\(\int \frac {1}{(4+x-2 x^2)^2 (2+3 x+5 x^2)^{5/2}} \, dx\) [162]

Optimal result

Integrand size = 25, antiderivative size = 246 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {1909421+953635 x}{222150588 \left (2+3 x+5 x^2\right )^{3/2}}-\frac {235-214 x}{15378 \left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{3/2}}+\frac {34495558451+65901205285 x}{3209187394248 \sqrt {2+3 x+5 x^2}}-\frac {\left (191614963+30645571 \sqrt {33}\right ) \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{3339424968 \sqrt {66 \left (107-11 \sqrt {33}\right )}}+\frac {\left (191614963-30645571 \sqrt {33}\right ) \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{3339424968 \sqrt {66 \left (107+11 \sqrt {33}\right )}} \] Output:

1/222150588*(1909421+953635*x)/(5*x^2+3*x+2)^(3/2)-1/15378*(235-214*x)/(-2 
*x^2+x+4)/(5*x^2+3*x+2)^(3/2)+1/3209187394248*(34495558451+65901205285*x)/ 
(5*x^2+3*x+2)^(1/2)-1/3339424968*(191614963+30645571*33^(1/2))*arctanh(1/2 
*(19-3*33^(1/2)+2*(11-5*33^(1/2))*x)/(214-22*33^(1/2))^(1/2)/(5*x^2+3*x+2) 
^(1/2))/(7062-726*33^(1/2))^(1/2)+1/3339424968*(191614963-30645571*33^(1/2 
))*arctanh(1/2*(19+3*33^(1/2)+2*(11+5*33^(1/2))*x)/(214+22*33^(1/2))^(1/2) 
/(5*x^2+3*x+2)^(1/2))/(7062+726*33^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.32 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {-337257025412-1137494801224 x-1536641704237 x^2-1190074919544 x^3+410856789795 x^4+659012052850 x^5}{3209187394248 \left (-4-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^{3/2}}+\frac {\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-8023187 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )+5896170 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+384758 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{50597348 \sqrt {5}}+\frac {\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {603087599 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )[phone] \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}[phone] \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{6678849936 \sqrt {5}} \] Input:

Integrate[1/((4 + x - 2*x^2)^2*(2 + 3*x + 5*x^2)^(5/2)),x]
 

Output:

(-337257025412 - 1137494801224*x - 1536641704237*x^2 - 1190074919544*x^3 + 
 410856789795*x^4 + 659012052850*x^5)/(3209187394248*(-4 - x + 2*x^2)*(2 + 
 3*x + 5*x^2)^(3/2)) + RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*# 
1^3 + 2*#1^4 & , (-8023187*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2 
] - #1] + 5896170*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 3847 
58*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5 
] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ]/(50597348*Sqrt[5]) + RootSum[-22 
+ 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (603087599*Sqrt[5] 
*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1] + 333008230*Log[-(Sqrt[5]* 
x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 10503086*Sqrt[5]*Log[-(Sqrt[5]*x) + 
Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4 
*#1^3) & ]/(6678849936*Sqrt[5])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1305, 27, 2135, 27, 2135, 27, 1365, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (-2 x^2+x+4\right )^2 \left (5 x^2+3 x+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1305

\(\displaystyle \frac {\int \frac {8560 x^2-7470 x+4483}{2 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{5/2}}dx}{15378}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {8560 x^2-7470 x+4483}{\left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{5/2}}dx}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\frac {\int \frac {3 \left (-7629080 x^2-15665426 x+59121207\right )}{2 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{21669}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-7629080 x^2-15665426 x+59121207}{\left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\frac {\frac {\int \frac {961 (111130267-61291142 x)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{7223}+\frac {65901205285 x+34495558451}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {31}{466} \int \frac {111130267-61291142 x}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx+\frac {65901205285 x+34495558451}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {\frac {\frac {31}{466} \left (-\frac {2}{33} \left (1011303843+191614963 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx-\frac {2}{33} \left (1011303843-191614963 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx\right )+\frac {65901205285 x+34495558451}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {31}{466} \left (\frac {4}{33} \left (1011303843+191614963 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )+\frac {4}{33} \left (1011303843-191614963 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )\right )+\frac {65901205285 x+34495558451}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {31}{466} \left (-\frac {1}{33} \sqrt {\frac {2}{107-11 \sqrt {33}}} \left (1011303843+191614963 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )-\frac {1}{33} \left (1011303843-191614963 \sqrt {33}\right ) \sqrt {\frac {2}{107+11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )\right )+\frac {65901205285 x+34495558451}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {953635 x+1909421}{7223 \left (5 x^2+3 x+2\right )^{3/2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}\)

Input:

Int[1/((4 + x - 2*x^2)^2*(2 + 3*x + 5*x^2)^(5/2)),x]
 

Output:

-1/15378*(235 - 214*x)/((4 + x - 2*x^2)*(2 + 3*x + 5*x^2)^(3/2)) + ((19094 
21 + 953635*x)/(7223*(2 + 3*x + 5*x^2)^(3/2)) + ((34495558451 + 6590120528 
5*x)/(7223*Sqrt[2 + 3*x + 5*x^2]) + (31*(-1/33*(Sqrt[2/(107 - 11*Sqrt[33]) 
]*(1011303843 + 191614963*Sqrt[33])*ArcTanh[(19 - 3*Sqrt[33] + 2*(11 - 5*S 
qrt[33])*x)/(2*Sqrt[2*(107 - 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])]) - ((10 
11303843 - 191614963*Sqrt[33])*Sqrt[2/(107 + 11*Sqrt[33])]*ArcTanh[(19 + 3 
*Sqrt[33] + 2*(11 + 5*Sqrt[33])*x)/(2*Sqrt[2*(107 + 11*Sqrt[33])]*Sqrt[2 + 
 3*x + 5*x^2])])/33))/466)/14446)/30756
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1305

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] - Simp[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*Si 
mp[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 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*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] && PosQ[b^2 - 4*a*c]
 

rule 2135

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, Simp[(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)))*( 
(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), x] + Simp[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]] /; F 
reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && 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]
 

Maple [A] (verified)

Time = 4.69 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {659012052850 x^{5}[phone] x^{4}-1190074919544 x^{3}-1536641704237 x^{2}-1137494801224 x -337257025412}{3209187394248 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}} \left (2 x^{2}-x -4\right )}-\frac {\left (-191614963+30645571 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214+22 \sqrt {33}+8 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )}{\sqrt {214+22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )^{2}+16 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )+214+22 \sqrt {33}}}\right )}{110201023944 \sqrt {214+22 \sqrt {33}}}-\frac {\left (191614963+30645571 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214-22 \sqrt {33}+8 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )}{\sqrt {214-22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}+16 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )+214-22 \sqrt {33}}}\right )}{110201023944 \sqrt {214-22 \sqrt {33}}}\)	\(241\)
trager	\(\text {Expression too large to display}\)	\(492\)
default	\(\text {Expression too large to display}\)	\(2154\)

Input:

int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3209187394248*(659012052850*x^5+410856789795*x^4-1190074919544*x^3-15366 
41704237*x^2-1137494801224*x-337257025412)/(5*x^2+3*x+2)^(3/2)/(2*x^2-x-4) 
-1/110201023944*(-191614963+30645571*33^(1/2))*33^(1/2)/(214+22*33^(1/2))^ 
(1/2)*arctanh(8*(107/4+11/4*33^(1/2)+(11/2+5/2*33^(1/2))*(x-1/4*33^(1/2)-1 
/4))/(214+22*33^(1/2))^(1/2)/(80*(x-1/4*33^(1/2)-1/4)^2+16*(11/2+5/2*33^(1 
/2))*(x-1/4*33^(1/2)-1/4)+214+22*33^(1/2))^(1/2))-1/110201023944*(19161496 
3+30645571*33^(1/2))*33^(1/2)/(214-22*33^(1/2))^(1/2)*arctanh(8*(107/4-11/ 
4*33^(1/2)+(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^(1/2)))/(214-22*33^(1/2))^(1/ 
2)/(80*(x-1/4+1/4*33^(1/2))^2+16*(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^(1/2))+ 
214-22*33^(1/2))^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (188) = 376\).

Time = 0.11 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x, algorithm="fricas")
 

Output:

1/25673499153984*(961*(50*x^6 + 35*x^5 - 72*x^4 - 125*x^3 - 120*x^2 - 52*x 
 - 16)*sqrt(15162357162557627/233*sqrt(33) + 2876991646833394763/7689)*log 
(-(sqrt(5*x^2 + 3*x + 2)*sqrt(15162357162557627/233*sqrt(33) + 28769916468 
33394763/7689)*(4566831233*sqrt(33) - 27060312081) + 715538790282752*sqrt( 
33)*(3*x + 4) - 55096486851771904*x - 14310775805655040)/x) - 961*(50*x^6 
+ 35*x^5 - 72*x^4 - 125*x^3 - 120*x^2 - 52*x - 16)*sqrt(15162357162557627/ 
233*sqrt(33) + 2876991646833394763/7689)*log((sqrt(5*x^2 + 3*x + 2)*sqrt(1 
5162357162557627/233*sqrt(33) + 2876991646833394763/7689)*(4566831233*sqrt 
(33) - 27060312081) - 715538790282752*sqrt(33)*(3*x + 4) + 550964868517719 
04*x + 14310775805655040)/x) + 961*(50*x^6 + 35*x^5 - 72*x^4 - 125*x^3 - 1 
20*x^2 - 52*x - 16)*sqrt(-15162357162557627/233*sqrt(33) + 287699164683339 
4763/7689)*log((sqrt(5*x^2 + 3*x + 2)*(4566831233*sqrt(33) + 27060312081)* 
sqrt(-15162357162557627/233*sqrt(33) + 2876991646833394763/7689) + 7155387 
90282752*sqrt(33)*(3*x + 4) + 55096486851771904*x + 14310775805655040)/x) 
- 961*(50*x^6 + 35*x^5 - 72*x^4 - 125*x^3 - 120*x^2 - 52*x - 16)*sqrt(-151 
62357162557627/233*sqrt(33) + 2876991646833394763/7689)*log(-(sqrt(5*x^2 + 
 3*x + 2)*(4566831233*sqrt(33) + 27060312081)*sqrt(-15162357162557627/233* 
sqrt(33) + 2876991646833394763/7689) - 715538790282752*sqrt(33)*(3*x + 4) 
- 55096486851771904*x - 14310775805655040)/x) + 8*(659012052850*x^5 + 4108 
56789795*x^4 - 1190074919544*x^3 - 1536641704237*x^2 - 1137494801224*x ...
 

Sympy [F]

\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (2 x^{2} - x - 4\right )^{2} \left (5 x^{2} + 3 x + 2\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(-2*x**2+x+4)**2/(5*x**2+3*x+2)**(5/2),x)
 

Output:

Integral(1/((2*x**2 - x - 4)**2*(5*x**2 + 3*x + 2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {5}{2}} {\left (2 \, x^{2} - x - 4\right )}^{2}} \,d x } \] Input:

integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((5*x^2 + 3*x + 2)^(5/2)*(2*x^2 - x - 4)^2), x)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=-\frac {10503086 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 38552366 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 70815609 \, \sqrt {5} x + 3544376 \, \sqrt {5} + 70815609 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{3339424968 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 91 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 44 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 22\right )}} + \frac {{\left (5 \, {\left (1566193045 \, x + 1686888069\right )} x + 6079914951\right )} x + 1961073075}{72936077142 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + 0.0000439911622979157 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.00204746962514176 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) - 0.0000439911622979157 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.00204746962514176 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:

integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x, algorithm="giac")
 

Output:

-1/3339424968*(10503086*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 38552366*s 
qrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 70815609*sqrt(5)*x + 354437 
6*sqrt(5) + 70815609*sqrt(5*x^2 + 3*x + 2))/(2*(sqrt(5)*x - sqrt(5*x^2 + 3 
*x + 2))^4 - 2*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 91*(sqrt(5) 
*x - sqrt(5*x^2 + 3*x + 2))^2 - 44*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 
 2)) - 22) + 1/72936077142*((5*(1566193045*x + 1686888069)*x + 6079914951) 
*x + 1961073075)/(5*x^2 + 3*x + 2)^(3/2) + 0.0000439911622979157*log(-sqrt 
(5)*x + sqrt(5*x^2 + 3*x + 2) + 8.38267526007000) - 0.00204746962514176*lo 
g(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.312157316296000) - 0.000043991162 
2979157*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.842024981991000) + 0.00 
204746962514176*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 4.99242498429000)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (-2\,x^2+x+4\right )}^2\,{\left (5\,x^2+3\,x+2\right )}^{5/2}} \,d x \] Input:

int(1/((x - 2*x^2 + 4)^2*(3*x + 5*x^2 + 2)^(5/2)),x)
 

Output:

int(1/((x - 2*x^2 + 4)^2*(3*x + 5*x^2 + 2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (-2 x^{2}+x +4\right )^{2} \left (5 x^{2}+3 x +2\right )^{\frac {5}{2}}}d x \] Input:

int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x)
 

Output:

int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(5/2),x)