1312966*x^7 - 16987736*x^6 + 65572040*x^5 - 85530240*x^4 + 112374
720*x^3 + 57314304*x^2 - sqrt(2)*(1011501*x^7 - 13081364*x^6 + 50391260*x^5 - 64806336*x^4 + 81634464*x^3 + 56
070144*x^2 - 56070144*x) - 57314304*x))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700) + 7590571938849196
*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 - 10368
00*x) + 3276288*x) + 345025997220418*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781
920*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(-(1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*sqrt(2*x^2 -
x + 3)*(sqrt(2)*(1867*x + 1425) - 3292*x - 442)*sqrt(3169333*sqrt(2) + 4530700) - 11567627293306*x^2 - 103872
57161336*sqrt(2)*(2*x^2 - x + 3) + 35647177985494*x - 47214805278800)/x^2) + 5572841103187343023219*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) + 2230
64064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34
615296*x^2 - 24772608*x + 18579456)) + 10421084*1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x
+ 2)*sqrt(3169333*sqrt(2) + 4530700)*arctan(1/172758074198807633719789*(64607782*sqrt(45307)*(2*1987037073032^
(3/4)*sqrt(62)*(2433118*x^7 - 9616349*x^6 + 20077988*x^5 - 32895253*x^4 + 16664280*x^3 - 8289000*x^2 - sqrt(2)
*(1842432*x^7 - 6916062*x^6 + 14611071*x^5 - 22920229*x^4 + 11367152*x^3 - 5107176*x^2 - 12897792*x + 8726400)
- 17452800*x + 12897792) + 1404517*1987037073032^(1/4)*sqrt(62)*(373384*x^7 - 5757834*x^6 + 30631880*x^5 - 70
476664*x^4 + 91370880*x^3 - 59457600*x^2 - sqrt(2)*(276977*x^7 - 4232733*x^6 + 22218448*x^5 - 50249260*x^4 + 6
4668384*x^3 - 39479328*x^2 - 46697472*x + 32016384) - 64032768*x + 46697472))*sqrt(2*x^2 - x + 3)*sqrt(3169333
*sqrt(2) + 4530700) - 490410017080486186043272*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 + 13207
10*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - sqrt(45307/2711)*(sqrt(45307)*(2
*1987037073032^(3/4)*sqrt(62)*(8480726*x^7 - 12210811*x^6 + 39548601*x^5 - 16962480*x^4 + 21434760*x^3 + 14432
256*x^2 - sqrt(2)*(6779042*x^7 - 9704193*x^6 + 31062363*x^5 - 11094928*x^4 + 12114072*x^3 + 16301952*x^2 - 163
01952*x) - 14432256*x) + 1404517*1987037073032^(1/4)*sqrt(62)*(1312966*x^7 - 16987736*x^6 + 65572040*x^5 - 855
30240*x^4 + 112374720*x^3 + 57314304*x^2 - sqrt(2)*(1011501*x^7 - 13081364*x^6 + 50391260*x^5 - 64806336*x^4 +
81634464*x^3 + 56070144*x^2 - 56070144*x) - 57314304*x))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700)
- 7590571938849196*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 + 1
209600*x^2 - 1036800*x) + 3276288*x) - 345025997220418*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 908
66808*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((1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt
(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1867*x + 1425) - 3292*x - 442)*sqrt(3169333*sqrt(2) + 4530700) + 1156762729
3306*x^2 + 10387257161336*sqrt(2)*(2*x^2 - x + 3) - 35647177985494*x + 47214805278800)/x^2) - 5572841103187343
023219*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37
981440*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)) + 1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*(226
53500*x^2 - 3169333*sqrt(2)*(5*x^2 + 3*x + 2) + 13592100*x + 9061400)*sqrt(3169333*sqrt(2) + 4530700)*log(1132
67500/2711*(1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1867*x + 1425) - 3
292*x - 442)*sqrt(3169333*sqrt(2) + 4530700) + 11567627293306*x^2 + 10387257161336*sqrt(2)*(2*x^2 - x + 3) - 3
5647177985494*x + 47214805278800)/x^2) - 1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*(22653500*x^2 - 316
9333*sqrt(2)*(5*x^2 + 3*x + 2) + 13592100*x + 9061400)*sqrt(3169333*sqrt(2) + 4530700)*log(-113267500/2711*(19
87037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1867*x + 1425) - 3292*x - 442)*s
qrt(3169333*sqrt(2) + 4530700) - 11567627293306*x^2 - 10387257161336*sqrt(2)*(2*x^2 - x + 3) + 35647177985494*
x - 47214805278800)/x^2) + 3629874229834144*sqrt(2)*(5*x^2 + 3*x + 2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x
- 1) - 32*x^2 + 16*x - 25) + 6440099440028320*sqrt(2*x^2 - x + 3)*(13*x + 7))/(5*x^2 + 3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError