3.98 \(\int \frac{1}{(3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^2} \, dx\)
Optimal. Leaf size=234 \[ -\frac{15101-8654 x}{1035276 \left (2 x^2-x+3\right )^{3/2}}-\frac{1352542 x+3133427}{523849656 \sqrt{2 x^2-x+3}}+\frac{65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}+\frac{625 \sqrt{\frac{1}{682} \left (30463+23600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (30463+23600 \sqrt{2}\right )}} \left (\left (687+445 \sqrt{2}\right ) x+242 \sqrt{2}+203\right )}{\sqrt{2 x^2-x+3}}\right )}{660176}-\frac{625 \sqrt{\frac{1}{682} \left (23600 \sqrt{2}-30463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (23600 \sqrt{2}-30463\right )}} \left (\left (687-445 \sqrt{2}\right ) x-242 \sqrt{2}+203\right )}{\sqrt{2 x^2-x+3}}\right )}{660176} \]
[Out]
-(15101 - 8654*x)/(1035276*(3 - x + 2*x^2)^(3/2)) - (3133427 + 1352542*x)/(523849656*Sqrt[3 - x + 2*x^2]) + (4
+ 65*x)/(682*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (625*Sqrt[(30463 + 23600*Sqrt[2])/682]*ArcTan[(Sqrt[1
1/(31*(30463 + 23600*Sqrt[2]))]*(203 + 242*Sqrt[2] + (687 + 445*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/660176 - (6
25*Sqrt[(-30463 + 23600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-30463 + 23600*Sqrt[2]))]*(203 - 242*Sqrt[2] + (68
7 - 445*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/660176
________________________________________________________________________________________
Rubi [A] time = 0.543007, antiderivative size = 234, normalized size of antiderivative = 1.,
number of steps used = 8, 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} \[ -\frac{15101-8654 x}{1035276 \left (2 x^2-x+3\right )^{3/2}}-\frac{1352542 x+3133427}{523849656 \sqrt{2 x^2-x+3}}+\frac{65 x+4}{682 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}+\frac{625 \sqrt{\frac{1}{682} \left (30463+23600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (30463+23600 \sqrt{2}\right )}} \left (\left (687+445 \sqrt{2}\right ) x+242 \sqrt{2}+203\right )}{\sqrt{2 x^2-x+3}}\right )}{660176}-\frac{625 \sqrt{\frac{1}{682} \left (23600 \sqrt{2}-30463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (23600 \sqrt{2}-30463\right )}} \left (\left (687-445 \sqrt{2}\right ) x-242 \sqrt{2}+203\right )}{\sqrt{2 x^2-x+3}}\right )}{660176} \]
Antiderivative was successfully verified.
[In]
Int[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)^2),x]
[Out]
-(15101 - 8654*x)/(1035276*(3 - x + 2*x^2)^(3/2)) - (3133427 + 1352542*x)/(523849656*Sqrt[3 - x + 2*x^2]) + (4
+ 65*x)/(682*(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)) + (625*Sqrt[(30463 + 23600*Sqrt[2])/682]*ArcTan[(Sqrt[1
1/(31*(30463 + 23600*Sqrt[2]))]*(203 + 242*Sqrt[2] + (687 + 445*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/660176 - (6
25*Sqrt[(-30463 + 23600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-30463 + 23600*Sqrt[2]))]*(203 - 242*Sqrt[2] + (68
7 - 445*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/660176
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 )^2} \, dx &=\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-1738+\frac{4411 x}{2}-5720 x^2}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx}{7502}\\ &=-\frac{15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}+\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{9406661}{2}+\frac{34397517 x}{4}-5235670 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{62634198}\\ &=-\frac{15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}-\frac{3133427+1352542 x}{523849656 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{22443155625}{4}+\frac{46206496875 x}{8}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{174310973034}\\ &=-\frac{15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}-\frac{3133427+1352542 x}{523849656 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\frac{14522041875}{8} \left (69+34 \sqrt{2}\right )+\frac{14522041875}{8} \left (1-35 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3834841406748 \sqrt{2}}-\frac{\int \frac{\frac{14522041875}{8} \left (69-34 \sqrt{2}\right )+\frac{14522041875}{8} \left (1+35 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3834841406748 \sqrt{2}}\\ &=-\frac{15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}-\frac{3133427+1352542 x}{523849656 