\(\int \frac {(5-2 x)^5}{(4+3 x) (2+3 x+x^2)^{3/2}} \, dx\) [629]

Optimal result

Integrand size = 27, antiderivative size = 105 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {8179-25435 x}{\sqrt {2+3 x+x^2}}+\frac {1832}{9} \sqrt {2+3 x+x^2}-\frac {16}{3} x \sqrt {2+3 x+x^2}-\frac {6436343 \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{27 \sqrt {2}}-\frac {90472}{27} \text {arctanh}\left (\frac {1+x}{\sqrt {2+3 x+x^2}}\right ) \] Output:

-(8179-25435*x)/(x^2+3*x+2)^(1/2)+1832/9*(x^2+3*x+2)^(1/2)-16/3*x*(x^2+3*x 
+2)^(1/2)-6436343/54*2^(1/2)*arctan(2^(1/2)*(1+x)/(x^2+3*x+2)^(1/2))-90472 
/27*arctanh((1+x)/(x^2+3*x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {\sqrt {2+3 x+x^2} \left (-69947+234315 x+1688 x^2-48 x^3\right )}{9 (1+x) (2+x)}+\frac {6436343 \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{27 \sqrt {2}}-\frac {90472}{27} \text {arctanh}\left (\frac {\sqrt {2+3 x+x^2}}{1+x}\right ) \] Input:

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

Output:

(Sqrt[2 + 3*x + x^2]*(-69947 + 234315*x + 1688*x^2 - 48*x^3))/(9*(1 + x)*( 
2 + x)) + (6436343*ArcTan[Sqrt[2 + 3*x + x^2]/(Sqrt[2]*(1 + x))])/(27*Sqrt 
[2]) - (90472*ArcTanh[Sqrt[2 + 3*x + x^2]/(1 + x)])/27
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1264, 27, 2184, 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.

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

\(\Big \downarrow \) 1264

\(\displaystyle -2 \int \frac {64 x^3-992 x^2+6848 x+249429}{4 (3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} \int \frac {64 x^3-992 x^2+6848 x+249429}{(3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{2} \left (-\frac {1}{54} \int \frac {6 \left (-11376 x^2+58240 x+2243325\right )}{(3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {32}{9} \sqrt {x^2+3 x+2} (3 x+4)\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {1}{9} \int \frac {-11376 x^2+58240 x+2243325}{(3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {32}{9} \sqrt {x^2+3 x+2} (3 x+4)\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (3792 \sqrt {x^2+3 x+2}-\frac {1}{9} \int \frac {9 (90472 x+2266077)}{(3 x+4) \sqrt {x^2+3 x+2}}dx\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (3792 \sqrt {x^2+3 x+2}-\int \frac {90472 x+2266077}{(3 x+4) \sqrt {x^2+3 x+2}}dx\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {90472}{3} \int \frac {1}{\sqrt {x^2+3 x+2}}dx-\frac {6436343}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx+3792 \sqrt {x^2+3 x+2}\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {6436343}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {180944}{3} \int \frac {1}{4-\frac {(2 x+3)^2}{x^2+3 x+2}}d\frac {2 x+3}{\sqrt {x^2+3 x+2}}+3792 \sqrt {x^2+3 x+2}\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {6436343}{3} \int \frac {1}{(3 x+4) \sqrt {x^2+3 x+2}}dx-\frac {90472}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )+3792 \sqrt {x^2+3 x+2}\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (\frac {12872686}{3} \int \frac {1}{-\frac {x^2}{x^2+3 x+2}-8}d\frac {x}{\sqrt {x^2+3 x+2}}-\frac {90472}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )+3792 \sqrt {x^2+3 x+2}\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} \left (\frac {1}{9} \left (-\frac {6436343 \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{3 \sqrt {2}}-\frac {90472}{3} \text {arctanh}\left (\frac {2 x+3}{2 \sqrt {x^2+3 x+2}}\right )+3792 \sqrt {x^2+3 x+2}\right )-\frac {32}{9} (3 x+4) \sqrt {x^2+3 x+2}\right )-\frac {8179-25435 x}{\sqrt {x^2+3 x+2}}\)

Input:

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

Output:

-((8179 - 25435*x)/Sqrt[2 + 3*x + x^2]) + ((-32*(4 + 3*x)*Sqrt[2 + 3*x + x 
^2])/9 + (3792*Sqrt[2 + 3*x + x^2] - (6436343*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 
 3*x + x^2])])/(3*Sqrt[2]) - (90472*ArcTanh[(3 + 2*x)/(2*Sqrt[2 + 3*x + x^ 
2])])/3)/9)/2
 

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 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, 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] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(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, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

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 = 1.40 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {48 x^{3}-1688 x^{2}-234315 x +69947}{9 \sqrt {x^{2}+3 x +2}}-\frac {45236 \ln \left (x +\frac {3}{2}+\sqrt {x^{2}+3 x +2}\right )}{27}-\frac {6436343 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{108}\)	\(71\)
trager	\(-\frac {48 x^{3}-1688 x^{2}-234315 x +69947}{9 \sqrt {x^{2}+3 x +2}}+\frac {45236 \ln \left (2 \sqrt {x^{2}+3 x +2}-3-2 x \right )}{27}+\frac {6436343 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}-x}{3 x +4}\right )}{108}\)	\(89\)
default	\(\frac {\frac {12355724 x}{243}+\frac {6177862}{81}}{\sqrt {x^{2}+3 x +2}}-\frac {371042}{81 \sqrt {x^{2}+3 x +2}}+\frac {45236 x}{27 \sqrt {x^{2}+3 x +2}}-\frac {45236 \ln \left (x +\frac {3}{2}+\sqrt {x^{2}+3 x +2}\right )}{27}+\frac {1688 x^{2}}{9 \sqrt {x^{2}+3 x +2}}-\frac {16 x^{3}}{3 \sqrt {x^{2}+3 x +2}}-\frac {6436343}{162 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}-\frac {6436343 \left (2 x +3\right )}{486 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}-\frac {6436343 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{108}\)	\(149\)

Input:

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

Output:

-1/9*(48*x^3-1688*x^2-234315*x+69947)/(x^2+3*x+2)^(1/2)-45236/27*ln(x+3/2+ 
(x^2+3*x+2)^(1/2))-6436343/108*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 = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.11 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {6436343 \, \sqrt {2} {\left (x^{2} + 3 \, x + 2\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) - 1381896 \, x^{2} - 90472 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (-2 \, x + 2 \, \sqrt {x^{2} + 3 \, x + 2} - 3\right ) + 6 \, {\left (48 \, x^{3} - 1688 \, x^{2} - 234315 \, x + 69947\right )} \sqrt {x^{2} + 3 \, x + 2} - 4145688 \, x - 2763792}{54 \, {\left (x^{2} + 3 \, x + 2\right )}} \] Input:

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

Output:

-1/54*(6436343*sqrt(2)*(x^2 + 3*x + 2)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2 
*sqrt(2)*sqrt(x^2 + 3*x + 2)) - 1381896*x^2 - 90472*(x^2 + 3*x + 2)*log(-2 
*x + 2*sqrt(x^2 + 3*x + 2) - 3) + 6*(48*x^3 - 1688*x^2 - 234315*x + 69947) 
*sqrt(x^2 + 3*x + 2) - 4145688*x - 2763792)/(x^2 + 3*x + 2)
 

Sympy [F]

\[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=- \int \frac {6250 x}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {5000 x^{2}}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\right )\, dx - \int \frac {2000 x^{3}}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {400 x^{4}}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\right )\, dx - \int \frac {32 x^{5}}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\, dx - \int \left (- \frac {3125}{3 x^{3} \sqrt {x^{2} + 3 x + 2} + 13 x^{2} \sqrt {x^{2} + 3 x + 2} + 18 x \sqrt {x^{2} + 3 x + 2} + 8 \sqrt {x^{2} + 3 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(6250*x/(3*x**3*sqrt(x**2 + 3*x + 2) + 13*x**2*sqrt(x**2 + 3*x + 
2) + 18*x*sqrt(x**2 + 3*x + 2) + 8*sqrt(x**2 + 3*x + 2)), x) - Integral(-5 
000*x**2/(3*x**3*sqrt(x**2 + 3*x + 2) + 13*x**2*sqrt(x**2 + 3*x + 2) + 18* 
x*sqrt(x**2 + 3*x + 2) + 8*sqrt(x**2 + 3*x + 2)), x) - Integral(2000*x**3/ 
(3*x**3*sqrt(x**2 + 3*x + 2) + 13*x**2*sqrt(x**2 + 3*x + 2) + 18*x*sqrt(x* 
*2 + 3*x + 2) + 8*sqrt(x**2 + 3*x + 2)), x) - Integral(-400*x**4/(3*x**3*s 
qrt(x**2 + 3*x + 2) + 13*x**2*sqrt(x**2 + 3*x + 2) + 18*x*sqrt(x**2 + 3*x 
+ 2) + 8*sqrt(x**2 + 3*x + 2)), x) - Integral(32*x**5/(3*x**3*sqrt(x**2 + 
3*x + 2) + 13*x**2*sqrt(x**2 + 3*x + 2) + 18*x*sqrt(x**2 + 3*x + 2) + 8*sq 
rt(x**2 + 3*x + 2)), x) - Integral(-3125/(3*x**3*sqrt(x**2 + 3*x + 2) + 13 
*x**2*sqrt(x**2 + 3*x + 2) + 18*x*sqrt(x**2 + 3*x + 2) + 8*sqrt(x**2 + 3*x 
 + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {16 \, x^{3}}{3 \, \sqrt {x^{2} + 3 \, x + 2}} + \frac {1688 \, x^{2}}{9 \, \sqrt {x^{2} + 3 \, x + 2}} - \frac {6436343}{108} \, \sqrt {2} \arcsin \left (\frac {x}{{\left | 3 \, x + 4 \right |}}\right ) + \frac {26035 \, x}{\sqrt {x^{2} + 3 \, x + 2}} - \frac {69947}{9 \, \sqrt {x^{2} + 3 \, x + 2}} - \frac {45236}{27} \, \log \left (\frac {2}{3} \, x + \frac {2}{3} \, \sqrt {x^{2} + 3 \, x + 2} + 1\right ) \] Input:

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

Output:

-16/3*x^3/sqrt(x^2 + 3*x + 2) + 1688/9*x^2/sqrt(x^2 + 3*x + 2) - 6436343/1 
08*sqrt(2)*arcsin(x/abs(3*x + 4)) + 26035*x/sqrt(x^2 + 3*x + 2) - 69947/9/ 
sqrt(x^2 + 3*x + 2) - 45236/27*log(2/3*x + 2/3*sqrt(x^2 + 3*x + 2) + 1)
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.72 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {6436343}{54} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 3 \, x + 2} + 4\right )}\right ) - \frac {{\left (8 \, {\left (6 \, x - 211\right )} x - 234315\right )} x + 69947}{9 \, \sqrt {x^{2} + 3 \, x + 2}} + \frac {45236}{27} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + 3 \, x + 2} - 3 \right |}\right ) \] Input:

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

Output:

-6436343/54*sqrt(2)*arctan(-1/2*sqrt(2)*(3*x - 3*sqrt(x^2 + 3*x + 2) + 4)) 
 - 1/9*((8*(6*x - 211)*x - 234315)*x + 69947)/sqrt(x^2 + 3*x + 2) + 45236/ 
27*log(abs(-2*x + 2*sqrt(x^2 + 3*x + 2) - 3))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.07 \[ \int \frac {(5-2 x)^5}{(4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {-6436343 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{2}-19309029 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x -12872686 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )-288 \sqrt {x^{2}+3 x +2}\, x^{3}+10128 \sqrt {x^{2}+3 x +2}\, x^{2}+1405890 \sqrt {x^{2}+3 x +2}\, x -419682 \sqrt {x^{2}+3 x +2}-90472 \,\mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+2 x +3\right ) x^{2}-271416 \,\mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+2 x +3\right ) x -180944 \,\mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+2 x +3\right )+1373490 x^{2}+4120470 x +2746980}{54 x^{2}+162 x +108} \] Input:

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

Output:

( - 6436343*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x**2 
- 19309029*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x - 12 
872686*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) - 288*sqrt 
(x**2 + 3*x + 2)*x**3 + 10128*sqrt(x**2 + 3*x + 2)*x**2 + 1405890*sqrt(x** 
2 + 3*x + 2)*x - 419682*sqrt(x**2 + 3*x + 2) - 90472*log(2*sqrt(x**2 + 3*x 
 + 2) + 2*x + 3)*x**2 - 271416*log(2*sqrt(x**2 + 3*x + 2) + 2*x + 3)*x - 1 
80944*log(2*sqrt(x**2 + 3*x + 2) + 2*x + 3) + 1373490*x**2 + 4120470*x + 2 
746980)/(54*(x**2 + 3*x + 2))