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

Optimal. Leaf size=255 \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}+\frac{4}{155} (4-5 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(2240 x+1277) \sqrt{2 x^2-x+3}}{7750}+\frac{11 \sqrt{\frac{11}{31} \left (224510383+194487500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (224510383+194487500 \sqrt{2}\right )}} \left (\left (87710+54423 \sqrt{2}\right ) x+33287 \sqrt{2}+21136\right )}{\sqrt{2 x^2-x+3}}\right )}{38750}-\frac{11 \sqrt{\frac{11}{31} \left (194487500 \sqrt{2}-224510383\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (194487500 \sqrt{2}-224510383\right )}} \left (\left (87710-54423 \sqrt{2}\right ) x-33287 \sqrt{2}+21136\right )}{\sqrt{2 x^2-x+3}}\right )}{38750}-\frac{4799 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2500 \sqrt{2}} \]

[Out]

-((1277 + 2240*x)*Sqrt[3 - x + 2*x^2])/7750 + (4*(4 - 5*x)*(3 - x + 2*x^2)^(3/2))/155 + ((3 + 10*x)*(3 - x + 2
*x^2)^(5/2))/(31*(2 + 3*x + 5*x^2)) - (4799*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2500*Sqrt[2]) + (11*Sqrt[(11*(224510
383 + 194487500*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(224510383 + 194487500*Sqrt[2]))]*(21136 + 33287*Sqrt[2] + (
87710 + 54423*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/38750 - (11*Sqrt[(11*(-224510383 + 194487500*Sqrt[2]))/31]*Ar
cTanh[(Sqrt[11/(62*(-224510383 + 194487500*Sqrt[2]))]*(21136 - 33287*Sqrt[2] + (87710 - 54423*Sqrt[2])*x))/Sqr
t[3 - x + 2*x^2]])/38750

________________________________________________________________________________________

Rubi [A]  time = 0.659844, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {971, 1066, 1076, 619, 215, 1035, 1029, 206, 204} \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}+\frac{4}{155} (4-5 x) \left (2 x^2-x+3\right )^{3/2}-\frac{(2240 x+1277) \sqrt{2 x^2-x+3}}{7750}+\frac{11 \sqrt{\frac{11}{31} \left (224510383+194487500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (224510383+194487500 \sqrt{2}\right )}} \left (\left (87710+54423 \sqrt{2}\right ) x+33287 \sqrt{2}+21136\right )}{\sqrt{2 x^2-x+3}}\right )}{38750}-\frac{11 \sqrt{\frac{11}{31} \left (194487500 \sqrt{2}-224510383\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (194487500 \sqrt{2}-224510383\right )}} \left (\left (87710-54423 \sqrt{2}\right ) x-33287 \sqrt{2}+21136\right )}{\sqrt{2 x^2-x+3}}\right )}{38750}-\frac{4799 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2500 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1277 + 2240*x)*Sqrt[3 - x + 2*x^2])/7750 + (4*(4 - 5*x)*(3 - x + 2*x^2)^(3/2))/155 + ((3 + 10*x)*(3 - x + 2
*x^2)^(5/2))/(31*(2 + 3*x + 5*x^2)) - (4799*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2500*Sqrt[2]) + (11*Sqrt[(11*(224510
383 + 194487500*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(224510383 + 194487500*Sqrt[2]))]*(21136 + 33287*Sqrt[2] + (
87710 + 54423*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/38750 - (11*Sqrt[(11*(-224510383 + 194487500*Sqrt[2]))/31]*Ar
cTanh[(Sqrt[11/(62*(-224510383 + 194487500*Sqrt[2]))]*(21136 - 33287*Sqrt[2] + (87710 - 54423*Sqrt[2])*x))/Sqr
t[3 - x + 2*x^2]])/38750

Rule 971

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

Rule 1066

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

Rule 1076

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c
, 0] && NeQ[e^2 - 4*d*f, 0]

Rule 619

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

Rule 215

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

Rule 1035

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (-\frac{75}{2}+15 x+80 x^2\right )}{2+3 x+5 x^2} \, dx\\ &=\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\left (87660-54300 x-53760 x^2\right ) \sqrt{3-x+2 x^2}}{2+3 x+5 x^2} \, dx}{18600}\\ &=-\frac{(1277+2240 x) \sqrt{3-x+2 x^2}}{7750}+\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-29217120+21064200 x-17852280 x^2}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{1860000}\\ &=-\frac{(1277+2240 x) \sqrt{3-x+2 x^2}}{7750}+\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-110381040+158877840 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{9300000}+\frac{4799 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2500}\\ &=-\frac{(1277+2240 x) \sqrt{3-x+2 x^2}}{7750}+\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{319440 \left (9272+3801 \sqrt{2}\right )+319440 \left (1670-5471 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{204600000 \sqrt{2}}-\frac{\int \frac{319440 \left (9272-3801 \sqrt{2}\right )+319440 \left (1670+5471 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{204600000 \sqrt{2}}+\frac{4799 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2500 \sqrt{46}}\\ &=-\frac{(1277+2240 x) \sqrt{3-x+2 x^2}}{7750}+\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{4799 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2500 \sqrt{2}}-\frac{\left (3865224 \left (388975000-224510383 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6326598643200 \left (224510383-194487500 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{319440 \left (21136-33287 \sqrt{2}\right )+319440 \left (87710-54423 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3875}-\frac{\left (3865224 \left (388975000+224510383 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6326598643200 \left (224510383+194487500 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{319440 \left (21136+33287 \sqrt{2}\right )+319440 \left (87710+54423 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{3875}\\ &=-\frac{(1277+2240 x) \sqrt{3-x+2 x^2}}{7750}+\frac{4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac{(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{4799 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2500 \sqrt{2}}+\frac{11 \sqrt{\frac{11}{31} \left (224510383+194487500 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (224510383+194487500 \sqrt{2}\right )}} \left (21136+33287 \sqrt{2}+\left (87710+54423 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{38750}-\frac{11 \sqrt{\frac{11}{31} \left (-224510383+194487500 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{62 \left (-224510383+194487500 \sqrt{2}\right )}} \left (21136-33287 \sqrt{2}+\left (87710-54423 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{38750}\\ \end{align*}

Mathematica [C]  time = 1.69943, size = 685, normalized size = 2.69 \[ -\frac{-1922000 \sqrt{2 x^2-x+3} x^3+7784100 \sqrt{2 x^2-x+3} x^2-5759180 \sqrt{2 x^2-x+3} x-5577520 \sqrt{2 x^2-x+3}-4611839 \sqrt{2} \left (5 x^2+3 x+2\right ) \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )+284735 \sqrt{286-22 i \sqrt{31}} x^2 \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+482405 i \sqrt{682 \left (13-i \sqrt{31}\right )} x^2 \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+170841 \sqrt{286-22 i \sqrt{31}} x \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+289443 i \sqrt{682 \left (13-i \sqrt{31}\right )} x \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+11 i \sqrt{286+22 i \sqrt{31}} \left (8771 \sqrt{31}+5177 i\right ) \left (5 x^2+3 x+2\right ) \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x-22 x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+113894 \sqrt{286-22 i \sqrt{31}} \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )+192962 i \sqrt{682 \left (13-i \sqrt{31}\right )} \tanh ^{-1}\left (\frac{-4 i \sqrt{31} x+22 x+i \sqrt{31}-63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{4805000 \left (5 x^2+3 x+2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-5577520*Sqrt[3 - x + 2*x^2] - 5759180*x*Sqrt[3 - x + 2*x^2] + 7784100*x^2*Sqrt[3 - x + 2*x^2] - 1922000*x^3
*Sqrt[3 - x + 2*x^2] - 4611839*Sqrt[2]*(2 + 3*x + 5*x^2)*ArcSinh[(-1 + 4*x)/Sqrt[23]] + (11*I)*Sqrt[286 + (22*
I)*Sqrt[31]]*(5177*I + 8771*Sqrt[31])*(2 + 3*x + 5*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])] + (192962*I)*Sqrt[682*(13 - I*Sqrt[31])]*ArcTanh[(-63 + I*S
qrt[31] + 22*x - (4*I)*Sqrt[31]*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] + 113894*Sqrt[286 - (2
2*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])] + (289443*I)*Sqrt[682*(13 - 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])] + 170841*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])] + (482405*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])] + 284735*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])])/(4805000*(2 + 3*x + 5*x^2))

________________________________________________________________________________________

Maple [B]  time = 0.414, size = 40028, normalized size = 157. \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)^2,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 )}^{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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 5.25698, size = 9335, normalized size = 36.61 \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)^2,x, algorithm="fricas")

[Out]

1/1322759922435707900000*(38925001324*1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt
(224510383*sqrt(2) + 388975000)*arctan(1/296975447063866363819995875*(110935670*sqrt(155590)*(4*1464599010050^
(3/4)*sqrt(62)*(18997882*x^7 - 82713851*x^6 + 169131062*x^5 - 298338397*x^4 + 156222120*x^3 - 89116200*x^2 - s
qrt(2)*(18111018*x^7 - 62947113*x^6 + 135463929*x^5 - 197908246*x^4 + 94500248*x^3 - 34095024*x^2 - 122404608*
x + 71452800) - 142905600*x + 122404608) + 2411645*1464599010050^(1/4)*sqrt(62)*(3035566*x^7 - 47612316*x^6 +
259553720*x^5 - 615321136*x^4 + 807721920*x^3 - 579888000*x^2 - sqrt(2)*(2643323*x^7 - 39854517*x^6 + 20495015
2*x^5 - 451004140*x^4 + 573424416*x^3 - 311722272*x^2 - 434377728*x + 268655616) - 537311232*x + 434377728))*s
qrt(2*x^2 - x + 3)*sqrt(224510383*sqrt(2) + 388975000) + 843027075536136774714827000*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 - 4561
92) - sqrt(77795/920561)*(sqrt(155590)*(4*1464599010050^(3/4)*sqrt(62)*(58767374*x^7 - 85793239*x^6 + 28553994
9*x^5 - 168939120*x^4 + 253241640*x^3 + 601344*x^2 - 4*sqrt(2)*(17889302*x^7 - 25424283*x^6 + 80174553*x^5 - 2
1241168*x^4 + 15593832*x^3 + 58564512*x^2 - 58564512*x) - 601344*x) + 2411645*1464599010050^(1/4)*sqrt(62)*(98
91184*x^7 - 128099264*x^6 + 496592960*x^5 - 666984960*x^4 + 949582080*x^3 + 183223296*x^2 - sqrt(2)*(10181049*
x^7 - 131588036*x^6 + 505509740*x^5 - 637596864*x^4 + 754818336*x^3 + 725677056*x^2 - 725677056*x) - 183223296
*x))*sqrt(2*x^2 - x + 3)*sqrt(224510383*sqrt(2) + 388975000) + 7599242656001778100*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 - 1180
51*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) + 3454201
20727353550*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(-(1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(9733*x
+ 29025) - 38758*x + 19292)*sqrt(224510383*sqrt(2) + 388975000) - 6744561519183110*x^2 - 6056340956001160*sqr
t(2)*(2*x^2 - x + 3) + 20784261008094890*x - 27528822527278000)/x^2) + 9579853131092463349032125*sqrt(31)*(282
8123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sq
rt(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) + 2230640
64*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615
296*x^2 - 24772608*x + 18579456)) + 38925001324*1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x
+ 2)*sqrt(224510383*sqrt(2) + 388975000)*arctan(1/296975447063866363819995875*(110935670*sqrt(155590)*(4*1464
599010050^(3/4)*sqrt(62)*(18997882*x^7 - 82713851*x^6 + 169131062*x^5 - 298338397*x^4 + 156222120*x^3 - 891162
00*x^2 - sqrt(2)*(18111018*x^7 - 62947113*x^6 + 135463929*x^5 - 197908246*x^4 + 94500248*x^3 - 34095024*x^2 -
122404608*x + 71452800) - 142905600*x + 122404608) + 2411645*1464599010050^(1/4)*sqrt(62)*(3035566*x^7 - 47612
316*x^6 + 259553720*x^5 - 615321136*x^4 + 807721920*x^3 - 579888000*x^2 - sqrt(2)*(2643323*x^7 - 39854517*x^6
+ 204950152*x^5 - 451004140*x^4 + 573424416*x^3 - 311722272*x^2 - 434377728*x + 268655616) - 537311232*x + 434
377728))*sqrt(2*x^2 - x + 3)*sqrt(224510383*sqrt(2) + 388975000) - 843027075536136774714827000*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) + 115430
4*x - 456192) - sqrt(77795/920561)*(sqrt(155590)*(4*1464599010050^(3/4)*sqrt(62)*(58767374*x^7 - 85793239*x^6
+ 285539949*x^5 - 168939120*x^4 + 253241640*x^3 + 601344*x^2 - 4*sqrt(2)*(17889302*x^7 - 25424283*x^6 + 801745
53*x^5 - 21241168*x^4 + 15593832*x^3 + 58564512*x^2 - 58564512*x) - 601344*x) + 2411645*1464599010050^(1/4)*sq
rt(62)*(9891184*x^7 - 128099264*x^6 + 496592960*x^5 - 666984960*x^4 + 949582080*x^3 + 183223296*x^2 - sqrt(2)*
(10181049*x^7 - 131588036*x^6 + 505509740*x^5 - 637596864*x^4 + 754818336*x^3 + 725677056*x^2 - 725677056*x) -
183223296*x))*sqrt(2*x^2 - x + 3)*sqrt(224510383*sqrt(2) + 388975000) - 7599242656001778100*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)
- 345420120727353550*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219
328*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((1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2
)*(9733*x + 29025) - 38758*x + 19292)*sqrt(224510383*sqrt(2) + 388975000) + 6744561519183110*x^2 + 60563409560
01160*sqrt(2)*(2*x^2 - x + 3) - 20784261008094890*x + 27528822527278000)/x^2) - 9579853131092463349032125*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)) + 11*1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(31)*(1944875000
*x^2 - 224510383*sqrt(2)*(5*x^2 + 3*x + 2) + 1166925000*x + 777950000)*sqrt(224510383*sqrt(2) + 388975000)*log
(14708117187500/920561*(1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(9733*
x + 29025) - 38758*x + 19292)*sqrt(224510383*sqrt(2) + 388975000) + 6744561519183110*x^2 + 6056340956001160*sq
rt(2)*(2*x^2 - x + 3) - 20784261008094890*x + 27528822527278000)/x^2) - 11*1464599010050^(1/4)*sqrt(155590)*sq
rt(62)*sqrt(31)*(1944875000*x^2 - 224510383*sqrt(2)*(5*x^2 + 3*x + 2) + 1166925000*x + 777950000)*sqrt(2245103
83*sqrt(2) + 388975000)*log(-14708117187500/920561*(1464599010050^(1/4)*sqrt(155590)*sqrt(62)*sqrt(31)*sqrt(2*
x^2 - x + 3)*(sqrt(2)*(9733*x + 29025) - 38758*x + 19292)*sqrt(224510383*sqrt(2) + 388975000) - 67445615191831
10*x^2 - 6056340956001160*sqrt(2)*(2*x^2 - x + 3) + 20784261008094890*x - 27528822527278000)/x^2) + 6347924867
76896221210*sqrt(2)*(5*x^2 + 3*x + 2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 170
678699669123600*(3100*x^3 - 12555*x^2 + 9289*x + 8996)*sqrt(2*x^2 - x + 3))/(5*x^2 + 3*x + 2)

________________________________________________________________________________________

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 )^{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)**2,x)

[Out]

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

[Out]

Exception raised: TypeError