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac{\left (6819140625 \left (47200-30463 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{6537580706796858984375}{64} \left (30463-23600 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{14522041875}{8} \left (203-242 \sqrt{2}\right )+\frac{14522041875}{8} \left (687-445 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{7936}-\frac{\left (6819140625 \left (47200+30463 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{6537580706796858984375}{64} \left (30463+23600 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{14522041875}{8} \left (203+242 \sqrt{2}\right )+\frac{14522041875}{8} \left (687+445 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{7936}\\ &=-\frac{15101-8654 x}{1035276 \left (3-x+2 x^2\right )^{3/2}}-\frac{3133427+1352542 x}{523849656 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{682 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac{625 \sqrt{\frac{1}{682} \left (30463+23600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (30463+23600 \sqrt{2}\right )}} \left (203+242 \sqrt{2}+\left (687+445 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{660176}-\frac{625 \sqrt{\frac{1}{682} \left (-30463+23600 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-30463+23600 \sqrt{2}\right )}} \left (203-242 \sqrt{2}+\left (687-445 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{660176}\\ \end{align*}
Mathematica [C] time = 1.16162, size = 296, normalized size = 1.26 \[ -\frac{5456 \left (13525420 x^5+32686812 x^4+2879479 x^3+84671384 x^2-5712309 x+31010342\right )+198375 i \sqrt{286+22 i \sqrt{31}} \left (687 \sqrt{31}+31 i\right ) \sqrt{2 x^2-x+3} \left (10 x^4+x^3+16 x^2+7 x+6\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+198375 \sqrt{286-22 i \sqrt{31}} \left (31+687 i \sqrt{31}\right ) \sqrt{2 x^2-x+3} \left (10 x^4+x^3+16 x^2+7 x+6\right ) \tanh ^{-1}\left (\frac{\left (22-4 i \sqrt{31}\right ) x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{2858123723136 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)^2),x]
[Out]
-(5456*(31010342 - 5712309*x + 84671384*x^2 + 2879479*x^3 + 32686812*x^4 + 13525420*x^5) + (198375*I)*Sqrt[286
+ (22*I)*Sqrt[31]]*(31*I + 687*Sqrt[31])*Sqrt[3 - x + 2*x^2]*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)*ArcTanh[(63 +
I*Sqrt[31] + (-22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] + 198375*Sqrt[286
- (22*I)*Sqrt[31]]*(31 + (687*I)*Sqrt[31])*Sqrt[3 - x + 2*x^2]*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)*ArcTanh[(-63
+ I*Sqrt[31] + (22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(2858123723136*(
3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2))
________________________________________________________________________________________
Maple [B] time = 0.175, size = 5975, normalized size = 25.5 \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)^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{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)^2,x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^2*(2*x^2 - x + 3)^(5/2)), x)
________________________________________________________________________________________
Fricas [B] time = 4.83618, size = 8498, normalized size = 36.32 \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)^2,x, algorithm="fricas")
[Out]
1/25604335602537914112*(301208632500*6962^(1/4)*sqrt(341)*sqrt(118)*sqrt(2)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 5
3*x^2 + 15*x + 18)*sqrt(30463*sqrt(2) + 47200)*arctan(1/11117215998613*(168268*sqrt(118)*(22*6962^(3/4)*sqrt(3
41)*(321084*x^7 - 1338894*x^6 + 2762802*x^5 - 4721048*x^4 + 2438224*x^3 - 1317312*x^2 - sqrt(2)*(277258*x^7 -
994619*x^6 + 2123978*x^5 - 3198193*x^4 + 1552680*x^3 - 621000*x^2 - 1900800*x + 1181952) - 2363904*x + 1900800
) + 1829*6962^(1/4)*sqrt(341)*(25187*x^7 - 392073*x^6 + 2114488*x^5 - 4948060*x^4 + 6460704*x^3 - 4452768*x^2
- sqrt(2)*(20477*x^7 - 310452*x^6 + 1610140*x^5 - 3584192*x^4 + 4580640*x^3 - 2620800*x^2 - 3400704*x + 219801
6) - 4396032*x + 3400704))*sqrt(2*x^2 - x + 3)*sqrt(30463*sqrt(2) + 47200) + 31558548641224*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(118/79)*(sqrt(118)*(22*6962^(3/4)*sqrt(341)*(1050904*x^7 - 1523916*x^6 + 5005956*x^5 - 257
