\(\int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx\) [538]

Optimal result

Integrand size = 27, antiderivative size = 92 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-\frac {8}{9} \sqrt {2+3 x+x^2}+\frac {12167 \sqrt {2+3 x+x^2}}{18 (4+3 x)}-\frac {529 \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{54 \sqrt {2}}+\frac {560}{27} \text {arctanh}\left (\frac {1+x}{\sqrt {2+3 x+x^2}}\right ) \] Output:

-8/9*(x^2+3*x+2)^(1/2)+12167*(x^2+3*x+2)^(1/2)/(72+54*x)-529/108*2^(1/2)*a 
rctan(2^(1/2)*(1+x)/(x^2+3*x+2)^(1/2))+560/27*arctanh((1+x)/(x^2+3*x+2)^(1 
/2))
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {(12103-48 x) \sqrt {2+3 x+x^2}}{18 (4+3 x)}+\frac {529 \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{54 \sqrt {2}}+\frac {560}{27} \text {arctanh}\left (\frac {\sqrt {2+3 x+x^2}}{1+x}\right ) \] Input:

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

Output:

((12103 - 48*x)*Sqrt[2 + 3*x + x^2])/(18*(4 + 3*x)) + (529*ArcTan[Sqrt[2 + 
 3*x + x^2]/(Sqrt[2]*(1 + x))])/(54*Sqrt[2]) + (560*ArcTanh[Sqrt[2 + 3*x + 
 x^2]/(1 + x)])/27
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1266, 27, 2184, 27, 1269, 1092, 219, 1154, 217}

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.

\(\displaystyle \int \frac {(5-2 x)^3}{(3 x+4)^2 \sqrt {x^2+3 x+2}} \, dx\)

\(\Big \downarrow \) 1266

\(\displaystyle \frac {1}{2} \int \frac {-96 x^2+848 x+1125}{18 (3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{36} \int \frac {-96 x^2+848 x+1125}{(3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{36} \left (\frac {1}{9} \int \frac {9 (1120 x+1317)}{(3 x+4) \sqrt {x^2+3 x+2}}dx-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{36} \left (\int \frac {1120 x+1317}{(3 x+4) \sqrt {x^2+3 x+2}}dx-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{36} \left (\frac {1120}{3} \int \frac {1}{\sqrt {x^2+3 x+2}}dx-\frac {529}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{36} \left (-\frac {529}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {2240}{3} \int \frac {1}{4-\frac {(2 x+3)^2}{x^2+3 x+2}}d\frac {2 x+3}{\sqrt {x^2+3 x+2}}-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{36} \left (-\frac {529}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx+\frac {1120}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{36} \left (\frac {1058}{3} \int \frac {1}{-\frac {x^2}{x^2+3 x+2}-8}d\frac {x}{\sqrt {x^2+3 x+2}}+\frac {1120}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{36} \left (-\frac {529 \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{3 \sqrt {2}}+\frac {1120}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )-32 \sqrt {x^2+3 x+2}\right )+\frac {12167 \sqrt {x^2+3 x+2}}{18 (3 x+4)}\)

Input:

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

Output:

(12167*Sqrt[2 + 3*x + x^2])/(18*(4 + 3*x)) + (-32*Sqrt[2 + 3*x + x^2] - (5 
29*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/(3*Sqrt[2]) + (1120*ArcTanh[ 
(3 + 2*x)/(2*Sqrt[2 + 3*x + x^2])])/3)/36
 

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 1092

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

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 1266

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^ 
n, d + e*x, x], R = PolynomialRemainder[(f + g*x)^n, 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)*Q + 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, f, g, p}, x] && IGtQ[n, 1] && ILtQ[m, -1] && NeQ[c*d^2 - b* 
d*e + a*e^2, 0] && (NeQ[m + n, 0] || EqQ[p, -2^(-1)])
 

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 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {280 \ln \left (x +\frac {3}{2}+\sqrt {x^{2}+3 x +2}\right )}{27}-\frac {8 \sqrt {x^{2}+3 x +2}}{9}+\frac {12167 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}{54 \left (x +\frac {4}{3}\right )}-\frac {529 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{216}\)	\(75\)
risch	\(-\frac {48 x^{3}-11959 x^{2}-36213 x -24206}{18 \left (3 x +4\right ) \sqrt {x^{2}+3 x +2}}+\frac {280 \ln \left (x +\frac {3}{2}+\sqrt {x^{2}+3 x +2}\right )}{27}-\frac {529 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{216}\)	\(78\)
trager	\(-\frac {\left (-12103+48 x \right ) \sqrt {x^{2}+3 x +2}}{18 \left (3 x +4\right )}-\frac {280 \ln \left (2 \sqrt {x^{2}+3 x +2}-3-2 x \right )}{27}+\frac {529 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}-x}{3 x +4}\right )}{216}\)	\(86\)

Input:

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

Output:

280/27*ln(x+3/2+(x^2+3*x+2)^(1/2))-8/9*(x^2+3*x+2)^(1/2)+12167/54/(x+4/3)* 
((x+4/3)^2+1/3*x+2/9)^(1/2)-529/216*2^(1/2)*arctan(3/4*x*2^(1/2)/(9*(x+4/3 
)^2+3*x+2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-\frac {529 \, \sqrt {2} {\left (3 \, x + 4\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) + 1120 \, {\left (3 \, x + 4\right )} \log \left (-2 \, x + 2 \, \sqrt {x^{2} + 3 \, x + 2} - 3\right ) + 6 \, \sqrt {x^{2} + 3 \, x + 2} {\left (48 \, x - 12103\right )} - 72786 \, x - 97048}{108 \, {\left (3 \, x + 4\right )}} \] Input:

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

Output:

-1/108*(529*sqrt(2)*(3*x + 4)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2*sqrt(2)* 
sqrt(x^2 + 3*x + 2)) + 1120*(3*x + 4)*log(-2*x + 2*sqrt(x^2 + 3*x + 2) - 3 
) + 6*sqrt(x^2 + 3*x + 2)*(48*x - 12103) - 72786*x - 97048)/(3*x + 4)
 

Sympy [F]

\[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=- \int \frac {150 x}{9 x^{2} \sqrt {x^{2} + 3 x + 2} + 24 x \sqrt {x^{2} + 3 x + 2} + 16 \sqrt {x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {60 x^{2}}{9 x^{2} \sqrt {x^{2} + 3 x + 2} + 24 x \sqrt {x^{2} + 3 x + 2} + 16 \sqrt {x^{2} + 3 x + 2}}\right )\, dx - \int \frac {8 x^{3}}{9 x^{2} \sqrt {x^{2} + 3 x + 2} + 24 x \sqrt {x^{2} + 3 x + 2} + 16 \sqrt {x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {125}{9 x^{2} \sqrt {x^{2} + 3 x + 2} + 24 x \sqrt {x^{2} + 3 x + 2} + 16 \sqrt {x^{2} + 3 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(150*x/(9*x**2*sqrt(x**2 + 3*x + 2) + 24*x*sqrt(x**2 + 3*x + 2) + 
 16*sqrt(x**2 + 3*x + 2)), x) - Integral(-60*x**2/(9*x**2*sqrt(x**2 + 3*x 
+ 2) + 24*x*sqrt(x**2 + 3*x + 2) + 16*sqrt(x**2 + 3*x + 2)), x) - Integral 
(8*x**3/(9*x**2*sqrt(x**2 + 3*x + 2) + 24*x*sqrt(x**2 + 3*x + 2) + 16*sqrt 
(x**2 + 3*x + 2)), x) - Integral(-125/(9*x**2*sqrt(x**2 + 3*x + 2) + 24*x* 
sqrt(x**2 + 3*x + 2) + 16*sqrt(x**2 + 3*x + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-\frac {529}{216} \, \sqrt {2} \arcsin \left (\frac {x}{{\left | 3 \, x + 4 \right |}}\right ) - \frac {8}{9} \, \sqrt {x^{2} + 3 \, x + 2} + \frac {12167 \, \sqrt {x^{2} + 3 \, x + 2}}{18 \, {\left (3 \, x + 4\right )}} + \frac {280}{27} \, \log \left (\frac {2}{3} \, x + \frac {2}{3} \, \sqrt {x^{2} + 3 \, x + 2} + 1\right ) \] Input:

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

Output:

-529/216*sqrt(2)*arcsin(x/abs(3*x + 4)) - 8/9*sqrt(x^2 + 3*x + 2) + 12167/ 
18*sqrt(x^2 + 3*x + 2)/(3*x + 4) + 280/27*log(2/3*x + 2/3*sqrt(x^2 + 3*x + 
 2) + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (72) = 144\).

Time = 0.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.54 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-\frac {529 \, \sqrt {2} \arcsin \left (-\frac {4}{3 \, {\left (3 \, x + 4\right )}} + \frac {1}{3}\right )}{216 \, \mathrm {sgn}\left (\frac {1}{3 \, x + 4}\right )} - \frac {280 \, \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (2 \, \sqrt {2} \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1} - 3\right )}}{\frac {4}{3 \, x + 4} - 1} + 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (2 \, \sqrt {2} \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1} - 3\right )}}{\frac {4}{3 \, x + 4} - 1} + 6 \right |}}\right )}{27 \, \mathrm {sgn}\left (\frac {1}{3 \, x + 4}\right )} + \frac {12167 \, \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1}}{54 \, \mathrm {sgn}\left (\frac {1}{3 \, x + 4}\right )} - \frac {16 \, {\left (\sqrt {2} - \frac {3 \, \sqrt {2} {\left (2 \, \sqrt {2} \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1} - 3\right )}}{\frac {4}{3 \, x + 4} - 1}\right )}}{9 \, {\left (\frac {{\left (2 \, \sqrt {2} \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1} - 3\right )}^{2}}{{\left (\frac {4}{3 \, x + 4} - 1\right )}^{2}} - \frac {6 \, {\left (2 \, \sqrt {2} \sqrt {\frac {1}{3 \, x + 4} - \frac {2}{{\left (3 \, x + 4\right )}^{2}} + 1} - 3\right )}}{\frac {4}{3 \, x + 4} - 1} + 1\right )} \mathrm {sgn}\left (\frac {1}{3 \, x + 4}\right )} \] Input:

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

Output:

-529/216*sqrt(2)*arcsin(-4/3/(3*x + 4) + 1/3)/sgn(1/(3*x + 4)) - 280/27*lo 
g(abs(-4*sqrt(2) - 2*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3) 
/(4/(3*x + 4) - 1) + 6)/abs(4*sqrt(2) - 2*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/ 
(3*x + 4)^2 + 1) - 3)/(4/(3*x + 4) - 1) + 6))/sgn(1/(3*x + 4)) + 12167/54* 
sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1)/sgn(1/(3*x + 4)) - 16/9*(sqrt(2) - 3 
*sqrt(2)*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/(4/(3*x + 4 
) - 1))/(((2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)^2/(4/(3*x 
+ 4) - 1)^2 - 6*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/(4/( 
3*x + 4) - 1) + 1)*sgn(1/(3*x + 4)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.35 \[ \int \frac {(5-2 x)^3}{(4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {-1587 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x -2116 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )-288 \sqrt {x^{2}+3 x +2}\, x +72618 \sqrt {x^{2}+3 x +2}+3360 \,\mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+2 x +3\right ) x +4480 \,\mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+2 x +3\right )}{324 x +432} \] Input:

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

Output:

( - 1587*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x - 2116 
*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) - 288*sqrt(x**2 
+ 3*x + 2)*x + 72618*sqrt(x**2 + 3*x + 2) + 3360*log(2*sqrt(x**2 + 3*x + 2 
) + 2*x + 3)*x + 4480*log(2*sqrt(x**2 + 3*x + 2) + 2*x + 3))/(108*(3*x + 4 
))