\(\int \frac {1+3 x+4 x^2}{(1+2 x)^3 (2-x+3 x^2)^{5/2}} \, dx\) [94]

Optimal result

Integrand size = 32, antiderivative size = 135 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}-\frac {8 \sqrt {2-x+3 x^2}}{2197 (1+2 x)^2}-\frac {144 \sqrt {2-x+3 x^2}}{28561 (1+2 x)}-\frac {2084 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{28561 \sqrt {13}} \] Output:

2/151593*(2363+3693*x)/(3*x^2-x+2)^(3/2)+12/15108769*(25771+103526*x)/(3*x 
^2-x+2)^(1/2)-8/2197*(3*x^2-x+2)^(1/2)/(1+2*x)^2-144*(3*x^2-x+2)^(1/2)/(28 
561+57122*x)-2084/371293*13^(1/2)*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2 
)^(1/2))
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {2-x+3 x^2} \left (847141+10777477 x+21890266 x^2+19381992 x^3+20074356 x^4+20304864 x^5\right )}{45326307 \left (2+3 x+x^2+6 x^3\right )^2}+\frac {4168 \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )}{28561 \sqrt {13}} \] Input:

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

Output:

(2*Sqrt[2 - x + 3*x^2]*(847141 + 10777477*x + 21890266*x^2 + 19381992*x^3 
+ 20074356*x^4 + 20304864*x^5))/(45326307*(2 + 3*x + x^2 + 6*x^3)^2) + (41 
68*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqrt[13]])/(285 
61*Sqrt[13])
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2177, 27, 2177, 27, 2181, 27, 1228, 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.

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

\(\Big \downarrow \) 2177

\(\displaystyle \frac {2}{69} \int \frac {3 \left (19696 x^3+53372 x^2+35610 x+10811\right )}{2197 (2 x+1)^3 \left (3 x^2-x+2\right )^{3/2}}dx+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {19696 x^3+53372 x^2+35610 x+10811}{(2 x+1)^3 \left (3 x^2-x+2\right )^{3/2}}dx}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2177

\(\displaystyle \frac {2 \left (\frac {2}{23} \int \frac {2116 \left (488 x^2+527 x+226\right )}{13 (2 x+1)^3 \sqrt {3 x^2-x+2}}dx+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {184}{13} \int \frac {488 x^2+527 x+226}{(2 x+1)^3 \sqrt {3 x^2-x+2}}dx+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {2 \left (\frac {184}{13} \left (-\frac {1}{26} \int -\frac {13 (898 x+683)}{2 (2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {13 \sqrt {3 x^2-x+2}}{2 (2 x+1)^2}\right )+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {184}{13} \left (\frac {1}{4} \int \frac {898 x+683}{(2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {13 \sqrt {3 x^2-x+2}}{2 (2 x+1)^2}\right )+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {2 \left (\frac {184}{13} \left (\frac {1}{4} \left (521 \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {36 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{2 (2 x+1)^2}\right )+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {2 \left (\frac {184}{13} \left (\frac {1}{4} \left (-1042 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}-\frac {36 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{2 (2 x+1)^2}\right )+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \left (\frac {184}{13} \left (\frac {1}{4} \left (-\frac {521 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{\sqrt {13}}-\frac {36 \sqrt {3 x^2-x+2}}{2 x+1}\right )-\frac {13 \sqrt {3 x^2-x+2}}{2 (2 x+1)^2}\right )+\frac {6 (103526 x+25771)}{299 \sqrt {3 x^2-x+2}}\right )}{50531}+\frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}\)

Input:

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

Output:

(2*(2363 + 3693*x))/(151593*(2 - x + 3*x^2)^(3/2)) + (2*((6*(25771 + 10352 
6*x))/(299*Sqrt[2 - x + 3*x^2]) + (184*((-13*Sqrt[2 - x + 3*x^2])/(2*(1 + 
2*x)^2) + ((-36*Sqrt[2 - x + 3*x^2])/(1 + 2*x) - (521*ArcTanh[(9 - 8*x)/(2 
*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/Sqrt[13])/4))/13))/50531
 

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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2177

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* 
x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], 
 x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], 
x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p 
 + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^ 
m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x 
)^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, 
 d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* 
e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58

method	result	size
risch	\(\frac {\frac {13536576}{15108769} x^{5}+\frac {13382904}{15108769} x^{4}+\frac {12921328}{15108769} x^{3}+\frac {43780532}{45326307} x^{2}+\frac {21554954}{45326307} x +\frac {1694282}{45326307}}{\left (1+2 x \right )^{2} \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {2084 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{371293}\)	\(78\)
trager	\(\frac {2 \left (20304864 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +847141\right ) \sqrt {3 x^{2}-x +2}}{45326307 \left (6 x^{3}+x^{2}+3 x +2\right )^{2}}-\frac {2084 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{371293}\)	\(105\)
default	\(\frac {521}{13182 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}+\frac {\frac {1772 x}{50531}-\frac {886}{151593}}{\left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}+\frac {\frac {1128048 x}{15108769}-\frac {188008}{15108769}}{\sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x}}+\frac {1042}{28561 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x}}-\frac {2084 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{371293}-\frac {1}{104 \left (\frac {1}{2}+x \right )^{2} \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}-\frac {1}{338 \left (\frac {1}{2}+x \right ) \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}\)	\(148\)

Input:

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

Output:

2/45326307*(20304864*x^5+20074356*x^4+19381992*x^3+21890266*x^2+10777477*x 
+847141)/(3*x^2-x+2)^(3/2)/(1+2*x)^2-2084/371293*13^(1/2)*arctanh(2/13*(9/ 
2-4*x)*13^(1/2)/(12*(1/2+x)^2+5-16*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.16 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (826827 \, \sqrt {13} {\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 13 \, {\left (20304864 \, x^{5} + 20074356 \, x^{4} + 19381992 \, x^{3} + 21890266 \, x^{2} + 10777477 \, x + 847141\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{589241991 \, {\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

2/589241991*(826827*sqrt(13)*(36*x^6 + 12*x^5 + 37*x^4 + 30*x^3 + 13*x^2 + 
 12*x + 4)*log(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) + 220*x^2 - 196* 
x + 185)/(4*x^2 + 4*x + 1)) + 13*(20304864*x^5 + 20074356*x^4 + 19381992*x 
^3 + 21890266*x^2 + 10777477*x + 847141)*sqrt(3*x^2 - x + 2))/(36*x^6 + 12 
*x^5 + 37*x^4 + 30*x^3 + 13*x^2 + 12*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.29 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2084}{371293} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {1128048 \, x}{15108769 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {363210}{15108769 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {1772 \, x}{50531 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {1}{26 \, {\left (4 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{2} + 4 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1}{169 \, {\left (2 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}\right )}} + \frac {10211}{303186 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

2084/371293*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/ 
abs(2*x + 1)) + 1128048/15108769*x/sqrt(3*x^2 - x + 2) + 363210/15108769/s 
qrt(3*x^2 - x + 2) + 1772/50531*x/(3*x^2 - x + 2)^(3/2) - 1/26/(4*(3*x^2 - 
 x + 2)^(3/2)*x^2 + 4*(3*x^2 - x + 2)^(3/2)*x + (3*x^2 - x + 2)^(3/2)) - 1 
/169/(2*(3*x^2 - x + 2)^(3/2)*x + (3*x^2 - x + 2)^(3/2)) + 10211/303186/(3 
*x^2 - x + 2)^(3/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (109) = 218\).

Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.73 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2084}{371293} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2 \, {\left (3 \, {\left (6 \, {\left (310578 \, x - 26213\right )} x + 1455755\right )} x + 1634293\right )}}{45326307 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (66 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} + 21 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} - 1015 \, \sqrt {3} x + 431 \, \sqrt {3} + 1015 \, \sqrt {3 \, x^{2} - x + 2}\right )}}{28561 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \] Input:

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

Output:

2084/371293*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 
4*sqrt(3*x^2 - x + 2))/(2*sqrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - 
x + 2))) + 2/45326307*(3*(6*(310578*x - 26213)*x + 1455755)*x + 1634293)/( 
3*x^2 - x + 2)^(3/2) - 8/28561*(66*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^3 + 2 
1*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 - 1015*sqrt(3)*x + 431*sqrt( 
3) + 1015*sqrt(3*x^2 - x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 2* 
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) - 5)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.99 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {527926464 \sqrt {3 x^{2}-x +2}\, x^{5}[phone] \sqrt {3 x^{2}-x +2}\, x^{4}[phone] \sqrt {3 x^{2}-x +2}\, x^{3}[phone] \sqrt {3 x^{2}-x +2}\, x^{2}[phone] \sqrt {3 x^{2}-x +2}\, x [phone] \sqrt {3 x^{2}-x +2}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{6}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{5}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{4}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{3}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{2}[phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x [phone] \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right )-119063088 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{6}-39687696 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{5}-122370396 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{4}-99219240 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{3}-42995004 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{2}-39687696 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x -13229232 \sqrt {13}\, \mathrm {log}\left (2 x +1\right )}{21212711676 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(2*(263963232*sqrt(3*x**2 - x + 2)*x**5 + 260966628*sqrt(3*x**2 - x + 2)*x 
**4 + 251965896*sqrt(3*x**2 - x + 2)*x**3 + 284573458*sqrt(3*x**2 - x + 2) 
*x**2 + 140107201*sqrt(3*x**2 - x + 2)*x + 11012833*sqrt(3*x**2 - x + 2) + 
 59531544*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**6 + 1 
9843848*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**5 + 611 
85198*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**4 + 49609 
620*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**3 + 2149750 
2*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**2 + 19843848* 
sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x + 6614616*sqrt(1 
3)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9) - 59531544*sqrt(13)*log( 
2*x + 1)*x**6 - 19843848*sqrt(13)*log(2*x + 1)*x**5 - 61185198*sqrt(13)*lo 
g(2*x + 1)*x**4 - 49609620*sqrt(13)*log(2*x + 1)*x**3 - 21497502*sqrt(13)* 
log(2*x + 1)*x**2 - 19843848*sqrt(13)*log(2*x + 1)*x - 6614616*sqrt(13)*lo 
g(2*x + 1)))/(589241991*(36*x**6 + 12*x**5 + 37*x**4 + 30*x**3 + 13*x**2 + 
 12*x + 4))