\(\int \frac {(2-x+3 x^2)^{3/2} (1+3 x+4 x^2)}{(1+2 x)^3} \, dx\) [56]

Optimal result

Integrand size = 32, antiderivative size = 138 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {1}{624} (1858-771 x) \sqrt {2-x+3 x^2}+\frac {(151+122 x) \left (2-x+3 x^2\right )^{3/2}}{312 (1+2 x)}-\frac {\left (2-x+3 x^2\right )^{5/2}}{26 (1+2 x)^2}+\frac {1519 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{192 \sqrt {3}}-\frac {1153 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{64 \sqrt {13}} \] Output:

1/624*(1858-771*x)*(3*x^2-x+2)^(1/2)+(151+122*x)*(3*x^2-x+2)^(3/2)/(312+62 
4*x)-1/26*(3*x^2-x+2)^(5/2)/(1+2*x)^2+1519/576*arcsinh(1/23*(1-6*x)*23^(1/ 
2))*3^(1/2)-1153/832*13^(1/2)*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1 
/2))
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {\frac {156 \sqrt {2-x+3 x^2} \left (182+627 x+390 x^2-68 x^3+96 x^4\right )}{(1+2 x)^2}+20754 \sqrt {13} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )+19747 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{7488} \] Input:

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

Output:

((156*Sqrt[2 - x + 3*x^2]*(182 + 627*x + 390*x^2 - 68*x^3 + 96*x^4))/(1 + 
2*x)^2 + 20754*Sqrt[13]*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3* 
x^2])/Sqrt[13]] + 19747*Sqrt[3]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/74 
88
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2181, 27, 1230, 27, 1231, 27, 1269, 1090, 222, 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.

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

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {1}{26} \int -\frac {(122 x+31) \left (3 x^2-x+2\right )^{3/2}}{2 (2 x+1)^2}dx-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{52} \int \frac {(122 x+31) \left (3 x^2-x+2\right )^{3/2}}{(2 x+1)^2}dx-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {1}{52} \left (\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}-\frac {1}{8} \int -\frac {2 (639-1028 x) \sqrt {3 x^2-x+2}}{2 x+1}dx\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \int \frac {(639-1028 x) \sqrt {3 x^2-x+2}}{2 x+1}dx+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {1}{3} (1858-771 x) \sqrt {3 x^2-x+2}-\frac {1}{48} \int -\frac {104 (970-1519 x)}{(2 x+1) \sqrt {3 x^2-x+2}}dx\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \int \frac {970-1519 x}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \left (\frac {3459}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {1519}{2} \int \frac {1}{\sqrt {3 x^2-x+2}}dx\right )+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \left (\frac {3459}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {1519 \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)}{2 \sqrt {69}}\right )+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \left (\frac {3459}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx-\frac {1519 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{2 \sqrt {3}}\right )+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \left (-3459 \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}-\frac {1519 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{2 \sqrt {3}}\right )+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{52} \left (\frac {1}{4} \left (\frac {13}{6} \left (-\frac {1519 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{2 \sqrt {3}}-\frac {3459 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2 \sqrt {13}}\right )+\frac {1}{3} \sqrt {3 x^2-x+2} (1858-771 x)\right )+\frac {(122 x+151) \left (3 x^2-x+2\right )^{3/2}}{6 (2 x+1)}\right )-\frac {\left (3 x^2-x+2\right )^{5/2}}{26 (2 x+1)^2}\)

Input:

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

Output:

-1/26*(2 - x + 3*x^2)^(5/2)/(1 + 2*x)^2 + (((151 + 122*x)*(2 - x + 3*x^2)^ 
(3/2))/(6*(1 + 2*x)) + (((1858 - 771*x)*Sqrt[2 - x + 3*x^2])/3 + (13*((-15 
19*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(2*Sqrt[3]) - (3459*ArcTanh[(9 - 8*x)/(2* 
Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(2*Sqrt[13])))/6)/4)/52
 

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 222

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

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[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 = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70

method	result	size
risch	\(\frac {288 x^{6}-300 x^{5}+1430 x^{4}+1355 x^{3}+699 x^{2}+1072 x +364}{48 \left (1+2 x \right )^{2} \sqrt {3 x^{2}-x +2}}-\frac {1519 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{576}-\frac {1153 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{832}\)	\(97\)
trager	\(\frac {\left (96 x^{4}-68 x^{3}+390 x^{2}+627 x +182\right ) \sqrt {3 x^{2}-x +2}}{48 \left (1+2 x \right )^{2}}+\frac {1153 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{832}-\frac {1519 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{576}\)	\(133\)
default	\(\frac {1153 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}{4056}-\frac {257 \left (6 x -1\right ) \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x}}{1248}-\frac {1519 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{576}+\frac {1153 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}{832}-\frac {1153 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{832}-\frac {\left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {5}{2}}}{104 \left (\frac {1}{2}+x \right )^{2}}+\frac {15 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {5}{2}}}{338 \left (\frac {1}{2}+x \right )}-\frac {15 \left (6 x -1\right ) \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x \right )^{\frac {3}{2}}}{676}\)	\(162\)

Input:

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

Output:

1/48*(288*x^6-300*x^5+1430*x^4+1355*x^3+699*x^2+1072*x+364)/(1+2*x)^2/(3*x 
^2-x+2)^(1/2)-1519/576*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-1153/832*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 = 159, normalized size of antiderivative = 1.15 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {19747 \, \sqrt {3} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 10377 \, \sqrt {13} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 312 \, {\left (96 \, x^{4} - 68 \, x^{3} + 390 \, x^{2} + 627 \, x + 182\right )} \sqrt {3 \, x^{2} - x + 2}}{14976 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \] Input:

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

Output:

1/14976*(19747*sqrt(3)*(4*x^2 + 4*x + 1)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2) 
*(6*x - 1) - 72*x^2 + 24*x - 25) + 10377*sqrt(13)*(4*x^2 + 4*x + 1)*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)) + 312*(96*x^4 - 68*x^3 + 390*x^2 + 627*x + 182)*sqrt(3*x^2 - x 
+ 2))/(4*x^2 + 4*x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {61}{312} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {{\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}}}{26 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} - \frac {257}{208} \, \sqrt {3 \, x^{2} - x + 2} x - \frac {1519}{576} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1153}{832} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {929}{312} \, \sqrt {3 \, x^{2} - x + 2} + \frac {15 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}}{52 \, {\left (2 \, x + 1\right )}} \] Input:

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

Output:

61/312*(3*x^2 - x + 2)^(3/2) - 1/26*(3*x^2 - x + 2)^(5/2)/(4*x^2 + 4*x + 1 
) - 257/208*sqrt(3*x^2 - x + 2)*x - 1519/576*sqrt(3)*arcsinh(6/23*sqrt(23) 
*x - 1/23*sqrt(23)) + 1153/832*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 
1) - 9/23*sqrt(23)/abs(2*x + 1)) + 929/312*sqrt(3*x^2 - x + 2) + 15/52*(3* 
x^2 - x + 2)^(3/2)/(2*x + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (111) = 222\).

Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.89 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {1}{96} \, {\left (2 \, {\left (24 \, x - 41\right )} x + 265\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {1519}{576} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac {1153}{832} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {446 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} - 85 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} - 1993 \, \sqrt {3} x + 1009 \, \sqrt {3} + 1993 \, \sqrt {3 \, x^{2} - x + 2}}{32 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \] Input:

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

Output:

1/96*(2*(24*x - 41)*x + 265)*sqrt(3*x^2 - x + 2) + 1519/576*sqrt(3)*log(-2 
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 1153/832*sqrt(13)*log(-1 
/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*sqrt(3*x^2 - x + 2))/(2*s 
qrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2))) + 1/32*(446*(sqrt( 
3)*x - sqrt(3*x^2 - x + 2))^3 - 85*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2 
))^2 - 1993*sqrt(3)*x + 1009*sqrt(3) + 1993*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. \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\int \frac {{\left (3\,x^2-x+2\right )}^{3/2}\,\left (4\,x^2+3\,x+1\right )}{{\left (2\,x+1\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.04 \[ \int \frac {\left (2-x+3 x^2\right )^{3/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^3} \, dx=\frac {14976 \sqrt {3 x^{2}-x +2}\, x^{4}-10608 \sqrt {3 x^{2}-x +2}\, x^{3}+60840 \sqrt {3 x^{2}-x +2}\, x^{2}+97812 \sqrt {3 x^{2}-x +2}\, x +28392 \sqrt {3 x^{2}-x +2}+41508 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{2}+41508 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x +10377 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right )-41508 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{2}-41508 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x -10377 \sqrt {13}\, \mathrm {log}\left (2 x +1\right )+78988 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}-6 x +1\right ) x^{2}+78988 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}-6 x +1\right ) x +19747 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}-6 x +1\right )}{29952 x^{2}+29952 x +7488} \] Input:

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

Output:

(14976*sqrt(3*x**2 - x + 2)*x**4 - 10608*sqrt(3*x**2 - x + 2)*x**3 + 60840 
*sqrt(3*x**2 - x + 2)*x**2 + 97812*sqrt(3*x**2 - x + 2)*x + 28392*sqrt(3*x 
**2 - x + 2) + 41508*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 
9)*x**2 + 41508*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x 
+ 10377*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9) - 41508*sq 
rt(13)*log(2*x + 1)*x**2 - 41508*sqrt(13)*log(2*x + 1)*x - 10377*sqrt(13)* 
log(2*x + 1) + 78988*sqrt(3)*log(2*sqrt(3*x**2 - x + 2)*sqrt(3) - 6*x + 1) 
*x**2 + 78988*sqrt(3)*log(2*sqrt(3*x**2 - x + 2)*sqrt(3) - 6*x + 1)*x + 19 
747*sqrt(3)*log(2*sqrt(3*x**2 - x + 2)*sqrt(3) - 6*x + 1))/(7488*(4*x**2 + 
 4*x + 1))