\(\int \frac {2+x}{(2+4 x-3 x^2) (1+3 x+2 x^2)^{5/2}} \, dx\) [106]

Optimal result

Integrand size = 30, antiderivative size = 197 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac {2 (273+230 x)}{15 \sqrt {1+3 x+2 x^2}}-\frac {1}{50} \sqrt {\frac {1}{3} \left (4885115+1544809 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (17-4 \sqrt {10}\right ) x}{2 \sqrt {55-17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right )+\frac {1}{50} \sqrt {\frac {1}{3} \left (4885115-1544809 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (17+4 \sqrt {10}\right ) x}{2 \sqrt {55+17 \sqrt {10}} \sqrt {1+3 x+2 x^2}}\right ) \] Output:

2/15*(21+22*x)/(2*x^2+3*x+1)^(3/2)+2/15*(273+230*x)/(2*x^2+3*x+1)^(1/2)-1/ 
150*(14655345+4634427*10^(1/2))^(1/2)*arctanh(1/2*(12-3*10^(1/2)+(17-4*10^ 
(1/2))*x)/(55-17*10^(1/2))^(1/2)/(2*x^2+3*x+1)^(1/2))+1/150*(14655345-4634 
427*10^(1/2))^(1/2)*arctanh(1/2*(12+3*10^(1/2)+(17+4*10^(1/2))*x)/(55+17*1 
0^(1/2))^(1/2)/(2*x^2+3*x+1)^(1/2))
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {1+3 x+2 x^2} \left (294+1071 x+1236 x^2+460 x^3\right )}{15 (1+x)^2 (1+2 x)^2}-\frac {1}{75} \sqrt {14655345+4634427 \sqrt {10}} \text {arctanh}\left (\frac {\sqrt {1-\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )+\frac {81 \text {arctanh}\left (\frac {\sqrt {1+\sqrt {\frac {2}{5}}} \sqrt {1+3 x+2 x^2}}{1+2 x}\right )}{5 \sqrt {24425575+7724045 \sqrt {10}}} \] Input:

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

Output:

(2*Sqrt[1 + 3*x + 2*x^2]*(294 + 1071*x + 1236*x^2 + 460*x^3))/(15*(1 + x)^ 
2*(1 + 2*x)^2) - (Sqrt[14655345 + 4634427*Sqrt[10]]*ArcTanh[(Sqrt[1 - Sqrt 
[2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/75 + (81*ArcTanh[(Sqrt[1 + Sqrt[ 
2/5]]*Sqrt[1 + 3*x + 2*x^2])/(1 + 2*x)])/(5*Sqrt[24425575 + 7724045*Sqrt[1 
0]])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1349, 27, 2135, 27, 1365, 27, 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.

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

\(\Big \downarrow \) 1349

\(\displaystyle \frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}-\frac {2}{45} \int -\frac {3 \left (-264 x^2+271 x+320\right )}{2 \left (-3 x^2+4 x+2\right ) \left (2 x^2+3 x+1\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \int \frac {-264 x^2+271 x+320}{\left (-3 x^2+4 x+2\right ) \left (2 x^2+3 x+1\right )^{3/2}}dx+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {1}{15} \left (\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}-\frac {2}{15} \int -\frac {45 (346-201 x)}{2 \left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (3 \int \frac {346-201 x}{\left (-3 x^2+4 x+2\right ) \sqrt {2 x^2+3 x+1}}dx+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3}{5} \left (335+106 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-\frac {3}{5} \left (335-106 \sqrt {10}\right ) \int \frac {1}{2 \left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3}{10} \left (335+106 \sqrt {10}\right ) \int \frac {1}{\left (-3 x-\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx-\frac {3}{10} \left (335-106 \sqrt {10}\right ) \int \frac {1}{\left (-3 x+\sqrt {10}+2\right ) \sqrt {2 x^2+3 x+1}}dx\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{15} \left (3 \left (\frac {3}{5} \left (335+106 \sqrt {10}\right ) \int \frac {1}{4 \left (55-17 \sqrt {10}\right )-\frac {\left (\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )+\frac {3}{5} \left (335-106 \sqrt {10}\right ) \int \frac {1}{4 \left (55+17 \sqrt {10}\right )-\frac {\left (\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )\right )^2}{2 x^2+3 x+1}}d\left (-\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{\sqrt {2 x^2+3 x+1}}\right )\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{15} \left (3 \left (-\frac {3 \left (335+106 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17-4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {55-17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55-17 \sqrt {10}}}-\frac {3 \left (335-106 \sqrt {10}\right ) \text {arctanh}\left (\frac {\left (17+4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {55+17 \sqrt {10}} \sqrt {2 x^2+3 x+1}}\right )}{10 \sqrt {55+17 \sqrt {10}}}\right )+\frac {2 (230 x+273)}{\sqrt {2 x^2+3 x+1}}\right )+\frac {2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}\)

Input:

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

Output:

(2*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + ((2*(273 + 230*x))/Sqrt[1 + 
 3*x + 2*x^2] + 3*((-3*(335 + 106*Sqrt[10])*ArcTanh[(3*(4 - Sqrt[10]) + (1 
7 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/(10* 
Sqrt[55 - 17*Sqrt[10]]) - (3*(335 - 106*Sqrt[10])*ArcTanh[(3*(4 + Sqrt[10] 
) + (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])] 
)/(10*Sqrt[55 + 17*Sqrt[10]])))/15
 

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 1349

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e 
_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> 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)))*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + 
b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b* 
c*d - 2*a*c*e + a*b*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*h - 2*g*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1 
) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*((-h)*c 
*e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g 
*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2 
*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))* 
(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a 
*((-h)*c*e)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h* 
c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e))*(2*p + 2*q + 5)*x^2, 
x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, 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])
 

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 = 3.46 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.40

method	result	size
trager	\(\frac {\frac {184}{3} x^{3}+\frac {824}{5} x^{2}+\frac {714}{5} x +\frac {196}{5}}{\left (2 x^{2}+3 x +1\right )^{\frac {3}{2}}}+\frac {2 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right ) \ln \left (-\frac {22695840000 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{5} x [phone] \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{3} x [phone] \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}[phone] \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{3}-5908432074489101275 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right ) x -12042322347390237 \sqrt {2 x^{2}+3 x +1}-4281865136972328150 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )}{1200 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-1795497 x +3089618}\right )}{5}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) \ln \left (-\frac {504352000 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )-20232850257120 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) x -11686912631520 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )-1601657971200 \sqrt {2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right ) x [phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+3600 \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-29310690\right )-3015957496555836 \sqrt {2 x^{2}+3 x +1}}{1200 x \operatorname {RootOf}\left (96000 \textit {\_Z}^{4}-781618400 \textit {\_Z}^{2}+6561\right )^{2}-7974733 x -3089618}\right )}{150}\)	\(472\)
default	\(\text {Expression too large to display}\)	\(878\)

Input:

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

Output:

2/15*(460*x^3+1236*x^2+1071*x+294)/(2*x^2+3*x+1)^(3/2)+2/5*RootOf(96000*_Z 
^4-781618400*_Z^2+6561)*ln(-(22695840000*RootOf(96000*_Z^4-781618400*_Z^2+ 
6561)^5*x[phone]*RootOf(96000*_Z^4-781618400*_Z^2+6561)^3*x+12012 
43478400*(2*x^2+3*x+1)^(1/2)*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2+5259 
11068418400*RootOf(96000*_Z^4-781618400*_Z^2+6561)^3-5908432074489101275*R 
ootOf(96000*_Z^4-781618400*_Z^2+6561)*x-12042322347390237*(2*x^2+3*x+1)^(1 
/2)-4281865136972328150*RootOf(96000*_Z^4-781618400*_Z^2+6561))/(1200*x*Ro 
otOf(96000*_Z^4-781618400*_Z^2+6561)^2-1795497*x+3089618))+1/150*RootOf(_Z 
^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)*ln(-(504352000* 
x*RootOf(96000*_Z^4-781618400*_Z^2+6561)^4*RootOf(_Z^2+3600*RootOf(96000*_ 
Z^4-781618400*_Z^2+6561)^2-29310690)-20232850257120*RootOf(96000*_Z^4-7816 
18400*_Z^2+6561)^2*RootOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561) 
^2-29310690)*x-11686912631520*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2*Roo 
tOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)-160165797 
1200*(2*x^2+3*x+1)^(1/2)*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2+10879105 
94433*RootOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-29310690)* 
x[phone]*RootOf(_Z^2+3600*RootOf(96000*_Z^4-781618400*_Z^2+6561)^2-2 
9310690)-3015957496555836*(2*x^2+3*x+1)^(1/2))/(1200*x*RootOf(96000*_Z^4-7 
81618400*_Z^2+6561)^2-7974733*x-3089618))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (141) = 282\).

Time = 0.08 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.04 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {23520 \, x^{4} + 70560 \, x^{3} + 3 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} \log \left (-\frac {81 \, \sqrt {10} x + {\left (893 \, \sqrt {10} x - 2824 \, x\right )} \sqrt {\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} + 162 \, x - 162 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 162}{x}\right ) - 3 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} \log \left (-\frac {81 \, \sqrt {10} x - {\left (893 \, \sqrt {10} x - 2824 \, x\right )} \sqrt {\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} + 162 \, x - 162 \, \sqrt {2 \, x^{2} + 3 \, x + 1} + 162}{x}\right ) + 3 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {-\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} \log \left (\frac {81 \, \sqrt {10} x + {\left (893 \, \sqrt {10} x + 2824 \, x\right )} \sqrt {-\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} - 162 \, x + 162 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 162}{x}\right ) - 3 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt {-\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} \log \left (\frac {81 \, \sqrt {10} x - {\left (893 \, \sqrt {10} x + 2824 \, x\right )} \sqrt {-\frac {1544809}{3} \, \sqrt {10} + \frac {4885115}{3}} - 162 \, x + 162 \, \sqrt {2 \, x^{2} + 3 \, x + 1} - 162}{x}\right ) + 76440 \, x^{2} + 20 \, {\left (460 \, x^{3} + 1236 \, x^{2} + 1071 \, x + 294\right )} \sqrt {2 \, x^{2} + 3 \, x + 1} + 35280 \, x + 5880}{150 \, {\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )}} \] Input:

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

Output:

1/150*(23520*x^4 + 70560*x^3 + 3*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt( 
1544809/3*sqrt(10) + 4885115/3)*log(-(81*sqrt(10)*x + (893*sqrt(10)*x - 28 
24*x)*sqrt(1544809/3*sqrt(10) + 4885115/3) + 162*x - 162*sqrt(2*x^2 + 3*x 
+ 1) + 162)/x) - 3*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(1544809/3*sqrt 
(10) + 4885115/3)*log(-(81*sqrt(10)*x - (893*sqrt(10)*x - 2824*x)*sqrt(154 
4809/3*sqrt(10) + 4885115/3) + 162*x - 162*sqrt(2*x^2 + 3*x + 1) + 162)/x) 
 + 3*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809/3*sqrt(10) + 488511 
5/3)*log((81*sqrt(10)*x + (893*sqrt(10)*x + 2824*x)*sqrt(-1544809/3*sqrt(1 
0) + 4885115/3) - 162*x + 162*sqrt(2*x^2 + 3*x + 1) - 162)/x) - 3*(4*x^4 + 
 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809/3*sqrt(10) + 4885115/3)*log((81* 
sqrt(10)*x - (893*sqrt(10)*x + 2824*x)*sqrt(-1544809/3*sqrt(10) + 4885115/ 
3) - 162*x + 162*sqrt(2*x^2 + 3*x + 1) - 162)/x) + 76440*x^2 + 20*(460*x^3 
 + 1236*x^2 + 1071*x + 294)*sqrt(2*x^2 + 3*x + 1) + 35280*x + 5880)/(4*x^4 
 + 12*x^3 + 13*x^2 + 6*x + 1)
 

Sympy [F]

\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=- \int \frac {x}{12 x^{6} \sqrt {2 x^{2} + 3 x + 1} + 20 x^{5} \sqrt {2 x^{2} + 3 x + 1} - 17 x^{4} \sqrt {2 x^{2} + 3 x + 1} - 58 x^{3} \sqrt {2 x^{2} + 3 x + 1} - 47 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 16 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {2 x^{2} + 3 x + 1} + 20 x^{5} \sqrt {2 x^{2} + 3 x + 1} - 17 x^{4} \sqrt {2 x^{2} + 3 x + 1} - 58 x^{3} \sqrt {2 x^{2} + 3 x + 1} - 47 x^{2} \sqrt {2 x^{2} + 3 x + 1} - 16 x \sqrt {2 x^{2} + 3 x + 1} - 2 \sqrt {2 x^{2} + 3 x + 1}}\, dx \] Input:

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

Output:

-Integral(x/(12*x**6*sqrt(2*x**2 + 3*x + 1) + 20*x**5*sqrt(2*x**2 + 3*x + 
1) - 17*x**4*sqrt(2*x**2 + 3*x + 1) - 58*x**3*sqrt(2*x**2 + 3*x + 1) - 47* 
x**2*sqrt(2*x**2 + 3*x + 1) - 16*x*sqrt(2*x**2 + 3*x + 1) - 2*sqrt(2*x**2 
+ 3*x + 1)), x) - Integral(2/(12*x**6*sqrt(2*x**2 + 3*x + 1) + 20*x**5*sqr 
t(2*x**2 + 3*x + 1) - 17*x**4*sqrt(2*x**2 + 3*x + 1) - 58*x**3*sqrt(2*x**2 
 + 3*x + 1) - 47*x**2*sqrt(2*x**2 + 3*x + 1) - 16*x*sqrt(2*x**2 + 3*x + 1) 
 - 2*sqrt(2*x**2 + 3*x + 1)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1276 vs. \(2 (141) = 282\).

Time = 0.16 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.48 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/300*sqrt(10)*(980*sqrt(10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55* 
(2*x^2 + 3*x + 1)^(3/2)) - 980*sqrt(10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^( 
3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 5292*sqrt(10)*x/(374*sqrt(10)*sqrt(2* 
x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1)) - 5292*sqrt(10)*x/(374*sqrt(1 
0)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 15680*sqrt(10)*x/ 
(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 15680*sqr 
t(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 3 
520*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) + 
 3520*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) 
 + 19008*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1 
)) + 19008*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 
 1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 
1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1 
)) + 750*sqrt(10)/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 
 1)^(3/2)) - 750*sqrt(10)/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 
 + 3*x + 1)^(3/2)) + 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1 
183*sqrt(2*x^2 + 3*x + 1)) - 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x 
+ 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 11760*sqrt(10)/(17*sqrt(10)*sqrt(2*x^ 
2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 11760*sqrt(10)/(17*sqrt(10)*sqr 
t(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 2760/(17*sqrt(10)*(2*x...
 

Giac [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.61 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (115 \, x + 309\right )} x + 1071\right )} x + 294\right )}}{15 \, {\left (2 \, x^{2} + 3 \, x + 1\right )}^{\frac {3}{2}}} + 0.00115890443051200 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} + 5.90976932712000\right ) - 36.0928986365600 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.176527156327000\right ) + 36.0928986365600 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 0.919278730509000\right ) - 0.00115890442441200 \, \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 3 \, x + 1} - 1.04272727395000\right ) \] Input:

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

Output:

2/15*((4*(115*x + 309)*x + 1071)*x + 294)/(2*x^2 + 3*x + 1)^(3/2) + 0.0011 
5890443051200*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) + 5.90976932712000) - 
 36.0928986365600*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.1765271563270 
00) + 36.0928986365600*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 0.91927873 
0509000) - 0.00115890442441200*log(-sqrt(2)*x + sqrt(2*x^2 + 3*x + 1) - 1. 
04272727395000)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{5/2}} \, dx=-\left (\int \frac {x}{12 \sqrt {2 x^{2}+3 x +1}\, x^{6}+20 \sqrt {2 x^{2}+3 x +1}\, x^{5}-17 \sqrt {2 x^{2}+3 x +1}\, x^{4}-58 \sqrt {2 x^{2}+3 x +1}\, x^{3}-47 \sqrt {2 x^{2}+3 x +1}\, x^{2}-16 \sqrt {2 x^{2}+3 x +1}\, x -2 \sqrt {2 x^{2}+3 x +1}}d x \right )-2 \left (\int \frac {1}{12 \sqrt {2 x^{2}+3 x +1}\, x^{6}+20 \sqrt {2 x^{2}+3 x +1}\, x^{5}-17 \sqrt {2 x^{2}+3 x +1}\, x^{4}-58 \sqrt {2 x^{2}+3 x +1}\, x^{3}-47 \sqrt {2 x^{2}+3 x +1}\, x^{2}-16 \sqrt {2 x^{2}+3 x +1}\, x -2 \sqrt {2 x^{2}+3 x +1}}d x \right ) \] Input:

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

Output:

 - int(x/(12*sqrt(2*x**2 + 3*x + 1)*x**6 + 20*sqrt(2*x**2 + 3*x + 1)*x**5 
- 17*sqrt(2*x**2 + 3*x + 1)*x**4 - 58*sqrt(2*x**2 + 3*x + 1)*x**3 - 47*sqr 
t(2*x**2 + 3*x + 1)*x**2 - 16*sqrt(2*x**2 + 3*x + 1)*x - 2*sqrt(2*x**2 + 3 
*x + 1)),x) - 2*int(1/(12*sqrt(2*x**2 + 3*x + 1)*x**6 + 20*sqrt(2*x**2 + 3 
*x + 1)*x**5 - 17*sqrt(2*x**2 + 3*x + 1)*x**4 - 58*sqrt(2*x**2 + 3*x + 1)* 
x**3 - 47*sqrt(2*x**2 + 3*x + 1)*x**2 - 16*sqrt(2*x**2 + 3*x + 1)*x - 2*sq 
rt(2*x**2 + 3*x + 1)),x)