\(\int \frac {1}{(4+x-2 x^2) (2+3 x+5 x^2)^{5/2}} \, dx\) [161]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {481+1035 x}{21669 \left (2+3 x+5 x^2\right )^{3/2}}+\frac {107 (28891+78685 x)}{104343458 \sqrt {2+3 x+5 x^2}}-\frac {\sqrt {\frac {4466971907+775844179 \sqrt {33}}{7689}} \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{434312}+\frac {\left (7721-1177 \sqrt {33}\right ) \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{108578 \sqrt {66 \left (107+11 \sqrt {33}\right )}} \] Output:

1/21669*(481+1035*x)/(5*x^2+3*x+2)^(3/2)+107/104343458*(28891+78685*x)/(5* 
x^2+3*x+2)^(1/2)-1/3339424968*(34346546992923+5965465892331*33^(1/2))^(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))+1/108578*(7721-1177*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))/(706 
2+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 = 0.97 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {25496548+93289413 x+122143710 x^2+126289425 x^3}{313030374 \left (2+3 x+5 x^2\right )^{3/2}}+\frac {\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-18055 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )+44490 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2354 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{217156 \sqrt {5}} \] Input:

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

Output:

(25496548 + 93289413*x + 122143710*x^2 + 126289425*x^3)/(313030374*(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 & , (-18055*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - # 
1] + 44490*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 2354*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) & ]/(217156*Sqrt[5])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.10, 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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}-\frac {\int -\frac {3 \left (-2760 x^2+698 x+7349\right )}{2 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{21669}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-2760 x^2+698 x+7349}{\left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\frac {\int \frac {961 (4449-2354 x)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{7223}+\frac {107 (78685 x+28891)}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {31}{466} \int \frac {4449-2354 x}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx+\frac {107 (78685 x+28891)}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {\frac {31}{466} \left (-\frac {2}{33} \left (38841+7721 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx-\frac {2}{33} \left (38841-7721 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx\right )+\frac {107 (78685 x+28891)}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {31}{466} \left (\frac {4}{33} \left (38841+7721 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )+\frac {4}{33} \left (38841-7721 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )\right )+\frac {107 (78685 x+28891)}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {31}{466} \left (-\frac {1}{33} \sqrt {\frac {2}{107-11 \sqrt {33}}} \left (38841+7721 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )-\frac {1}{33} \left (38841-7721 \sqrt {33}\right ) \sqrt {\frac {2}{107+11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )\right )+\frac {107 (78685 x+28891)}{7223 \sqrt {5 x^2+3 x+2}}}{14446}+\frac {1035 x+481}{21669 \left (5 x^2+3 x+2\right )^{3/2}}\)

Input:

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

Output:

(481 + 1035*x)/(21669*(2 + 3*x + 5*x^2)^(3/2)) + ((107*(28891 + 78685*x))/ 
(7223*Sqrt[2 + 3*x + 5*x^2]) + (31*(-1/33*(Sqrt[2/(107 - 11*Sqrt[33])]*(38 
841 + 7721*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])]) - ((38841 - 7721*Sqrt 
[33])*Sqrt[2/(107 + 11*Sqrt[33])]*ArcTanh[(19 + 3*Sqrt[33] + 2*(11 + 5*Sqr 
t[33])*x)/(2*Sqrt[2*(107 + 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])])/33))/466 
)/14446
 

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 [C] (warning: unable to verify)

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

Time = 4.49 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.29

method	result	size
trager	\(\text {Expression too large to display}\)	\(471\)
default	\(\text {Expression too large to display}\)	\(868\)

Input:

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

Output:

1/313030374*(126289425*x^3+122143710*x^2+93289413*x[phone])/(5*x^2+3*x+2 
)^(3/2)+1/3339424968*RootOf(_Z^2+60539618304*RootOf(129913344*_Z^4-1474100 
72931*_Z^2+188626913344)^2-68693093985846)*ln(-(-155073098028220416*x*Root 
Of(129913344*_Z^4-147410072931*_Z^2+188626913344)^4*RootOf(_Z^2+6053961830 
4*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2-68693093985846)+ 
185700032569843862400*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344 
)^2*RootOf(_Z^2+60539618304*RootOf(129913344*_Z^4-147410072931*_Z^2+188626 
913344)^2-68693093985846)*x+104732638156902355239720960*(5*x^2+3*x+2)^(1/2 
)*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2-7886774808737240 
064*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2*RootOf(_Z^2+60 
539618304*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2-68693093 
985846)-27828862679762090275069*RootOf(_Z^2+60539618304*RootOf(129913344*_ 
Z^4-147410072931*_Z^2+188626913344)^2-68693093985846)*x-120846116382610297 
241654369124*(5*x^2+3*x+2)^(1/2)+1385018122685360936344*RootOf(_Z^2+605396 
18304*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2-686930939858 
46))/(3936768*x*RootOf(129913344*_Z^4-147410072931*_Z^2+188626913344)^2-26 
21408043*x-3103376716))+4/54289*RootOf(129913344*_Z^4-147410072931*_Z^2+18 
8626913344)*ln((18066015920287678464*x*RootOf(129913344*_Z^4-147410072931* 
_Z^2+188626913344)^5-19364240028027826126272*RootOf(129913344*_Z^4-1474100 
72931*_Z^2+188626913344)^3*x+49589317309139372746080*(5*x^2+3*x+2)^(1/2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (150) = 300\).

Time = 0.14 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {2883 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} \log \left (-\frac {3 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} {\left (178291 \, \sqrt {\frac {11}{3}} - 358369\right )} + 5211744 \, \sqrt {\frac {11}{3}} {\left (3 \, x + 4\right )} - 133768096 \, x - 34744960}{x}\right ) - 2883 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} \log \left (\frac {3 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} {\left (178291 \, \sqrt {\frac {11}{3}} - 358369\right )} - 5211744 \, \sqrt {\frac {11}{3}} {\left (3 \, x + 4\right )} + 133768096 \, x + 34744960}{x}\right ) + 2883 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} \log \left (\frac {3 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (178291 \, \sqrt {\frac {11}{3}} + 358369\right )} \sqrt {-\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} + 5211744 \, \sqrt {\frac {11}{3}} {\left (3 \, x + 4\right )} + 133768096 \, x + 34744960}{x}\right ) - 2883 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} \log \left (-\frac {3 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (178291 \, \sqrt {\frac {11}{3}} + 358369\right )} \sqrt {-\frac {70531289}{233} \, \sqrt {\frac {11}{3}} + \frac {4466971907}{7689}} - 5211744 \, \sqrt {\frac {11}{3}} {\left (3 \, x + 4\right )} - 133768096 \, x - 34744960}{x}\right ) + 8 \, {\left (126289425 \, x^{3} + 122143710 \, x^{2} + 93289413 \, x + 25496548\right )} \sqrt {5 \, x^{2} + 3 \, x + 2}}{2504242992 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/2504242992*(2883*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(70531289/233 
*sqrt(11/3) + 4466971907/7689)*log(-(3*sqrt(5*x^2 + 3*x + 2)*sqrt(70531289 
/233*sqrt(11/3) + 4466971907/7689)*(178291*sqrt(11/3) - 358369) + 5211744* 
sqrt(11/3)*(3*x + 4) - 133768096*x - 34744960)/x) - 2883*(25*x^4 + 30*x^3 
+ 29*x^2 + 12*x + 4)*sqrt(70531289/233*sqrt(11/3) + 4466971907/7689)*log(( 
3*sqrt(5*x^2 + 3*x + 2)*sqrt(70531289/233*sqrt(11/3) + 4466971907/7689)*(1 
78291*sqrt(11/3) - 358369) - 5211744*sqrt(11/3)*(3*x + 4) + 133768096*x + 
34744960)/x) + 2883*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-70531289/2 
33*sqrt(11/3) + 4466971907/7689)*log((3*sqrt(5*x^2 + 3*x + 2)*(178291*sqrt 
(11/3) + 358369)*sqrt(-70531289/233*sqrt(11/3) + 4466971907/7689) + 521174 
4*sqrt(11/3)*(3*x + 4) + 133768096*x + 34744960)/x) - 2883*(25*x^4 + 30*x^ 
3 + 29*x^2 + 12*x + 4)*sqrt(-70531289/233*sqrt(11/3) + 4466971907/7689)*lo 
g(-(3*sqrt(5*x^2 + 3*x + 2)*(178291*sqrt(11/3) + 358369)*sqrt(-70531289/23 
3*sqrt(11/3) + 4466971907/7689) - 5211744*sqrt(11/3)*(3*x + 4) - 133768096 
*x - 34744960)/x) + 8*(126289425*x^3 + 122143710*x^2 + 93289413*x + 254965 
48)*sqrt(5*x^2 + 3*x + 2))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
 

Sympy [F]

\[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=- \int \frac {1}{50 x^{6} \sqrt {5 x^{2} + 3 x + 2} + 35 x^{5} \sqrt {5 x^{2} + 3 x + 2} - 72 x^{4} \sqrt {5 x^{2} + 3 x + 2} - 125 x^{3} \sqrt {5 x^{2} + 3 x + 2} - 120 x^{2} \sqrt {5 x^{2} + 3 x + 2} - 52 x \sqrt {5 x^{2} + 3 x + 2} - 16 \sqrt {5 x^{2} + 3 x + 2}}\, dx \] Input:

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

Output:

-Integral(1/(50*x**6*sqrt(5*x**2 + 3*x + 2) + 35*x**5*sqrt(5*x**2 + 3*x + 
2) - 72*x**4*sqrt(5*x**2 + 3*x + 2) - 125*x**3*sqrt(5*x**2 + 3*x + 2) - 12 
0*x**2*sqrt(5*x**2 + 3*x + 2) - 52*x*sqrt(5*x**2 + 3*x + 2) - 16*sqrt(5*x* 
*2 + 3*x + 2)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (150) = 300\).

Time = 0.15 (sec) , antiderivative size = 1099, normalized size of antiderivative = 5.33 \[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/95139*sqrt(33)*(6200*sqrt(33)*x/(11*sqrt(33)*(5*x^2 + 3*x + 2)^(3/2) + 1 
07*(5*x^2 + 3*x + 2)^(3/2)) - 6200*sqrt(33)*x/(11*sqrt(33)*(5*x^2 + 3*x + 
2)^(3/2) - 107*(5*x^2 + 3*x + 2)^(3/2)) + 74400*sqrt(33)*x/(1177*sqrt(33)* 
sqrt(5*x^2 + 3*x + 2) + 7721*sqrt(5*x^2 + 3*x + 2)) - 74400*sqrt(33)*x/(11 
77*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 7721*sqrt(5*x^2 + 3*x + 2)) + 8000*sqr 
t(33)*x/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) + 107*sqrt(5*x^2 + 3*x + 2)) - 
8000*sqrt(33)*x/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 107*sqrt(5*x^2 + 3*x 
+ 2)) + 13640*x/(11*sqrt(33)*(5*x^2 + 3*x + 2)^(3/2) + 107*(5*x^2 + 3*x + 
2)^(3/2)) + 13640*x/(11*sqrt(33)*(5*x^2 + 3*x + 2)^(3/2) - 107*(5*x^2 + 3* 
x + 2)^(3/2)) + 163680*x/(1177*sqrt(33)*sqrt(5*x^2 + 3*x + 2) + 7721*sqrt( 
5*x^2 + 3*x + 2)) + 163680*x/(1177*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 7721*s 
qrt(5*x^2 + 3*x + 2)) + 17600*x/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) + 107*s 
qrt(5*x^2 + 3*x + 2)) + 17600*x/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 107*s 
qrt(5*x^2 + 3*x + 2)) + 1860*sqrt(33)/(11*sqrt(33)*(5*x^2 + 3*x + 2)^(3/2) 
 + 107*(5*x^2 + 3*x + 2)^(3/2)) - 1860*sqrt(33)/(11*sqrt(33)*(5*x^2 + 3*x 
+ 2)^(3/2) - 107*(5*x^2 + 3*x + 2)^(3/2)) + 22320*sqrt(33)/(1177*sqrt(33)* 
sqrt(5*x^2 + 3*x + 2) + 7721*sqrt(5*x^2 + 3*x + 2)) - 22320*sqrt(33)/(1177 
*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 7721*sqrt(5*x^2 + 3*x + 2)) + 2400*sqrt( 
33)/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) + 107*sqrt(5*x^2 + 3*x + 2)) - 2400 
*sqrt(33)/(11*sqrt(33)*sqrt(5*x^2 + 3*x + 2) - 107*sqrt(5*x^2 + 3*x + 2...
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (4+x-2 x^2\right ) \left (2+3 x+5 x^2\right )^{5/2}} \, dx=\frac {3 \, {\left (535 \, {\left (78685 \, x + 76102\right )} x + 31096471\right )} x + 25496548}{313030374 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}}} + 0.0000833934028287498 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.00248050182749728 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) - 0.0000833934028286578 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.00248050182749728 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:

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

Output:

1/313030374*(3*(535*(78685*x + 76102)*x + 31096471)*x + 25496548)/(5*x^2 + 
 3*x + 2)^(3/2) + 0.0000833934028287498*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x 
+ 2) + 8.38267526007000) - 0.00248050182749728*log(-sqrt(5)*x + sqrt(5*x^2 
 + 3*x + 2) - 0.312157316296000) - 0.0000833934028286578*log(-sqrt(5)*x + 
sqrt(5*x^2 + 3*x + 2) - 0.842024981991000) + 0.00248050182749728*log(-sqrt 
(5)*x + sqrt(5*x^2 + 3*x + 2) - 4.99242498429000)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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