2736*x^4 + 3615264*x^3 + 877824*x^2 - sqrt(2)*(1065206*x^7 - 1518091*x^6 + 4815081*x^5 - 1448880*x^4 + 1303560
*x^3 + 3131136*x^2 - 3131136*x) - 877824*x) + 1829*6962^(1/4)*sqrt(341)*(84981*x^7 - 1100084*x^6 + 4256060*x^5
- 5639616*x^4 + 7745184*x^3 + 2571264*x^2 - 242*sqrt(2)*(319*x^7 - 4124*x^6 + 15860*x^5 - 20160*x^4 + 24480*x
^3 + 20736*x^2 - 20736*x) - 2571264*x))*sqrt(2*x^2 - x + 3)*sqrt(30463*sqrt(2) + 47200) + 187549318*sqrt(31)*s
qrt(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) + 327
6288*x) + 8524969*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 - 19
44*x) + 144820224*x))*sqrt(-(6962^(1/4)*sqrt(341)*sqrt(118)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(101*x + 176
) - 277*x + 75)*sqrt(30463*sqrt(2) + 47200) - 219481829*x^2 - 197085724*sqrt(2)*(2*x^2 - x + 3) + 676362371*x
- 895844200)/x^2) + 358619870923*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 25414659
2*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 + 49
0880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 301208632500*6962^(1/4)*sqrt
(341)*sqrt(118)*sqrt(2)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18)*sqrt(30463*sqrt(2) + 47200)*arcta
n(1/11117215998613*(168268*sqrt(118)*(22*6962^(3/4)*sqrt(341)*(321084*x^7 - 1338894*x^6 + 2762802*x^5 - 472104
8*x^4 + 2438224*x^3 - 1317312*x^2 - sqrt(2)*(277258*x^7 - 994619*x^6 + 2123978*x^5 - 3198193*x^4 + 1552680*x^3
- 621000*x^2 - 1900800*x + 1181952) - 2363904*x + 1900800) + 1829*6962^(1/4)*sqrt(341)*(25187*x^7 - 392073*x^
6 + 2114488*x^5 - 4948060*x^4 + 6460704*x^3 - 4452768*x^2 - sqrt(2)*(20477*x^7 - 310452*x^6 + 1610140*x^5 - 35
84192*x^4 + 4580640*x^3 - 2620800*x^2 - 3400704*x + 2198016) - 4396032*x + 3400704))*sqrt(2*x^2 - x + 3)*sqrt(
30463*sqrt(2) + 47200) - 31558548641224*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(118/79)*(sqrt(118)*(22*6962^(3/4
)*sqrt(341)*(1050904*x^7 - 1523916*x^6 + 5005956*x^5 - 2572736*x^4 + 3615264*x^3 + 877824*x^2 - sqrt(2)*(10652
06*x^7 - 1518091*x^6 + 4815081*x^5 - 1448880*x^4 + 1303560*x^3 + 3131136*x^2 - 3131136*x) - 877824*x) + 1829*6
962^(1/4)*sqrt(341)*(84981*x^7 - 1100084*x^6 + 4256060*x^5 - 5639616*x^4 + 7745184*x^3 + 2571264*x^2 - 242*sqr
t(2)*(319*x^7 - 4124*x^6 + 15860*x^5 - 20160*x^4 + 24480*x^3 + 20736*x^2 - 20736*x) - 2571264*x))*sqrt(2*x^2 -
x + 3)*sqrt(30463*sqrt(2) + 47200) - 187549318*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293
072*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) - 8524969*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((6962^(1/4)*sqrt(341)*sqr
t(118)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(101*x + 176) - 277*x + 75)*sqrt(30463*sqrt(2) + 47200) + 2194818
29*x^2 + 197085724*sqrt(2)*(2*x^2 - x + 3) - 676362371*x + 895844200)/x^2) - 358619870923*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)) + 991875*6962^(1/4)*sqrt(341)*sqrt(118)*sqrt(31)*(944000*x^6 - 377600*x^5 + 2879200
*x^4 + 47200*x^3 + 2501600*x^2 - 30463*sqrt(2)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2 + 15*x + 18) + 708000*x
+ 849600)*sqrt(30463*sqrt(2) + 47200)*log(7375000000000/79*(6962^(1/4)*sqrt(341)*sqrt(118)*sqrt(31)*sqrt(2*x^
2 - x + 3)*(sqrt(2)*(101*x + 176) - 277*x + 75)*sqrt(30463*sqrt(2) + 47200) + 219481829*x^2 + 197085724*sqrt(2
)*(2*x^2 - x + 3) - 676362371*x + 895844200)/x^2) - 991875*6962^(1/4)*sqrt(341)*sqrt(118)*sqrt(31)*(944000*x^6
- 377600*x^5 + 2879200*x^4 + 47200*x^3 + 2501600*x^2 - 30463*sqrt(2)*(20*x^6 - 8*x^5 + 61*x^4 + x^3 + 53*x^2
+ 15*x + 18) + 708000*x + 849600)*sqrt(30463*sqrt(2) + 47200)*log(-7375000000000/79*(6962^(1/4)*sqrt(341)*sqrt
(118)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(101*x + 176) - 277*x + 75)*sqrt(30463*sqrt(2) + 47200) - 21948182
9*x^2 - 197085724*sqrt(2)*(2*x^2 - x + 3) + 676362371*x - 895844200)/x^2) - 48877259552*(13525420*x^5 + 326868
12*x^4 + 2879479*x^3 + 84671384*x^2 - 5712309*x + 31010342)*sqrt(2*x^2 - x + 3))/(20*x^6 - 8*x^5 + 61*x^4 + x^
3 + 53*x^2 + 15*x + 18)
________________________________________________________________________________________
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 )^{2}}\, 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)**2,x)
[Out]
Integral(1/((2*x**2 - x + 3)**(5/2)*(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(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError