\(\int \frac {1}{(4+x-2 x^2)^3 \sqrt {2+3 x+5 x^2}} \, dx\) [152]

Optimal result

Integrand size = 25, antiderivative size = 233 \[ \int \frac {1}{\left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2}} \, dx=-\frac {(235-214 x) \sqrt {2+3 x+5 x^2}}{30756 \left (4+x-2 x^2\right )^2}-\frac {(1763725-1644506 x) \sqrt {2+3 x+5 x^2}}{315310512 \left (4+x-2 x^2\right )}-\frac {\left (22434211+746867 \sqrt {33}\right ) \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{105103504 \sqrt {66 \left (107-11 \sqrt {33}\right )}}+\frac {\left (22434211-746867 \sqrt {33}\right ) \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{105103504 \sqrt {66 \left (107+11 \sqrt {33}\right )}} \] Output:

-1/30756*(235-214*x)*(5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^2-(1763725-1644506*x 
)*(5*x^2+3*x+2)^(1/2)/(-630621024*x^2+315310512*x[phone]/105103504* 
(22434211+746867*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/ 
105103504*(22434211-746867*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 = 1.08 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2}} \, dx=\frac {-\frac {1922 \sqrt {2+3 x+5 x^2} \left (9464120-7008227 x-5171956 x^2+3289012 x^3\right )}{\left (4+x-2 x^2\right )^2}-108859442055 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]+8201600 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {1147243 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+757075 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-2 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {4671198570463 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}[phone] \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{606026804064} \] Input:

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

Output:

((-1922*Sqrt[2 + 3*x + 5*x^2]*(9464120 - 7008227*x - 5171956*x^2 + 3289012 
*x^3))/(4 + x - 2*x^2)^2 - 108859442055*RootSum[-22 + 44*Sqrt[5]*#1 - 91*# 
1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] 
 - #1]/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ] + 8201600*RootSu 
m[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (1147243*Sqr 
t[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 757075*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) & ] - 2*RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]* 
#1^3 + 2*#1^4 & , (4671198570463*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 
 5*x^2] - #1]*#1 + 3102459942439*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) & ])/6060268040 
64
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1305, 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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {\int \frac {4280 x^2-1700 x+14757}{2 \left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx}{30756}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4280 x^2-1700 x+14757}{\left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {-\frac {\int -\frac {9 (11590539-1493734 x)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{15378}-\frac {\sqrt {5 x^2+3 x+2} (1763725-1644506 x)}{5126 \left (-2 x^2+x+4\right )}}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {11590539-1493734 x}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{10252}-\frac {(1763725-1644506 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {\frac {3 \left (-\frac {2}{33} \left (24646611+22434211 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx-\frac {2}{33} \left (24646611-22434211 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx\right )}{10252}-\frac {(1763725-1644506 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {3 \left (\frac {4}{33} \left (24646611+22434211 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )+\frac {4}{33} \left (24646611-22434211 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )\right )}{10252}-\frac {(1763725-1644506 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 \left (-\frac {1}{33} \sqrt {\frac {2}{107-11 \sqrt {33}}} \left (24646611+22434211 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )-\frac {1}{33} \left (24646611-22434211 \sqrt {33}\right ) \sqrt {\frac {2}{107+11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )\right )}{10252}-\frac {(1763725-1644506 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}}{61512}-\frac {(235-214 x) \sqrt {5 x^2+3 x+2}}{30756 \left (-2 x^2+x+4\right )^2}\)

Input:

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

Output:

-1/30756*((235 - 214*x)*Sqrt[2 + 3*x + 5*x^2])/(4 + x - 2*x^2)^2 + (-1/512 
6*((1763725 - 1644506*x)*Sqrt[2 + 3*x + 5*x^2])/(4 + x - 2*x^2) + (3*(-1/3 
3*(Sqrt[2/(107 - 11*Sqrt[33])]*(24646611 + 22434211*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])]) - ((24646611 - 22434211*Sqrt[33])*Sqrt[2/(107 + 11*Sqr 
t[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))/10252)/61512
 

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 = 4.46 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {\left (3289012 x^{3}-5171956 x^{2}-7008227 x +9464120\right ) \sqrt {5 x^{2}+3 x +2}}{315310512 \left (2 x^{2}-x -4\right )^{2}}-\frac {\left (-22434211+746867 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214+22 \sqrt {33}+8 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )}{\sqrt {214+22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )^{2}+16 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )+214+22 \sqrt {33}}}\right )}{3468415632 \sqrt {214+22 \sqrt {33}}}-\frac {\left (22434211+746867 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214-22 \sqrt {33}+8 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )}{\sqrt {214-22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}+16 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )+214-22 \sqrt {33}}}\right )}{3468415632 \sqrt {214-22 \sqrt {33}}}\)	\(231\)
trager	\(\text {Expression too large to display}\)	\(484\)
default	\(\text {Expression too large to display}\)	\(1160\)

Input:

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

Output:

-1/315310512*(3289012*x^3-5171956*x^2-7008227*x+9464120)/(2*x^2-x-4)^2*(5* 
x^2+3*x+2)^(1/2)-1/3468415632*(-22434211+746867*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))-1/3468415632* 
(22434211+746867*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. 376 vs. \(2 (181) = 362\).

Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2}} \, dx=\frac {3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} \log \left (-\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} {\left (253413521 \, \sqrt {33} - 2418159777\right )} + 60610760346848 \, \sqrt {33} {\left (3 \, x + 4\right )} - 4667028546707296 \, x - 1212215206936960}{x}\right ) - 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} \log \left (\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} {\left (253413521 \, \sqrt {33} - 2418159777\right )} - 60610760346848 \, \sqrt {33} {\left (3 \, x + 4\right )} + 4667028546707296 \, x + 1212215206936960}{x}\right ) + 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {-\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} \log \left (\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} {\left (253413521 \, \sqrt {33} + 2418159777\right )} \sqrt {-\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} + 60610760346848 \, \sqrt {33} {\left (3 \, x + 4\right )} + 4667028546707296 \, x + 1212215206936960}{x}\right ) - 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {-\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} \log \left (-\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} {\left (253413521 \, \sqrt {33} + 2418159777\right )} \sqrt {-\frac {211917426800849}{699} \, \sqrt {33} + \frac {16996616820423467}{7689}} - 60610760346848 \, \sqrt {33} {\left (3 \, x + 4\right )} - 4667028546707296 \, x - 1212215206936960}{x}\right ) - 8 \, {\left (3289012 \, x^{3} - 5171956 \, x^{2} - 7008227 \, x + 9464120\right )} \sqrt {5 \, x^{2} + 3 \, x + 2}}{2522484096 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )}} \] Input:

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

Output:

1/2522484096*(3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(211917426800849/6 
99*sqrt(33) + 16996616820423467/7689)*log(-(sqrt(5*x^2 + 3*x + 2)*sqrt(211 
917426800849/699*sqrt(33) + 16996616820423467/7689)*(253413521*sqrt(33) - 
2418159777) + 60610760346848*sqrt(33)*(3*x + 4) - 4667028546707296*x - 121 
2215206936960)/x) - 3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(21191742680 
0849/699*sqrt(33) + 16996616820423467/7689)*log((sqrt(5*x^2 + 3*x + 2)*sqr 
t(211917426800849/699*sqrt(33) + 16996616820423467/7689)*(253413521*sqrt(3 
3) - 2418159777) - 60610760346848*sqrt(33)*(3*x + 4) + 4667028546707296*x 
+ 1212215206936960)/x) + 3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(-21191 
7426800849/699*sqrt(33) + 16996616820423467/7689)*log((sqrt(5*x^2 + 3*x + 
2)*(253413521*sqrt(33) + 2418159777)*sqrt(-211917426800849/699*sqrt(33) + 
16996616820423467/7689) + 60610760346848*sqrt(33)*(3*x + 4) + 466702854670 
7296*x + 1212215206936960)/x) - 3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt 
(-211917426800849/699*sqrt(33) + 16996616820423467/7689)*log(-(sqrt(5*x^2 
+ 3*x + 2)*(253413521*sqrt(33) + 2418159777)*sqrt(-211917426800849/699*sqr 
t(33) + 16996616820423467/7689) - 60610760346848*sqrt(33)*(3*x + 4) - 4667 
028546707296*x - 1212215206936960)/x) - 8*(3289012*x^3 - 5171956*x^2 - 700 
8227*x + 9464120)*sqrt(5*x^2 + 3*x + 2))/(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 1 
6)
 

Sympy [F]

\[ \int \frac {1}{\left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2}} \, dx=- \int \frac {1}{8 x^{6} \sqrt {5 x^{2} + 3 x + 2} - 12 x^{5} \sqrt {5 x^{2} + 3 x + 2} - 42 x^{4} \sqrt {5 x^{2} + 3 x + 2} + 47 x^{3} \sqrt {5 x^{2} + 3 x + 2} + 84 x^{2} \sqrt {5 x^{2} + 3 x + 2} - 48 x \sqrt {5 x^{2} + 3 x + 2} - 64 \sqrt {5 x^{2} + 3 x + 2}}\, dx \] Input:

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

Output:

-Integral(1/(8*x**6*sqrt(5*x**2 + 3*x + 2) - 12*x**5*sqrt(5*x**2 + 3*x + 2 
) - 42*x**4*sqrt(5*x**2 + 3*x + 2) + 47*x**3*sqrt(5*x**2 + 3*x + 2) + 84*x 
**2*sqrt(5*x**2 + 3*x + 2) - 48*x*sqrt(5*x**2 + 3*x + 2) - 64*sqrt(5*x**2 
+ 3*x + 2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (181) = 362\).

Time = 0.17 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2}} \, dx=-\frac {8962404 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{7} - 82986840 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{6} - 276608240 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{5} + 4781277004 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} + 20669978041 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} + 5000096948 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 700654966 \, \sqrt {5} x - 188637064 \, \sqrt {5} - 700654966 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{315310512 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 91 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 44 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 22\right )}^{2}} + 0.00162881405851131 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.00472864743789132 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) - 0.00162881405850180 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.00472864743789132 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:

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

Output:

-1/315310512*(8962404*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 82986840*sqr 
t(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 - 276608240*(sqrt(5)*x - sqrt(5 
*x^2 + 3*x + 2))^5 + 4781277004*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2) 
)^4 + 20669978041*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 + 5000096948*sqrt( 
5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 + 700654966*sqrt(5)*x - 188637064 
*sqrt(5) - 700654966*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)^2 + 0.00162881405851131*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) 
+ 8.38267526007000) - 0.00472864743789132*log(-sqrt(5)*x + sqrt(5*x^2 + 3* 
x + 2) - 0.312157316296000) - 0.00162881405850180*log(-sqrt(5)*x + sqrt(5* 
x^2 + 3*x + 2) - 0.842024981991000) + 0.00472864743789132*log(-sqrt(5)*x + 
 sqrt(5*x^2 + 3*x + 2) - 4.99242498429000)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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