\(\int \frac {\sqrt {3-x+2 x^2}}{(2+3 x+5 x^2)^3} \, dx\) [98]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [C] (warning: unable to verify)
Fricas [B] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F(-2)]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 27, antiderivative size = 223 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (509587+362788 \sqrt {2}+\left (1235163+872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136}-\frac {\sqrt {\frac {1}{682} \left (-112285869463+79399380740 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-112285869463+79399380740 \sqrt {2}\right )}} \left (509587-362788 \sqrt {2}+\left (1235163-872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136} \] Output:

1/62*(3+10*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^2+(3464+13665*x)*(2*x^2-x+3) 
^(1/2)/(422840*x^2+253704*x+169136)+1/115350752*(76578962973766+5415037766 
4680*2^(1/2))^(1/2)*arctan(11^(1/2)/(3480861953353+2461380802940*2^(1/2))^ 
(1/2)*(509587+362788*2^(1/2)+(1235163+872375*2^(1/2))*x)/(2*x^2-x+3)^(1/2) 
)-1/115350752*(-76578962973766+54150377664680*2^(1/2))^(1/2)*arctanh(11^(1 
/2)/(-3480861953353+2461380802940*2^(1/2))^(1/2)*(509587-362788*2^(1/2)+(1 
235163-872375*2^(1/2))*x)/(2*x^2-x+3)^(1/2))
 

Mathematica [C] (verified)

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

Time = 0.98 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {661250 \sqrt {3-x+2 x^2} \left (11020+51362 x+58315 x^2+68325 x^3\right )}{\left (2+3 x+5 x^2\right )^2}+\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-537295920831 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+120146195680 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-45923442075 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-248 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2139373897 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+277937160 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-228643025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{55920590000} \] Input:

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

Output:

((661250*Sqrt[3 - x + 2*x^2]*(11020 + 51362*x + 58315*x^2 + 68325*x^3))/(2 
 + 3*x + 5*x^2)^2 + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 
 - 5*#1^4 & , (-537295920831*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] 
+ 120146195680*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 4 
5923442075*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] 
 + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] - 248*RootSum[-56 - 26*Sqrt[2]*#1 
 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-2139373897*Log[-(Sqrt[2]*x) + S 
qrt[3 - x + 2*x^2] - #1] + 277937160*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x 
 + 2*x^2] - #1]*#1 - 228643025*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1 
]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/55920590000
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1302, 27, 2135, 27, 1368, 27, 1362, 217, 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.

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

\(\Big \downarrow \) 1302

\(\displaystyle \frac {(10 x+3) \sqrt {2 x^2-x+3}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {80 x^2-62 x+183}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{124} \int \frac {80 x^2-62 x+183}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}dx+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {1}{124} \left (\frac {\int \frac {11 (77456-32605 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{124} \left (\frac {\int \frac {77456-32605 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1368

\(\displaystyle \frac {1}{124} \left (\frac {\frac {\int -\frac {11 \left (-\left (\left (44851-32605 \sqrt {2}\right ) x\right )-77456 \sqrt {2}+110061\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (44851+32605 \sqrt {2}\right ) x\right )+77456 \sqrt {2}+110061\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{124} \left (\frac {\frac {\int \frac {-\left (\left (44851+32605 \sqrt {2}\right ) x\right )+77456 \sqrt {2}+110061}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (44851-32605 \sqrt {2}\right ) x\right )-77456 \sqrt {2}+110061}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1362

\(\displaystyle \frac {1}{124} \left (\frac {\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463-79399380740 \sqrt {2}\right )}d\frac {\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (112285869463+79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463+79399380740 \sqrt {2}\right )}d\frac {\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{124} \left (\frac {\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )^2}{2 x^2-x+3}-31 \left (112285869463-79399380740 \sqrt {2}\right )}d\frac {\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{124} \left (\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (112285869463-79399380740 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (79399380740 \sqrt {2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (79399380740 \sqrt {2}-112285869463\right )}}}{1364}+\frac {\sqrt {2 x^2-x+3} (13665 x+3464)}{682 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\)

Input:

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

Output:

((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + (((3464 + 1366 
5*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(112285869463 + 
79399380740*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 79399380740* 
Sqrt[2]))]*(509587 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[ 
3 - x + 2*x^2]] + ((112285869463 - 79399380740*Sqrt[2])*ArcTanh[(Sqrt[11/( 
31*(-112285869463 + 79399380740*Sqrt[2]))]*(509587 - 362788*Sqrt[2] + (123 
5163 - 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-112285869463 + 
 79399380740*Sqrt[2])])/1364)/124
 

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 217
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 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 1302
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e 
*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) 
   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p 
+ 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]
 

rule 1362
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h)   Subst[I 
nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ 
g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], 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] && Ne 
Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f 
), 0]
 

rule 1368
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d 
 - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q)   Int[Simp[h*(b*d - a*e) - g*(c*d - 
a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr 
t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q)   Int[Simp[h*(b*d - a*e) - g*(c* 
d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) 
*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[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 = 5.59 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17

method result size
trager \(\text {Expression too large to display}\) \(483\)
risch \(\frac {\left (68325 x^{3}+58315 x^{2}+51362 x +11020\right ) \sqrt {2 x^{2}-x +3}}{84568 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {\sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (33504619 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+47385010 \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+49124007834 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-69208569562 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{3575873312 \sqrt {\frac {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) \(726\)
default \(\text {Expression too large to display}\) \(44343\)

Input:

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

Output:

1/84568*(68325*x^3+58315*x^2+51362*x+11020)/(5*x^2+3*x+2)^2*(2*x^2-x+3)^(1 
/2)-1/115350752*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+612631703790128 
*_Z^2+394016353868467684225)^2+76578962973766)*ln(-(8901118918912*RootOf(_ 
Z^2+29767936*RootOf(238143488*_Z^4+612631703790128*_Z^2+394016353868467684 
225)^2+76578962973766)*RootOf(238143488*_Z^4+612631703790128*_Z^2+39401635 
3868467684225)^4*x+21996649948054194864*RootOf(238143488*_Z^4+612631703790 
128*_Z^2+394016353868467684225)^2*RootOf(_Z^2+29767936*RootOf(238143488*_Z 
^4+612631703790128*_Z^2+394016353868467684225)^2+76578962973766)*x-2031490 
73871924400*RootOf(238143488*_Z^4+612631703790128*_Z^2+3940163538684676842 
25)^2*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+612631703790128*_Z^2+3940 
16353868467684225)^2+76578962973766)+10558503141216967088325712*RootOf(238 
143488*_Z^4+612631703790128*_Z^2+394016353868467684225)^2*(2*x^2-x+3)^(1/2 
)+13558387834041967792583352*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+61 
2631703790128*_Z^2+394016353868467684225)^2+76578962973766)*x-238972679575 
677054341025*RootOf(_Z^2+29767936*RootOf(238143488*_Z^4+612631703790128*_Z 
^2+394016353868467684225)^2+76578962973766)+135771321202551192561528744000 
47*(2*x^2-x+3)^(1/2))/(21824*x*RootOf(238143488*_Z^4+612631703790128*_Z^2+ 
394016353868467684225)^2+27985547479*x-114559849))+1/21142*RootOf(23814348 
8*_Z^4+612631703790128*_Z^2+394016353868467684225)*ln((1139343221620736*x* 
RootOf(238143488*_Z^4+612631703790128*_Z^2+394016353868467684225)^5+304...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (170) = 340\).

Time = 0.10 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.63 \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/676544*(2*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(39699690370/341*sq 
rt(2) + 112285869463/682)*arctan(-22/114559849*(4*(3603694*x^3 - 8193804*x 
^2 - sqrt(2)*(2555895*x^3 - 5800781*x^2 - 1956912*x + 2641464) - 2804336*x 
 + 3717888)*sqrt(2*x^2 - x + 3) + (171*x^4 + 1212*x^3 - 1640*x^2 - 176*sqr 
t(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(39699690370/341*sqrt(2) 
 - 112285869463/682))*sqrt(39699690370/341*sqrt(2) + 112285869463/682)/(34 
3*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 2*(25*x^4 + 30*x^3 + 29*x^2 + 
 12*x + 4)*sqrt(39699690370/341*sqrt(2) + 112285869463/682)*arctan(22/1145 
59849*(4*(3603694*x^3 - 8193804*x^2 - sqrt(2)*(2555895*x^3 - 5800781*x^2 - 
 1956912*x + 2641464) - 2804336*x + 3717888)*sqrt(2*x^2 - x + 3) - (171*x^ 
4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 393 
6*x)*sqrt(39699690370/341*sqrt(2) - 112285869463/682))*sqrt(39699690370/34 
1*sqrt(2) + 112285869463/682)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576) 
) + (25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(39699690370/341*sqrt(2) - 1 
12285869463/682)*log((5613432601*x^2 + 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(32 
73511*x - 7920565) + 4647054*x - 11194076)*sqrt(39699690370/341*sqrt(2) - 
112285869463/682) + 5040633356*sqrt(2)*(2*x^2 - x + 3) - 17298537199*x + 2 
2911969800)/x^2) - (25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(39699690370/ 
341*sqrt(2) - 112285869463/682)*log((5613432601*x^2 - 22*sqrt(2*x^2 - x + 
3)*(sqrt(2)*(3273511*x - 7920565) + 4647054*x - 11194076)*sqrt(39699690...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, 
infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf 
inity,inf
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {-6400 \sqrt {2 x^{2}-x +3}\, x^{3}-9280 \sqrt {2 x^{2}-x +3}\, x^{2}-5144 \sqrt {2 x^{2}-x +3}\, x +3462 \sqrt {2 x^{2}-x +3}+19510425 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{4}+23412510 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{3}+22632093 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{2}+9365004 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x +3121668 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right )+7555625 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{4}+9066750 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{3}+8764525 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x^{2}+3626700 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right ) x +1208900 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{250 x^{8}+325 x^{7}+720 x^{6}+804 x^{5}+876 x^{4}+579 x^{3}+322 x^{2}+100 x +24}d x \right )}{5698125 x^{4}+6837750 x^{3}+6609825 x^{2}+2735100 x +911700} \] Input:

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

Output:

( - 6400*sqrt(2*x**2 - x + 3)*x**3 - 9280*sqrt(2*x**2 - x + 3)*x**2 - 5144 
*sqrt(2*x**2 - x + 3)*x + 3462*sqrt(2*x**2 - x + 3) + 19510425*int(sqrt(2* 
x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579* 
x**3 + 322*x**2 + 100*x + 24),x)*x**4 + 23412510*int(sqrt(2*x**2 - x + 3)/ 
(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x** 
2 + 100*x + 24),x)*x**3 + 22632093*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 32 
5*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24 
),x)*x**2 + 9365004*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x* 
*6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)*x + 312166 
8*int(sqrt(2*x**2 - x + 3)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 87 
6*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x) + 7555625*int((sqrt(2*x**2 - 
 x + 3)*x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579* 
x**3 + 322*x**2 + 100*x + 24),x)*x**4 + 9066750*int((sqrt(2*x**2 - x + 3)* 
x**2)/(250*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 3 
22*x**2 + 100*x + 24),x)*x**3 + 8764525*int((sqrt(2*x**2 - x + 3)*x**2)/(2 
50*x**8 + 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 
+ 100*x + 24),x)*x**2 + 3626700*int((sqrt(2*x**2 - x + 3)*x**2)/(250*x**8 
+ 325*x**7 + 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x 
+ 24),x)*x + 1208900*int((sqrt(2*x**2 - x + 3)*x**2)/(250*x**8 + 325*x**7 
+ 720*x**6 + 804*x**5 + 876*x**4 + 579*x**3 + 322*x**2 + 100*x + 24),x)...