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

Optimal. Leaf size=222 \[ -\frac{1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(226249-99620 x) \sqrt{2 x^2-x+3}}{80000}-\frac{121 \sqrt{\frac{11}{31} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (25000 \sqrt{2}-15457\right )}} \left (-\left (690+247 \sqrt{2}\right ) x-443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}+\frac{121 \sqrt{\frac{11}{31} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (15457+25000 \sqrt{2}\right )}} \left (-\left (690-247 \sqrt{2}\right ) x+443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}} \]

[Out]

-((226249 - 99620*x)*Sqrt[3 - x + 2*x^2])/80000 - ((103 - 60*x)*(3 - x + 2*x^2)^(3/2))/600 - (7216203*ArcSinh[
(1 - 4*x)/Sqrt[23]])/(800000*Sqrt[2]) - (121*Sqrt[(11*(-15457 + 25000*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(-1545
7 + 25000*Sqrt[2]))]*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125 + (121*Sqrt[(11*(
15457 + 25000*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2]))]*(196 + 443*Sqrt[2] - (690 - 247*Sqr
t[2])*x))/Sqrt[3 - x + 2*x^2]])/3125

________________________________________________________________________________________

Rubi [A]  time = 0.538982, antiderivative size = 222, 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 = {977, 1066, 1076, 619, 215, 1035, 1029, 206, 204} \[ -\frac{1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(226249-99620 x) \sqrt{2 x^2-x+3}}{80000}-\frac{121 \sqrt{\frac{11}{31} \left (25000 \sqrt{2}-15457\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (25000 \sqrt{2}-15457\right )}} \left (-\left (690+247 \sqrt{2}\right ) x-443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}+\frac{121 \sqrt{\frac{11}{31} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (15457+25000 \sqrt{2}\right )}} \left (-\left (690-247 \sqrt{2}\right ) x+443 \sqrt{2}+196\right )}{\sqrt{2 x^2-x+3}}\right )}{3125}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((226249 - 99620*x)*Sqrt[3 - x + 2*x^2])/80000 - ((103 - 60*x)*(3 - x + 2*x^2)^(3/2))/600 - (7216203*ArcSinh[
(1 - 4*x)/Sqrt[23]])/(800000*Sqrt[2]) - (121*Sqrt[(11*(-15457 + 25000*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(-1545
7 + 25000*Sqrt[2]))]*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3125 + (121*Sqrt[(11*(
15457 + 25000*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2]))]*(196 + 443*Sqrt[2] - (690 - 247*Sqr
t[2])*x))/Sqrt[3 - x + 2*x^2]])/3125

Rule 977

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((b*f
*(3*p + 2*q) - c*e*(2*p + q) + 2*c*f*(p + q)*x)*(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^(q + 1))/(2*f^2*(p
+ q)*(2*p + 2*q + 1)), x] - Dist[1/(2*f^2*(p + q)*(2*p + 2*q + 1)), Int[(a + b*x + c*x^2)^(p - 2)*(d + e*x +
f*x^2)^q*Simp[(b*d - a*e)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*(b^2*d*f*(1 - p) - a*(f*(b*e - 2*a*f)*(2*p +
2*q + 1) + c*(2*d*f - e^2*(2*p + q)))) + (2*(c*d - a*f)*(c*e - b*f)*(1 - p)*(2*p + q) - (p + q)*((b^2 - 4*a*c
)*e*f*(1 - p) + b*(c*(e^2 - 4*d*f)*(2*p + q) + f*(2*c*d - b*e + 2*a*f)*(2*p + 2*q + 1))))*x + ((c*e - b*f)^2*(
1 - p)*p + c*(p + q)*(f*(b*e - 2*a*f)*(4*p + 2*q - 1) - c*(2*d*f*(1 - 2*p) + e^2*(3*p + q - 1))))*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] && GtQ[p, 1] && NeQ[p + q
, 0] && NeQ[2*p + 2*q + 1, 0] &&  !IGtQ[p, 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 )^{5/2}}{2+3 x+5 x^2} \, dx &=-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac{1}{300} \int \frac{\left (-\frac{4731}{2}+\frac{6135 x}{4}-\frac{14943 x^2}{4}\right ) \sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx\\ &=-\frac{(226249-99620 x) \sqrt{3-x+2 x^2}}{80000}-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}+\frac{\int \frac{\frac{3205293}{8}-\frac{11339385 x}{16}+\frac{21648609 x^2}{16}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{30000}\\ &=-\frac{(226249-99620 x) \sqrt{3-x+2 x^2}}{80000}-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}+\frac{\int \frac{-702768-7602672 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{150000}+\frac{7216203 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{800000}\\ &=-\frac{(226249-99620 x) \sqrt{3-x+2 x^2}}{80000}-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac{\int \frac{-702768 \left (108-11 \sqrt{2}\right )-702768 \left (130-119 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3300000 \sqrt{2}}+\frac{\int \frac{-702768 \left (108+11 \sqrt{2}\right )-702768 \left (130+119 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3300000 \sqrt{2}}+\frac{7216203 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{800000 \sqrt{46}}\\ &=-\frac{(226249-99620 x) \sqrt{3-x+2 x^2}}{80000}-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}}+\frac{\left (935384208 \left (50000-15457 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{30620737433088 \left (15457-25000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{702768 \left (196-443 \sqrt{2}\right )-702768 \left (690+247 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3125}+\frac{\left (935384208 \left (50000+15457 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{30620737433088 \left (15457+25000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{702768 \left (196+443 \sqrt{2}\right )-702768 \left (690-247 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3125}\\ &=-\frac{(226249-99620 x) \sqrt{3-x+2 x^2}}{80000}-\frac{1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac{7216203 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{800000 \sqrt{2}}-\frac{121 \sqrt{\frac{11}{31} \left (-15457+25000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (-15457+25000 \sqrt{2}\right )}} \left (196-443 \sqrt{2}-\left (690+247 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{3125}+\frac{121 \sqrt{\frac{11}{31} \left (15457+25000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (15457+25000 \sqrt{2}\right )}} \left (196+443 \sqrt{2}-\left (690-247 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{3125}\\ \end{align*}

Mathematica [C]  time = 1.06988, size = 229, normalized size = 1.03 \[ \frac{620 \sqrt{2 x^2-x+3} \left (48000 x^3-106400 x^2+412060 x-802347\right )+46464 \sqrt{286+22 i \sqrt{31}} \left (403-69 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )-46464 i \sqrt{286-22 i \sqrt{31}} \left (69 \sqrt{31}-403 i\right ) \tanh ^{-1}\left (\frac{\left (22-4 i \sqrt{31}\right ) x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{148800000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(620*Sqrt[3 - x + 2*x^2]*(-802347 + 412060*x - 106400*x^2 + 48000*x^3) + 671106879*Sqrt[2]*ArcSinh[(-1 + 4*x)/
Sqrt[23]] + 46464*Sqrt[286 + (22*I)*Sqrt[31]]*(403 - (69*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])] - (46464*I)*Sqrt[286 - (22*I)*Sqrt[31]]*(-40
3*I + 69*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])])/148800000

________________________________________________________________________________________

Maple [B]  time = 0.179, size = 4860, normalized size = 21.9 \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)^(5/2)/(5*x^2+3*x+2),x)

[Out]

1/5*x^3*(2*x^2-x+3)^(1/2)-133/300*x^2*(2*x^2-x+3)^(1/2)+20603/12000*x*(2*x^2-x+3)^(1/2)-267449/80000*(2*x^2-x+
3)^(1/2)+7216203/1600000*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+4/33034375*(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)*(75195*(-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))+106294*(-775687+549362*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+22
379*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))[phone]*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)-158290154*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))/(-8866+6820*2^(1/2))^(1/2)+6/6606875*(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)*(10915*(-77
5687+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))+14918*(-775687+549362*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))-5052938*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)-51565338*ar
ctanh(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)/(-8
866+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))/(-8866+6820*2^(1
/2))^(1/2)-21/1321375*(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)*(4245*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arctan(1/11692487*(-77
5687+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))+61
54*(-775687+549362*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+8
2*(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))[phone]*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)-359414*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))/(-8866+6820*2^(1/2))^(1/2)-37/528550*(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)*(2365*(-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))+3338*(-775687+549362*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))+319244
2*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)-5264358*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))/(-8866+6820*2^(1/2))^(1/2)-63/105710*(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)*(285*(-775687+549362*2^(1/2))^(1/2)*2^(1/2)*(-88
66+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))+386*(-775687+549362*2^(1/2))^(1/2)*(-8866+6820*2^(1/2))^(1/2)*arcta
n(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))-274846*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)-1543366*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))/(-8866+6820*2^(1/2))^(1/2)+27/21142*(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)*(151*(-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))+218*(-775687+549362*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))+401698*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)-63426*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))/(-8866+6820*2^(1/2))^(1/2)+27/21142*(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)*(369*(-77568
7+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))+520*(-775687+549362*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))+465124*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)-866822*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))/(-8866+6820*2^(1/2))^(1/
2)

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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{5}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.90656, size = 7899, normalized size = 35.58 \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)^(5/2)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

121/96875000*6050^(1/4)*sqrt(31)*sqrt(2)*sqrt(-772850000*sqrt(2) + 2500000000)*arctan(1/254496437500*(72244150
0000*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) + 2300*(4*6050^(3/4)*sqrt(31)*(35898*x^7 - 441939*x^6 + 782418*x^5 - 2117233
*x^4 + 1272680*x^3 - 1081800*x^2 - sqrt(2)*(173702*x^7 - 453907*x^6 + 1056481*x^5 - 1083344*x^4 + 393672*x^3 +
152064*x^2 - 1043712*x + 259200) - 518400*x + 1043712) + 5*6050^(1/4)*sqrt(31)*(317294*x^7 - 5870544*x^6 + 38
857480*x^5 - 111531424*x^4 + 156761280*x^3 - 168192000*x^2 - sqrt(2)*(712757*x^7 - 10233303*x^6 + 48529768*x^5
- 94500260*x^4 + 113086944*x^3 - 22282848*x^2 - 106417152*x + 37407744) - 74815488*x + 106417152))*sqrt(2*x^2
- x + 3)*sqrt(-772850000*sqrt(2) + 2500000000) - sqrt(10/5711)*(314105000*sqrt(31)*sqrt(2)*(123408*x^8 - 9141
52*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) - (4*6050^(3/4)*s
qrt(31)*(167914*x^7 - 195429*x^6 + 331239*x^5 + 1685680*x^4 - 3693960*x^3 + 4195584*x^2 + 22*sqrt(2)*(37846*x^
7 - 52859*x^6 + 160569*x^5 - 4464*x^4 - 49464*x^3 + 202176*x^2 - 202176*x) - 4195584*x) - 5*6050^(1/4)*sqrt(31
)*(160956*x^7 - 2232176*x^6 + 11218640*x^5 - 38096640*x^4 + 139374720*x^3 - 296027136*x^2 - sqrt(2)*(3246491*x
^7 - 41888524*x^6 + 159670660*x^5 - 190080576*x^4 + 180496224*x^3 + 376648704*x^2 - 376648704*x) + 296027136*x
))*sqrt(2*x^2 - x + 3)*sqrt(-772850000*sqrt(2) + 2500000000) + 14277500*sqrt(31)*(254591*x^8 - 4815126*x^7 + 3
2303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 51
7*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((6050^(1/4)*sqrt(2*x^2 - x +
3)*(sqrt(2)*(163*x - 725) + 562*x - 888)*sqrt(-772850000*sqrt(2) + 2500000000) + 139919500*x^2 + 125642000*sqr
t(2)*(2*x^2 - x + 3) - 431180500*x + 571100000)/x^2) + 8209562500*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 533855
60*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 97
89*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
)) + 121/96875000*6050^(1/4)*sqrt(31)*sqrt(2)*sqrt(-772850000*sqrt(2) + 2500000000)*arctan(-1/254496437500*(72
2441500000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98
496*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 + 5
46048*x - 539136) + 1154304*x - 456192) - 2300*(4*6050^(3/4)*sqrt(31)*(35898*x^7 - 441939*x^6 + 782418*x^5 - 2
117233*x^4 + 1272680*x^3 - 1081800*x^2 - sqrt(2)*(173702*x^7 - 453907*x^6 + 1056481*x^5 - 1083344*x^4 + 393672
*x^3 + 152064*x^2 - 1043712*x + 259200) - 518400*x + 1043712) + 5*6050^(1/4)*sqrt(31)*(317294*x^7 - 5870544*x^
6 + 38857480*x^5 - 111531424*x^4 + 156761280*x^3 - 168192000*x^2 - sqrt(2)*(712757*x^7 - 10233303*x^6 + 485297
68*x^5 - 94500260*x^4 + 113086944*x^3 - 22282848*x^2 - 106417152*x + 37407744) - 74815488*x + 106417152))*sqrt
(2*x^2 - x + 3)*sqrt(-772850000*sqrt(2) + 2500000000) - sqrt(10/5711)*(314105000*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) + (4*6050^(
3/4)*sqrt(31)*(167914*x^7 - 195429*x^6 + 331239*x^5 + 1685680*x^4 - 3693960*x^3 + 4195584*x^2 + 22*sqrt(2)*(37
846*x^7 - 52859*x^6 + 160569*x^5 - 4464*x^4 - 49464*x^3 + 202176*x^2 - 202176*x) - 4195584*x) - 5*6050^(1/4)*s
qrt(31)*(160956*x^7 - 2232176*x^6 + 11218640*x^5 - 38096640*x^4 + 139374720*x^3 - 296027136*x^2 - sqrt(2)*(324
6491*x^7 - 41888524*x^6 + 159670660*x^5 - 190080576*x^4 + 180496224*x^3 + 376648704*x^2 - 376648704*x) + 29602
7136*x))*sqrt(2*x^2 - x + 3)*sqrt(-772850000*sqrt(2) + 2500000000) + 14277500*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(-(6050^(1/4)*sqrt(2*x^2
- x + 3)*(sqrt(2)*(163*x - 725) + 562*x - 888)*sqrt(-772850000*sqrt(2) + 2500000000) - 139919500*x^2 - 125642
000*sqrt(2)*(2*x^2 - x + 3) + 431180500*x - 571100000)/x^2) + 8209562500*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))/(25851
91*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 1
8579456)) - 121/2213012500000*6050^(1/4)*(15457*sqrt(2) + 50000)*sqrt(-772850000*sqrt(2) + 2500000000)*log(915
0625000/5711*(6050^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(163*x - 725) + 562*x - 888)*sqrt(-772850000*sqrt(2) + 2
500000000) + 139919500*x^2 + 125642000*sqrt(2)*(2*x^2 - x + 3) - 431180500*x + 571100000)/x^2) + 121/221301250
0000*6050^(1/4)*(15457*sqrt(2) + 50000)*sqrt(-772850000*sqrt(2) + 2500000000)*log(-9150625000/5711*(6050^(1/4)
*sqrt(2*x^2 - x + 3)*(sqrt(2)*(163*x - 725) + 562*x - 888)*sqrt(-772850000*sqrt(2) + 2500000000) - 139919500*x
^2 - 125642000*sqrt(2)*(2*x^2 - x + 3) + 431180500*x - 571100000)/x^2) + 1/240000*(48000*x^3 - 106400*x^2 + 41
2060*x - 802347)*sqrt(2*x^2 - x + 3) + 7216203/3200000*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) -
32*x^2 + 16*x - 25)

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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{5}{2}}}{5 x^{2} + 3 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError