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

Optimal. Leaf size=281 \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{3844 \left (5 x^2+3 x+2\right )}+\frac{(11359-12920 x) \sqrt{2 x^2-x+3}}{48050}+\frac{\sqrt{11 \left (1+4 \sqrt{2}\right )} \left (2937349+1978861 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3531015707557+2498852071250 \sqrt{2}\right )}} \left (\left (9832420+6895071 \sqrt{2}\right ) x+2937349 \sqrt{2}+3957722\right )}{\sqrt{2 x^2-x+3}}\right )}{29791000}-\frac{\left (2937349-1978861 \sqrt{2}\right ) \sqrt{11 \left (4 \sqrt{2}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (2498852071250 \sqrt{2}-3531015707557\right )}} \left (\left (9832420-6895071 \sqrt{2}\right ) x-2937349 \sqrt{2}+3957722\right )}{\sqrt{2 x^2-x+3}}\right )}{29791000}-\frac{4}{125} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

[Out]

((11359 - 12920*x)*Sqrt[3 - x + 2*x^2])/48050 + ((3 + 10*x)*(3 - x + 2*x^2)^(5/2))/(62*(2 + 3*x + 5*x^2)^2) +
((769 + 2336*x)*(3 - x + 2*x^2)^(3/2))/(3844*(2 + 3*x + 5*x^2)) - (4*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/125
+ (Sqrt[11*(1 + 4*Sqrt[2])]*(2937349 + 1978861*Sqrt[2])*ArcTan[(Sqrt[11/(62*(3531015707557 + 2498852071250*Sqr
t[2]))]*(3957722 + 2937349*Sqrt[2] + (9832420 + 6895071*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/29791000 - ((293734
9 - 1978861*Sqrt[2])*Sqrt[11*(-1 + 4*Sqrt[2])]*ArcTanh[(Sqrt[11/(62*(-3531015707557 + 2498852071250*Sqrt[2]))]
*(3957722 - 2937349*Sqrt[2] + (9832420 - 6895071*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/29791000

________________________________________________________________________________________

Rubi [A]  time = 0.654547, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {971, 1054, 1066, 1076, 619, 215, 1035, 1029, 206, 204} \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(2336 x+769) \left (2 x^2-x+3\right )^{3/2}}{3844 \left (5 x^2+3 x+2\right )}+\frac{(11359-12920 x) \sqrt{2 x^2-x+3}}{48050}+\frac{\sqrt{11 \left (1+4 \sqrt{2}\right )} \left (2937349+1978861 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3531015707557+2498852071250 \sqrt{2}\right )}} \left (\left (9832420+6895071 \sqrt{2}\right ) x+2937349 \sqrt{2}+3957722\right )}{\sqrt{2 x^2-x+3}}\right )}{29791000}-\frac{\left (2937349-1978861 \sqrt{2}\right ) \sqrt{11 \left (4 \sqrt{2}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (2498852071250 \sqrt{2}-3531015707557\right )}} \left (\left (9832420-6895071 \sqrt{2}\right ) x-2937349 \sqrt{2}+3957722\right )}{\sqrt{2 x^2-x+3}}\right )}{29791000}-\frac{4}{125} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11359 - 12920*x)*Sqrt[3 - x + 2*x^2])/48050 + ((3 + 10*x)*(3 - x + 2*x^2)^(5/2))/(62*(2 + 3*x + 5*x^2)^2) +
((769 + 2336*x)*(3 - x + 2*x^2)^(3/2))/(3844*(2 + 3*x + 5*x^2)) - (4*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/125
+ (Sqrt[11*(1 + 4*Sqrt[2])]*(2937349 + 1978861*Sqrt[2])*ArcTan[(Sqrt[11/(62*(3531015707557 + 2498852071250*Sqr
t[2]))]*(3957722 + 2937349*Sqrt[2] + (9832420 + 6895071*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/29791000 - ((293734
9 - 1978861*Sqrt[2])*Sqrt[11*(-1 + 4*Sqrt[2])]*ArcTanh[(Sqrt[11/(62*(-3531015707557 + 2498852071250*Sqrt[2]))]
*(3957722 - 2937349*Sqrt[2] + (9832420 - 6895071*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/29791000

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 1054

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*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x
^2)^(p + 1)*(d + e*x + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(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[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3)
+ C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(
q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*
x^2, x], 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] && 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 )^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac{1}{62} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (-\frac{195}{2}+35 x+40 x^2\right )}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\left (\frac{66735}{4}-7375 x-25840 x^2\right ) \sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx}{9610}\\ &=\frac{(11359-12920 x) \sqrt{3-x+2 x^2}}{48050}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-6782705+2898425 x-307520 x^2}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{961000}\\ &=\frac{(11359-12920 x) \sqrt{3-x+2 x^2}}{48050}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-33298485+15414685 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{4805000}+\frac{8}{125} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{(11359-12920 x) \sqrt{3-x+2 x^2}}{48050}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{1}{125} \left (4 \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{605 \left (885694-605427 \sqrt{2}\right )-605 \left (325160-280267 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{105710000 \sqrt{2}}+\frac{\int \frac{605 \left (885694+605427 \sqrt{2}\right )-605 \left (325160+280267 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{105710000 \sqrt{2}}\\ &=\frac{(11359-12920 x) \sqrt{3-x+2 x^2}}{48050}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac{4}{125} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )-\frac{\left (1331 \left (4997704142500-3531015707557 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-22693550 \left (3531015707557-2498852071250 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{605 \left (3957722-2937349 \sqrt{2}\right )+605 \left (9832420-6895071 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{192200}-\frac{\left (1331 \left (4997704142500+3531015707557 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-22693550 \left (3531015707557+2498852071250 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{605 \left (3957722+2937349 \sqrt{2}\right )+605 \left (9832420+6895071 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{192200}\\ &=\frac{(11359-12920 x) \sqrt{3-x+2 x^2}}{48050}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(769+2336 x) \left (3-x+2 x^2\right )^{3/2}}{3844 \left (2+3 x+5 x^2\right )}-\frac{4}{125} \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )+\frac{\sqrt{\frac{11}{31} \left (3531015707557+2498852071250 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (3531015707557+2498852071250 \sqrt{2}\right )}} \left (3957722+2937349 \sqrt{2}+\left (9832420+6895071 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{961000}-\frac{\sqrt{\frac{11}{31} \left (-3531015707557+2498852071250 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (-3531015707557+2498852071250 \sqrt{2}\right )}} \left (3957722-2937349 \sqrt{2}+\left (9832420-6895071 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{961000}\\ \end{align*}

Mathematica [C]  time = 2.11623, size = 1009, normalized size = 3.59 \[ \frac{12599950 \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 ) x^4-12290525 i \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 ) x^4+15119940 \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 ) x^3-14748630 i \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 ) x^3+662597100 \sqrt{2 x^2-x+3} x^3+14615942 \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 ) x^2-14257009 i \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 ) x^2+640207040 \sqrt{2 x^2-x+3} x^2+6047976 \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 ) x-5899452 i \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 ) x+474815220 \sqrt{2 x^2-x+3} x+1906624 \sqrt{2} \left (5 x^2+3 x+2\right )^2 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )-i \sqrt{286+22 i \sqrt{31}} \left (-503998 i+491621 \sqrt{31}\right ) \left (5 x^2+3 x+2\right )^2 \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 )+2015992 \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 )-1966484 i \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 )[phone] \sqrt{2 x^2-x+3}}{59582000 \left (5 x^2+3 x+2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(153804640*Sqrt[3 - x + 2*x^2] + 474815220*x*Sqrt[3 - x + 2*x^2] + 640207040*x^2*Sqrt[3 - x + 2*x^2] + 6625971
00*x^3*Sqrt[3 - x + 2*x^2] + 1906624*Sqrt[2]*(2 + 3*x + 5*x^2)^2*ArcSinh[(-1 + 4*x)/Sqrt[23]] - I*Sqrt[286 + (
22*I)*Sqrt[31]]*(-503998*I + 491621*Sqrt[31])*(2 + 3*x + 5*x^2)^2*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])] - (1966484*I)*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])] + 2015992*S
qrt[286 - (22*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])] - (5899452*I)*Sqrt[682*(13 - I*Sqrt[31])]*x*ArcTanh[(-63 + I*Sqrt[31] + 22*x - (4*I)*Sq
rt[31]*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] + 6047976*Sqrt[286 - (22*I)*Sqrt[31]]*x*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])] - (14257009
*I)*Sqrt[682*(13 - I*Sqrt[31])]*x^2*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])] + 14615942*Sqrt[286 - (22*I)*Sqrt[31]]*x^2*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])] - (14748630*I)*Sqrt[682*(13 - I*Sqrt[31
])]*x^3*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
])] + 15119940*Sqrt[286 - (22*I)*Sqrt[31]]*x^3*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])] - (12290525*I)*Sqrt[682*(13 - I*Sqrt[31])]*x^4*ArcTanh[(-63 + I*Sqr
t[31] + 22*x - (4*I)*Sqrt[31]*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] + 12599950*Sqrt[286 - (2
2*I)*Sqrt[31]]*x^4*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])])/(59582000*(2 + 3*x + 5*x^2)^2)

________________________________________________________________________________________

Maple [B]  time = 0.69, size = 119458, normalized size = 425.1 \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)^3,x)

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 5.49537, size = 10927, normalized size = 38.89 \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)^3,x, algorithm="fricas")

[Out]

1/758714159921174808909075728000*(3184949732636*3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(2)
*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(3531015707557*sqrt(2) + 4997704142500)*arctan(1/453548488062940310
3991789624695893204150231*(2850690442882*sqrt(1999081657)*(2*3868444992270541948232^(3/4)*sqrt(62)*(2627559914
*x^7 - 10187615527*x^6 + 21362956024*x^5 - 34451465819*x^4 + 17321103240*x^3 - 8320757400*x^2 - sqrt(2)*(18933
66636*x^7 - 7237484076*x^6 + 15226003533*x^5 - 24262105817*x^4 + 12127036096*x^3 - 5664787848*x^2 - 1336758681
6*x + 9338025600) - 18676051200*x + 13367586816) + 61971531367*3868444992270541948232^(1/4)*sqrt(62)*(40011633
2*x^7 - 6149336082*x^6 + 32552996440*x^5 - 74427496472*x^4 + 96235107840*x^3 - 61219656000*x^2 - sqrt(2)*(2866
85371*x^7 - 4395067059*x^6 + 23180544704*x^5 - 52748573780*x^4 + 68065744032*x^3 - 42544702944*x^2 - 486258370
56*x + 34092306432) - 68184612864*x + 48625837056))*sqrt(2*x^2 - x + 3)*sqrt(3531015707557*sqrt(2) + 499770414
2500) + 12874924822431853972621854418491567805329688*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1
385256*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(1999081657/828550919)
*(sqrt(1999081657)*(2*3868444992270541948232^(3/4)*sqrt(62)*(9351066298*x^7 - 13433496653*x^6 + 43310345823*x^
5 - 17374572240*x^4 + 20927636280*x^3 + 18483199488*x^2 - sqrt(2)*(6839273266*x^7 - 9809465289*x^6 + 315240996
99*x^5 - 12024617744*x^4 + 13914887256*x^3 + 14839341696*x^2 - 14839341696*x) - 18483199488*x) + 61971531367*3
868444992270541948232^(1/4)*sqrt(62)*(1427210918*x^7 - 18462714328*x^6 + 71210222920*x^5 - 92387041920*x^4 + 1
19489780160*x^3 + 68726817792*x^2 - sqrt(2)*(1033310523*x^7 - 13365477772*x^6 + 51521534980*x^5 - 66583614528*
x^4 + 85122955872*x^3 + 53108877312*x^2 - 53108877312*x) - 68726817792*x))*sqrt(2*x^2 - x + 3)*sqrt(3531015707
557*sqrt(2) + 4997704142500) + 4516423329856721284677540671884*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 157
8888*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) + 205291969538941876576251848
722*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 16895692
8*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) + 144820
224*x))*sqrt(-(3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2
141441*x + 1076175) - 3217616*x - 1065266)*sqrt(3531015707557*sqrt(2) + 4997704142500) - 155990877430002205517
374*x^2 - 140073440957553000872744*sqrt(2)*(2*x^2 - x + 3) + 480706581467965980267826*x - 63669745889796818578
5200)/x^2) + 146305963891271067870702891119222361424201*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 1
42835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 1
0070*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)) + 31849
49732636*3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*
sqrt(3531015707557*sqrt(2) + 4997704142500)*arctan(1/4535484880629403103991789624695893204150231*(285069044288
2*sqrt(1999081657)*(2*3868444992270541948232^(3/4)*sqrt(62)*(2627559914*x^7 - 10187615527*x^6 + 21362956024*x^
5 - 34451465819*x^4 + 17321103240*x^3 - 8320757400*x^2 - sqrt(2)*(1893366636*x^7 - 7237484076*x^6 + 1522600353
3*x^5 - 24262105817*x^4 + 12127036096*x^3 - 5664787848*x^2 - 13367586816*x + 9338025600) - 18676051200*x + 133
67586816) + 61971531367*3868444992270541948232^(1/4)*sqrt(62)*(400116332*x^7 - 6149336082*x^6 + 32552996440*x^
5 - 74427496472*x^4 + 96235107840*x^3 - 61219656000*x^2 - sqrt(2)*(286685371*x^7 - 4395067059*x^6 + 2318054470
4*x^5 - 52748573780*x^4 + 68065744032*x^3 - 42544702944*x^2 - 48625837056*x + 34092306432) - 68184612864*x + 4
8625837056))*sqrt(2*x^2 - x + 3)*sqrt(3531015707557*sqrt(2) + 4997704142500) - 1287492482243185397262185441849
1567805329688*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(1999081657/828550919)*(sqrt(1999081657)*(2*38684449922705419
48232^(3/4)*sqrt(62)*(9351066298*x^7 - 13433496653*x^6 + 43310345823*x^5 - 17374572240*x^4 + 20927636280*x^3 +
18483199488*x^2 - sqrt(2)*(6839273266*x^7 - 9809465289*x^6 + 31524099699*x^5 - 12024617744*x^4 + 13914887256*
x^3 + 14839341696*x^2 - 14839341696*x) - 18483199488*x) + 61971531367*3868444992270541948232^(1/4)*sqrt(62)*(1
427210918*x^7 - 18462714328*x^6 + 71210222920*x^5 - 92387041920*x^4 + 119489780160*x^3 + 68726817792*x^2 - sqr
t(2)*(1033310523*x^7 - 13365477772*x^6 + 51521534980*x^5 - 66583614528*x^4 + 85122955872*x^3 + 53108877312*x^2
- 53108877312*x) - 68726817792*x))*sqrt(2*x^2 - x + 3)*sqrt(3531015707557*sqrt(2) + 4997704142500) - 45164233
29856721284677540671884*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 7
98336*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) - 205291969538941876576251848722*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((3868444992270541948232^(1
/4)*sqrt(1999081657)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2141441*x + 1076175) - 3217616*x - 106526
6)*sqrt(3531015707557*sqrt(2) + 4997704142500) + 155990877430002205517374*x^2 + 140073440957553000872744*sqrt(
2)*(2*x^2 - x + 3) - 480706581467965980267826*x + 636697458897968185785200)/x^2) - 146305963891271067870702891
119222361424201*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 - 135629
44*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 3868444992270541948232^(1/4)*sqrt(1999081657)
*sqrt(62)*sqrt(31)*(124942603562500*x^4 + 149931124275000*x^3 + 144933420132500*x^2 - 3531015707557*sqrt(2)*(2
5*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 59972449710000*x + 19990816570000)*sqrt(3531015707557*sqrt(2) + 49977041
42500)*log(3123565089062500/828550919*(3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(31)*sqrt(2*
x^2 - x + 3)*(sqrt(2)*(2141441*x + 1076175) - 3217616*x - 1065266)*sqrt(3531015707557*sqrt(2) + 4997704142500)
+ 155990877430002205517374*x^2 + 140073440957553000872744*sqrt(2)*(2*x^2 - x + 3) - 480706581467965980267826*
x + 636697458897968185785200)/x^2) - 3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(31)*(12494260
3562500*x^4 + 149931124275000*x^3 + 144933420132500*x^2 - 3531015707557*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12
*x + 4) + 59972449710000*x + 19990816570000)*sqrt(3531015707557*sqrt(2) + 4997704142500)*log(-3123565089062500
/828550919*(3868444992270541948232^(1/4)*sqrt(1999081657)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2141
441*x + 1076175) - 3217616*x - 1065266)*sqrt(3531015707557*sqrt(2) + 4997704142500) - 155990877430002205517374
*x^2 - 140073440957553000872744*sqrt(2)*(2*x^2 - x + 3) + 480706581467965980267826*x - 63669745889796818578520
0)/x^2) + 12139426558738796942545211648*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(-4*sqrt(2)*sqrt(2*x^
2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 86845533393682860541101280*(97155*x^3 + 93872*x^2 + 69621*x + 225
52)*sqrt(2*x^2 - x + 3))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

________________________________________________________________________________________

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

[Out]

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

[Out]

Exception raised: TypeError