\(\int \frac {2+x}{(2+4 x-3 x^2) (1+3 x-2 x^2)^{5/2}} \, dx\) [27]

Optimal result

Integrand size = 30, antiderivative size = 193 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \arctan \left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \text {arctanh}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \]

[Out]

-2/51*(15+14*x)/(-2*x^2+3*x+1)^(3/2)-2/867*(291+4814*x)/(-2*x^2+3*x+1)^(1/2)+9/10*arctan(1/2*(12-3*10^(1/2)+x*
(1+4*10^(1/2)))/(-2*x^2+3*x+1)^(1/2)/(1+10^(1/2))^(1/2))*(-265+85*10^(1/2))^(1/2)+9/10*arctanh(1/2*(x*(1-4*10^
(1/2))+12+3*10^(1/2))/(-2*x^2+3*x+1)^(1/2)/(-1+10^(1/2))^(1/2))*(265+85*10^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1030, 1074, 1046, 738, 210, 212} \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=\frac {9}{2} \sqrt {\frac {1}{5} \left (17 \sqrt {10}-53\right )} \arctan \left (\frac {\left (1+4 \sqrt {10}\right ) x+3 \left (4-\sqrt {10}\right )}{2 \sqrt {1+\sqrt {10}} \sqrt {-2 x^2+3 x+1}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \text {arctanh}\left (\frac {\left (1-4 \sqrt {10}\right ) x+3 \left (4+\sqrt {10}\right )}{2 \sqrt {\sqrt {10}-1} \sqrt {-2 x^2+3 x+1}}\right )-\frac {2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac {2 (4814 x+291)}{867 \sqrt {-2 x^2+3 x+1}} \]

[In]

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

[Out]

(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) - (2*(291 + 4814*x))/(867*Sqrt[1 + 3*x - 2*x^2]) + (9*Sqrt[(-53
+ 17*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])]
)/2 + (9*Sqrt[(53 + 17*Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqr
t[1 + 3*x - 2*x^2])])/2

Rule 210

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1030

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> 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] + Dist[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 1046

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(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 1074

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((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)))*((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] + Dist[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] /; FreeQ[{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] && 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}+\frac {2}{51} \int \frac {-56+\frac {235 x}{2}+84 x^2}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {4}{867} \int \frac {\frac {7803}{2}+\frac {23409 x}{4}}{\left (2+4 x-3 x^2\right ) \sqrt {1+3 x-2 x^2}} \, dx \\ & = -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {1}{5} \left (27 \left (5-2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4-2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx+\frac {1}{5} \left (27 \left (5+2 \sqrt {10}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {10}-6 x\right ) \sqrt {1+3 x-2 x^2}} \, dx \\ & = -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}-\frac {1}{5} \left (54 \left (5-2 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4-2 \sqrt {10}\right )-8 \left (4-2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4-2 \sqrt {10}\right )-\left (18-4 \left (4-2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right )-\frac {1}{5} \left (54 \left (5+2 \sqrt {10}\right )\right ) \text {Subst}\left (\int \frac {1}{144+72 \left (4+2 \sqrt {10}\right )-8 \left (4+2 \sqrt {10}\right )^2-x^2} \, dx,x,\frac {-12-3 \left (4+2 \sqrt {10}\right )-\left (18-4 \left (4+2 \sqrt {10}\right )\right ) x}{\sqrt {1+3 x-2 x^2}}\right ) \\ & = -\frac {2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {2 (291+4814 x)}{867 \sqrt {1+3 x-2 x^2}}+\frac {9}{2} \sqrt {\frac {1}{5} \left (-53+17 \sqrt {10}\right )} \tan ^{-1}\left (\frac {3 \left (4-\sqrt {10}\right )+\left (1+4 \sqrt {10}\right ) x}{2 \sqrt {1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right )+\frac {9}{2} \sqrt {\frac {1}{5} \left (53+17 \sqrt {10}\right )} \tanh ^{-1}\left (\frac {3 \left (4+\sqrt {10}\right )+\left (1-4 \sqrt {10}\right ) x}{2 \sqrt {-1+\sqrt {10}} \sqrt {1+3 x-2 x^2}}\right ) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.45 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.95 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {2 \left (546+5925 x+13860 x^2-9628 x^3\right )}{867 \left (1+3 x-2 x^2\right )^{3/2}}-\frac {9}{2} \text {RootSum}\left [5+20 \text {$\#$1}+8 \text {$\#$1}^2-8 \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-13 \log (x)+13 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}-6 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^2+2 \log \left (-1+\sqrt {1+3 x-2 x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{5+4 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

(-2*(546 + 5925*x + 13860*x^2 - 9628*x^3))/(867*(1 + 3*x - 2*x^2)^(3/2)) - (9*RootSum[5 + 20*#1 + 8*#1^2 - 8*#
1^3 + 2*#1^4 & , (-13*Log[x] + 13*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1] + 6*Log[x]*#1 - 6*Log[-1 + Sqrt[1 + 3
*x - 2*x^2] - x*#1]*#1 - 2*Log[x]*#1^2 + 2*Log[-1 + Sqrt[1 + 3*x - 2*x^2] - x*#1]*#1^2)/(5 + 4*#1 - 6*#1^2 + 2
*#1^3) & ])/2

Maple [C] (warning: unable to verify)

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

Time = 2.27 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.51

method	result	size
trager	\(\frac {2 \left (9628 x^{3}-13860 x^{2}-5925 x -546\right ) \sqrt {-2 x^{2}+3 x +1}}{867 \left (2 x^{2}-3 x -1\right )^{2}}-18 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) \ln \left (-\frac {-2105600 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{5}+5362400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3} x +74880 \sqrt {-2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}+473760 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{3}-3406349 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right ) x -99945 \sqrt {-2 x^{2}+3 x +1}-632106 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )}{80 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-87 x -34}\right )+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) \ln \left (-\frac {2105600 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-217440 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +473760 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-1497600 \sqrt {-2 x^{2}+3 x +1}\, \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-2187 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right ) x +4374 \operatorname {RootOf}\left (\textit {\_Z}^{2}+400 \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-530\right )-14580 \sqrt {-2 x^{2}+3 x +1}}{80 x \operatorname {RootOf}\left (6400 \textit {\_Z}^{4}-8480 \textit {\_Z}^{2}-81\right )^{2}-19 x +34}\right )}{10}\)	\(484\)
default	\(\text {Expression too large to display}\)	\(868\)

[In]

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

[Out]

2/867*(9628*x^3-13860*x^2-5925*x-546)/(2*x^2-3*x-1)^2*(-2*x^2+3*x+1)^(1/2)-18*RootOf(6400*_Z^4-8480*_Z^2-81)*l
n(-(-2105600*x*RootOf(6400*_Z^4-8480*_Z^2-81)^5+5362400*RootOf(6400*_Z^4-8480*_Z^2-81)^3*x+74880*(-2*x^2+3*x+1
)^(1/2)*RootOf(6400*_Z^4-8480*_Z^2-81)^2+473760*RootOf(6400*_Z^4-8480*_Z^2-81)^3-3406349*RootOf(6400*_Z^4-8480
*_Z^2-81)*x-99945*(-2*x^2+3*x+1)^(1/2)-632106*RootOf(6400*_Z^4-8480*_Z^2-81))/(80*x*RootOf(6400*_Z^4-8480*_Z^2
-81)^2-87*x-34))+9/10*RootOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*ln(-(2105600*x*RootOf(6400*_Z^4-84
80*_Z^2-81)^4*RootOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)-217440*RootOf(6400*_Z^4-8480*_Z^2-81)^2*Ro
otOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*x+473760*RootOf(6400*_Z^4-8480*_Z^2-81)^2*RootOf(_Z^2+400*
RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)-1497600*(-2*x^2+3*x+1)^(1/2)*RootOf(6400*_Z^4-8480*_Z^2-81)^2-2187*RootO
f(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)*x+4374*RootOf(_Z^2+400*RootOf(6400*_Z^4-8480*_Z^2-81)^2-530)-
14580*(-2*x^2+3*x+1)^(1/2))/(80*x*RootOf(6400*_Z^4-8480*_Z^2-81)^2-19*x+34))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (137) = 274\).

Time = 0.31 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.27 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=-\frac {43680 \, x^{4} - 131040 \, x^{3} - 867 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1377 \, \sqrt {10} + 4293} \log \left (-\frac {405 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x + 40 \, \sqrt {5} x\right )} \sqrt {-1377 \, \sqrt {10} + 4293} + 810 \, x - 810 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 810}{x}\right ) + 867 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {-1377 \, \sqrt {10} + 4293} \log \left (-\frac {405 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x + 40 \, \sqrt {5} x\right )} \sqrt {-1377 \, \sqrt {10} + 4293} + 810 \, x - 810 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} + 810}{x}\right ) - 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x + {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 7803 \, \sqrt {5} {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt {17 \, \sqrt {10} + 53} \log \left (\frac {9 \, {\left (45 \, \sqrt {10} x - {\left (13 \, \sqrt {10} \sqrt {5} x - 40 \, \sqrt {5} x\right )} \sqrt {17 \, \sqrt {10} + 53} - 90 \, x + 90 \, \sqrt {-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 54600 \, x^{2} - 20 \, {\left (9628 \, x^{3} - 13860 \, x^{2} - 5925 \, x - 546\right )} \sqrt {-2 \, x^{2} + 3 \, x + 1} + 65520 \, x + 10920}{8670 \, {\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )}} \]

[In]

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

[Out]

-1/8670*(43680*x^4 - 131040*x^3 - 867*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(-1377*sqrt(10) + 4293)*l
og(-(405*sqrt(10)*x + (13*sqrt(10)*sqrt(5)*x + 40*sqrt(5)*x)*sqrt(-1377*sqrt(10) + 4293) + 810*x - 810*sqrt(-2
*x^2 + 3*x + 1) + 810)/x) + 867*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(-1377*sqrt(10) + 4293)*log(-(4
05*sqrt(10)*x - (13*sqrt(10)*sqrt(5)*x + 40*sqrt(5)*x)*sqrt(-1377*sqrt(10) + 4293) + 810*x - 810*sqrt(-2*x^2 +
3*x + 1) + 810)/x) - 7803*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10
)*x + (13*sqrt(10)*sqrt(5)*x - 40*sqrt(5)*x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x
) + 7803*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10)*x - (13*sqrt(10)
*sqrt(5)*x - 40*sqrt(5)*x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 54600*x^2 - 20
*(9628*x^3 - 13860*x^2 - 5925*x - 546)*sqrt(-2*x^2 + 3*x + 1) + 65520*x + 10920)/(4*x^4 - 12*x^3 + 5*x^2 + 6*x
+ 1)

Sympy [F]

\[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=- \int \frac {x}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx - \int \frac {2}{12 x^{6} \sqrt {- 2 x^{2} + 3 x + 1} - 52 x^{5} \sqrt {- 2 x^{2} + 3 x + 1} + 55 x^{4} \sqrt {- 2 x^{2} + 3 x + 1} + 22 x^{3} \sqrt {- 2 x^{2} + 3 x + 1} - 31 x^{2} \sqrt {- 2 x^{2} + 3 x + 1} - 16 x \sqrt {- 2 x^{2} + 3 x + 1} - 2 \sqrt {- 2 x^{2} + 3 x + 1}}\, dx \]

[In]

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

[Out]

-Integral(x/(12*x**6*sqrt(-2*x**2 + 3*x + 1) - 52*x**5*sqrt(-2*x**2 + 3*x + 1) + 55*x**4*sqrt(-2*x**2 + 3*x +
1) + 22*x**3*sqrt(-2*x**2 + 3*x + 1) - 31*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) - 52*x**5*sqrt(-2*x**2 + 3*x + 1) + 55*
x**4*sqrt(-2*x**2 + 3*x + 1) + 22*x**3*sqrt(-2*x**2 + 3*x + 1) - 31*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. \(2 (137) = 274\).

Time = 0.32 (sec) , antiderivative size = 1276, normalized size of antiderivative = 6.61 \[ \int \frac {2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/17340*sqrt(10)*(2108*sqrt(10)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2108*sqrt(1
0)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) - 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2
+ 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^
2 + 3*x + 1)) + 1984*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 1984*sqrt(10)*x/(
sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 70227*sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs
(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(1
0) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) + 1) + 11*sqrt(sqrt(10) + 1)) -
2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3
/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 58752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 58
752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1
) + sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 561816*arcsi
n(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(1
7)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) +
1) + 11*sqrt(sqrt(10) + 1)) - 714*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) + 71
4*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2
*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) - 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*
x^2 + 3*x + 1)) - 1488*sqrt(10)/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1488*sqrt(10)/(sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 +
3*x + 1)^(3/2)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 143208/(2*sqrt(10)*sqr
t(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 143208/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2
+ 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3
*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 70227*sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10
) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/
18)/(sqrt(10) - 1)^(5/2) + 561816*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x -
2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/(sqrt(10) - 1
)^(5/2))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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