3.91 \(\int \frac{1}{(3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^2} \, dx\)
Optimal. Leaf size=211 \[ -\frac{6315-2306 x}{345092 \sqrt{2 x^2-x+3}}+\frac{65 x+4}{682 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (129694447+103775000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (129694447+103775000 \sqrt{2}\right )}} \left (\left (45519+29065 \sqrt{2}\right ) x+16454 \sqrt{2}+12611\right )}{\sqrt{2 x^2-x+3}}\right )}{30008}-\frac{\sqrt{\frac{1}{682} \left (103775000 \sqrt{2}-129694447\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (103775000 \sqrt{2}-129694447\right )}} \left (\left (45519-29065 \sqrt{2}\right ) x-16454 \sqrt{2}+12611\right )}{\sqrt{2 x^2-x+3}}\right )}{30008} \]
[Out]
-(6315 - 2306*x)/(345092*Sqrt[3 - x + 2*x^2]) + (4 + 65*x)/(682*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (Sqrt
[(129694447 + 103775000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(129694447 + 103775000*Sqrt[2]))]*(12611 + 16454*Sqr
t[2] + (45519 + 29065*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/30008 - (Sqrt[(-129694447 + 103775000*Sqrt[2])/682]*A
rcTanh[(Sqrt[11/(31*(-129694447 + 103775000*Sqrt[2]))]*(12611 - 16454*Sqrt[2] + (45519 - 29065*Sqrt[2])*x))/Sq
rt[3 - x + 2*x^2]])/30008
________________________________________________________________________________________
Rubi [A] time = 0.473226, antiderivative size = 211, normalized size of antiderivative = 1.,
number of steps used = 7, 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} \[ -\frac{6315-2306 x}{345092 \sqrt{2 x^2-x+3}}+\frac{65 x+4}{682 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (129694447+103775000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (129694447+103775000 \sqrt{2}\right )}} \left (\left (45519+29065 \sqrt{2}\right ) x+16454 \sqrt{2}+12611\right )}{\sqrt{2 x^2-x+3}}\right )}{30008}-\frac{\sqrt{\frac{1}{682} \left (103775000 \sqrt{2}-129694447\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (103775000 \sqrt{2}-129694447\right )}} \left (\left (45519-29065 \sqrt{2}\right ) x-16454 \sqrt{2}+12611\right )}{\sqrt{2 x^2-x+3}}\right )}{30008} \]
Antiderivative was successfully verified.
[In]
Int[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2),x]
[Out]
-(6315 - 2306*x)/(345092*Sqrt[3 - x + 2*x^2]) + (4 + 65*x)/(682*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (Sqrt
[(129694447 + 103775000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(129694447 + 103775000*Sqrt[2]))]*(12611 + 16454*Sqr
t[2] + (45519 + 29065*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/30008 - (Sqrt[(-129694447 + 103775000*Sqrt[2])/682]*A
rcTanh[(Sqrt[11/(31*(-129694447 + 103775000*Sqrt[2]))]*(12611 - 16454*Sqrt[2] + (45519 - 29065*Sqrt[2])*x))/Sq
rt[3 - x + 2*x^2]])/30008
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 )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac{4+65 x}{682 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-1782+\frac{3333 x}{2}-2860 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{7502}\\ &=-\frac{6315-2306 x}{345092 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{3002857}{2}+\frac{6943585 x}{4}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{20878066}\\ &=-\frac{6315-2306 x}{345092 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\frac{30613}{4} \left (4653+2158 \sqrt{2}\right )+\frac{30613}{4} \left (337-2495 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{459317452 \sqrt{2}}-\frac{\int \frac{\frac{30613}{4} \left (4653-2158 \sqrt{2}\right )+\frac{30613}{4} \left (337+2495 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{459317452 \sqrt{2}}\\ &=-\frac{6315-2306 x}{345092 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\left (253 \left (207550000-129694447 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{29051828839}{16} \left (129694447-103775000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{30613}{4} \left (12611-16454 \sqrt{2}\right )+\frac{30613}{4} \left (45519-29065 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{1984}-\frac{\left (253 \left (207550000+129694447 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{29051828839}{16} \left (129694447+103775000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{30613}{4} \left (12611+16454 \sqrt{2}\right )+\frac{30613}{4} \left (45519+29065 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{1984}\\ &=-\frac{6315-2306 x}{345092 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac{\sqrt{\frac{1}{682} \left (129694447+103775000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (129694447+103775000 \sqrt{2}\right )}} \left (12611+16454 \sqrt{2}+\left (45519+29065 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{30008}-\frac{\sqrt{\frac{1}{682} \left (-129694447+103775000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-129694447+103775000 \sqrt{2}\right )}} \left (12611-16454 \sqrt{2}+\left (45519-29065 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{30008}\\ \end{align*}
Mathematica [C] time = 1.48804, size = 740, normalized size = 3.51 \[ \frac{100 \left (\frac{682 \left (\left (22-4 i \sqrt{31}\right ) x+i \sqrt{31}+52\right )}{\left (\sqrt{31}+13 i\right ) \left (10 i x+\sqrt{31}+3 i\right ) \sqrt{2 x^2-x+3}}+\frac{682 \left (\left (22+4 i \sqrt{31}\right ) x-i \sqrt{31}+52\right )}{\left (\sqrt{31}-13 i\right ) \left (-10 i x+\sqrt{31}-3 i\right ) \sqrt{2 x^2-x+3}}+\frac{22 \left (2 \left (11 \sqrt{31}-62 i\right ) x+52 \sqrt{31}+31 i\right )}{\left (\sqrt{31}+13 i\right ) \sqrt{2 x^2-x+3}}+\frac{22 \left (2 \left (11 \sqrt{31}+62 i\right ) x+52 \sqrt{31}-31 i\right )}{\left (\sqrt{31}-13 i\right ) \sqrt{2 x^2-x+3}}+\frac{575 i \sqrt{682 \left (13+i \sqrt{31}\right )} \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\left (\sqrt{31}-13 i\right )^2}+\frac{155 \left (44 \left (16353+581 i \sqrt{31}\right ) \sqrt{2 x^2-x+3}+345 \sqrt{286+22 i \sqrt{31}} \left (10 \left (11+2 i \sqrt{31}\right ) x+17 i \sqrt{31}-29\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )\right )}{22 \left (\sqrt{31}-13 i\right )^3 \left (-10 i x+\sqrt{31}-3 i\right )}+\frac{155 \left (44 \left (16353-581 i \sqrt{31}\right ) \sqrt{2 x^2-x+3}+345 \sqrt{286-22 i \sqrt{31}} \left (\left (-110+20 i \sqrt{31}\right ) x+17 i \sqrt{31}+29\right ) \tanh ^{-1}\left (\frac{\left (22-4 i \sqrt{31}\right ) x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )\right )}{22 \left (\sqrt{31}+13 i\right )^3 \left (10 i x+\sqrt{31}+3 i\right )}-\frac{575 i \sqrt{682 \left (13-i \sqrt{31}\right )} \tanh ^{-1}\left (\frac{\left (-22+4 i \sqrt{31}\right ) x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\left (\sqrt{31}+13 i\right )^2}\right )}{2674463} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2),x]
[Out]
(100*((682*(52 + I*Sqrt[31] + (22 - (4*I)*Sqrt[31])*x))/((13*I + Sqrt[31])*(3*I + Sqrt[31] + (10*I)*x)*Sqrt[3
- x + 2*x^2]) + (682*(52 - I*Sqrt[31] + (22 + (4*I)*Sqrt[31])*x))/((-13*I + Sqrt[31])*(-3*I + Sqrt[31] - (10*I
)*x)*Sqrt[3 - x + 2*x^2]) + (22*(31*I + 52*Sqrt[31] + 2*(-62*I + 11*Sqrt[31])*x))/((13*I + Sqrt[31])*Sqrt[3 -
x + 2*x^2]) + (22*(-31*I + 52*Sqrt[31] + 2*(62*I + 11*Sqrt[31])*x))/((-13*I + Sqrt[31])*Sqrt[3 - x + 2*x^2]) +
((575*I)*Sqrt[682*(13 + 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])])/(-13*I + Sqrt[31])^2 + (155*(44*(16353 + (581*I)*Sqrt[31])*Sqrt[3 - x + 2*x
^2] + 345*Sqrt[286 + (22*I)*Sqrt[31]]*(-29 + (17*I)*Sqrt[31] + 10*(11 + (2*I)*Sqrt[31])*x)*ArcTanh[(63 + I*Sqr
t[31] + (-22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/(22*(-13*I + Sqrt[31]
)^3*(-3*I + Sqrt[31] - (10*I)*x)) + (155*(44*(16353 - (581*I)*Sqrt[31])*Sqrt[3 - x + 2*x^2] + 345*Sqrt[286 - (
22*I)*Sqrt[31]]*(29 + (17*I)*Sqrt[31] + (-110 + (20*I)*Sqrt[31])*x)*ArcTanh[(-63 + I*Sqrt[31] + (22 - (4*I)*Sq
rt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/(22*(13*I + Sqrt[31])^3*(3*I + Sqrt[31] + (1
0*I)*x)) - ((575*I)*Sqrt[682*(13 - I*Sqrt[31])]*ArcTanh[(63 - I*Sqrt[31] + (-22 + (4*I)*Sqrt[31])*x)/(2*Sqrt[2
86 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(13*I + Sqrt[31])^2))/2674463
________________________________________________________________________________________
Maple [B] time = 0.179, size = 5942, normalized size = 28.2 \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)^(3/2)/(5*x^2+3*x+2)^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}{\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)^2,x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^2*(2*x^2 - x + 3)^(3/2)), x)
________________________________________________________________________________________
Fricas [B] time = 5.23242, size = 9287, normalized size = 44.01 \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)^2,x, algorithm="fricas")
[Out]
1/29889247038841109870720*(35183643812*3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(2)*(10*x^4 + x^3 + 16*x^2 +
7*x + 6)*sqrt(129694447*sqrt(2) + 207550000)*arctan(1/2437871055247532640924125*(59193260*sqrt(20755)*(11*344
6160200^(3/4)*sqrt(341)*(20748108*x^7 - 87744678*x^6 + 180517074*x^5 - 311740976*x^4 + 161753488*x^3 - 8904614
4*x^2 - sqrt(2)*(18515146*x^7 - 65709803*x^6 + 140687186*x^5 - 209710441*x^4 + 101256360*x^3 - 39198600*x^2 -
126316800*x + 76909824) - 153819648*x + 126316800) + 643405*3446160200^(1/4)*sqrt(341)*(1637219*x^7 - 25548801
*x^6 + 138274456*x^5 - 324967420*x^4 + 425065248*x^3 - 297030816*x^2 - sqrt(2)*(1361849*x^7 - 20608224*x^6 + 1
06575580*x^5 - 236322704*x^4 + 301502880*x^3 - 169632000*x^2 - 225358848*x + 143534592) - 287069184*x + 225358
848))*sqrt(2*x^2 - x + 3)*sqrt(129694447*sqrt(2) + 207550000) + 6920408156831705561333000*sqrt(31)*sqrt(2)*(28
180*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 - 1
02335*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(41510/397951)*(sqrt(20755)*(11*3446160200^(3/4)*sqrt(341)*(66710248*x^7 - 96938292*x^6 + 319
739772*x^5 - 172116032*x^4 + 247423968*x^3 + 38700288*x^2 - sqrt(2)*(71827622*x^7 - 102266467*x^6 + 323714097*
x^5 - 93357360*x^4 + 79054920*x^3 + 219532032*x^2 - 219532032*x) - 38700288*x) + 643405*3446160200^(1/4)*sqrt(
341)*(5462397*x^7 - 70721108*x^6 + 273784220*x^5 - 364358592*x^4 + 506287008*x^3 + 144903168*x^2 - 2*sqrt(2)*(
2586013*x^7 - 33428948*x^6 + 128512220*x^5 - 162918720*x^4 + 196126560*x^3 + 173705472*x^2 - 173705472*x) - 14
4903168*x))*sqrt(2*x^2 - x + 3)*sqrt(129694447*sqrt(2) + 207550000) + 116912097033204550*sqrt(31)*sqrt(2)*(123
408*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) + 5
314186228782025*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(-(3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(6137*
x + 12812) - 18949*x + 6675)*sqrt(129694447*sqrt(2) + 207550000) - 388930324332445*x^2 - 349243556543420*sqrt(
2)*(2*x^2 - x + 3) + 1198540387228555*x - 1587470711561000)/x^2) + 78641001782178472287875*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 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^
2 - 24772608*x + 18579456)) + 35183643812*3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(2)*(10*x^4 + x^3 + 16*x^
2 + 7*x + 6)*sqrt(129694447*sqrt(2) + 207550000)*arctan(1/2437871055247532640924125*(59193260*sqrt(20755)*(11*
3446160200^(3/4)*sqrt(341)*(20748108*x^7 - 87744678*x^6 + 180517074*x^5 - 311740976*x^4 + 161753488*x^3 - 8904
6144*x^2 - sqrt(2)*(18515146*x^7 - 65709803*x^6 + 140687186*x^5 - 209710441*x^4 + 101256360*x^3 - 39198600*x^2
- 126316800*x + 76909824) - 153819648*x + 126316800) + 643405*3446160200^(1/4)*sqrt(341)*(1637219*x^7 - 25548
801*x^6 + 138274456*x^5 - 324967420*x^4 + 425065248*x^3 - 297030816*x^2 - sqrt(2)*(1361849*x^7 - 20608224*x^6
+ 106575580*x^5 - 236322704*x^4 + 301502880*x^3 - 169632000*x^2 - 225358848*x + 143534592) - 287069184*x + 225
358848))*sqrt(2*x^2 - x + 3)*sqrt(129694447*sqrt(2) + 207550000) - 6920408156831705561333000*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(41510/397951)*(sqrt(20755)*(11*3446160200^(3/4)*sqrt(341)*(66710248*x^7 - 96938292*x^6 +
319739772*x^5 - 172116032*x^4 + 247423968*x^3 + 38700288*x^2 - sqrt(2)*(71827622*x^7 - 102266467*x^6 + 3237140
97*x^5 - 93357360*x^4 + 79054920*x^3 + 219532032*x^2 - 219532032*x) - 38700288*x) + 643405*3446160200^(1/4)*sq
rt(341)*(5462397*x^7 - 70721108*x^6 + 273784220*x^5 - 364358592*x^4 + 506287008*x^3 + 144903168*x^2 - 2*sqrt(2
)*(2586013*x^7 - 33428948*x^6 + 128512220*x^5 - 162918720*x^4 + 196126560*x^3 + 173705472*x^2 - 173705472*x) -
144903168*x))*sqrt(2*x^2 - x + 3)*sqrt(129694447*sqrt(2) + 207550000) - 116912097033204550*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)
- 5314186228782025*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 - 1
944*x) + 144820224*x))*sqrt((3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(613
7*x + 12812) - 18949*x + 6675)*sqrt(129694447*sqrt(2) + 207550000) + 388930324332445*x^2 + 349243556543420*sqr
t(2)*(2*x^2 - x + 3) - 1198540387228555*x + 1587470711561000)/x^2) - 78641001782178472287875*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 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*
x^2 - 24772608*x + 18579456)) + 23*3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*(2075500000*x^4 + 207550000
*x^3 + 3320800000*x^2 - 129694447*sqrt(2)*(10*x^4 + x^3 + 16*x^2 + 7*x + 6) + 1452850000*x + 1245300000)*sqrt(
129694447*sqrt(2) + 207550000)*log(1037750000000/397951*(3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*sqrt(
2*x^2 - x + 3)*(sqrt(2)*(6137*x + 12812) - 18949*x + 6675)*sqrt(129694447*sqrt(2) + 207550000) + 3889303243324
45*x^2 + 349243556543420*sqrt(2)*(2*x^2 - x + 3) - 1198540387228555*x + 1587470711561000)/x^2) - 23*3446160200
^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*(2075500000*x^4 + 207550000*x^3 + 3320800000*x^2 - 129694447*sqrt(2)*(10
*x^4 + x^3 + 16*x^2 + 7*x + 6) + 1452850000*x + 1245300000)*sqrt(129694447*sqrt(2) + 207550000)*log(-103775000
0000/397951*(3446160200^(1/4)*sqrt(20755)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(6137*x + 12812) - 1
8949*x + 6675)*sqrt(129694447*sqrt(2) + 207550000) - 388930324332445*x^2 - 349243556543420*sqrt(2)*(2*x^2 - x
+ 3) + 1198540387228555*x - 1587470711561000)/x^2) + 86612402022768160*(11530*x^3 - 24657*x^2 + 18557*x - 1060
6)*sqrt(2*x^2 - x + 3))/(10*x^4 + x^3 + 16*x^2 + 7*x + 6)
________________________________________________________________________________________
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 )^{2}}\, 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)**2,x)
[Out]
Integral(1/((2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2), x)
________________________________________________________________________________________
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)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError