\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^2} \, dx\) [946]

Optimal result

Integrand size = 27, antiderivative size = 141 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {1}{512} (3865-8082 x) \sqrt {2+5 x+3 x^2}-\frac {1}{192} (65-1194 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+5 x+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {41053 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{512 \sqrt {3}}-\frac {1325}{64} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right ) \] Output:

-1/512*(3865-8082*x)*(3*x^2+5*x+2)^(1/2)-1/192*(65-1194*x)*(3*x^2+5*x+2)^( 
3/2)-(34+x)*(3*x^2+5*x+2)^(5/2)/(30+20*x)+41053/1536*arctanh(3^(1/2)*(1+x) 
/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1325/64*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2 
+5*x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {-\frac {\sqrt {2+5 x+3 x^2} \left (293973+40412 x-118996 x^2-80064 x^3-28512 x^4+6912 x^5\right )}{3+2 x}-159000 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+205265 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{7680} \] Input:

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

Output:

(-((Sqrt[2 + 5*x + 3*x^2]*(293973 + 40412*x - 118996*x^2 - 80064*x^3 - 285 
12*x^4 + 6912*x^5))/(3 + 2*x)) - 159000*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3* 
x^2)/5]/(1 + x)] + 205265*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x 
)])/7680
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1230, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {1}{8} \int -\frac {2 (199 x+166) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {(199 x+166) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{4} \left (-\frac {1}{96} \int -\frac {3 (8082 x+6823) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \int \frac {(8082 x+6823) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (-\frac {1}{48} \int -\frac {6 (82106 x+70159)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{4} \sqrt {3 x^2+5 x+2} (3865-8082 x)\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \int \frac {82106 x+70159}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \left (41053 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-53000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \left (82106 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-53000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \left (\frac {41053 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-53000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \left (106000 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {41053 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} \left (\frac {1}{32} \left (\frac {1}{8} \left (\frac {41053 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-10600 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {1}{4} (3865-8082 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{48} (65-1194 x) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {(x+34) \left (3 x^2+5 x+2\right )^{5/2}}{10 (2 x+3)}\)

Input:

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

Output:

-1/10*((34 + x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x) + (-1/48*((65 - 1194*x) 
*(2 + 5*x + 3*x^2)^(3/2)) + (-1/4*((3865 - 8082*x)*Sqrt[2 + 5*x + 3*x^2]) 
+ ((41053*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] - 
10600*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/8)/32) 
/4
 

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 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 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]
 

Maple [A] (verified)

Time = 1.87 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {20736 x^{7}-50976 x^{6}-368928 x^{5}-814332 x^{4}-633872 x^{3}+845987 x^{2}+1550689 x +587946}{7680 \left (2 x +3\right ) \sqrt {3 x^{2}+5 x +2}}+\frac {41053 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3072}+\frac {1325 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{128}\)	\(117\)
trager	\(-\frac {\left (6912 x^{5}-28512 x^{4}-80064 x^{3}-118996 x^{2}+40412 x +293973\right ) \sqrt {3 x^{2}+5 x +2}}{7680 \left (2 x +3\right )}-\frac {1325 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{2 x +3}\right )}{128}+\frac {41053 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{3072}\)	\(138\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{10 \left (x +\frac {3}{2}\right )}-\frac {53 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{20}+\frac {199 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{192}+\frac {1347 \left (6 x +5\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{512}+\frac {41053 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{3072}-\frac {265 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{48}-\frac {1325 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{128}+\frac {1325 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{128}+\frac {13 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{20}\)	\(195\)

Input:

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

Output:

-1/7680*(20736*x^7-50976*x^6-368928*x^5-814332*x^4-633872*x^3+845987*x^2+1 
550689*x+587946)/(2*x+3)/(3*x^2+5*x+2)^(1/2)+41053/3072*ln(1/3*(5/2+3*x)*3 
^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+1325/128*5^(1/2)*arctanh(2/5*(-7/2-4*x 
)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.06 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {205265 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 159000 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (6912 \, x^{5} - 28512 \, x^{4} - 80064 \, x^{3} - 118996 \, x^{2} + 40412 \, x + 293973\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{30720 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

1/30720*(205265*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x 
 + 5) + 72*x^2 + 120*x + 49) + 159000*sqrt(5)*(2*x + 3)*log(-(4*sqrt(5)*sq 
rt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12*x + 9)) 
- 4*(6912*x^5 - 28512*x^4 - 80064*x^3 - 118996*x^2 + 40412*x + 293973)*sqr 
t(3*x^2 + 5*x + 2))/(2*x + 3)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \] Input:

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

Output:

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-9 
6*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-165*x**2*sq 
rt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-113*x**3*sqrt(3*x 
**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-15*x**4*sqrt(3*x**2 + 5 
*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/( 
4*x**2 + 12*x + 9), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {1}{20} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {199}{32} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {65}{192} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {4041}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {41053}{3072} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {1325}{128} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {3865}{512} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

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

Output:

-1/20*(3*x^2 + 5*x + 2)^(5/2) + 199/32*(3*x^2 + 5*x + 2)^(3/2)*x - 65/192* 
(3*x^2 + 5*x + 2)^(3/2) - 13/4*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 4041/25 
6*sqrt(3*x^2 + 5*x + 2)*x + 41053/3072*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5* 
x + 2) + 3*x + 5/2) + 1325/128*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/a 
bs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 3865/512*sqrt(3*x^2 + 5*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (113) = 226\).

Time = 0.73 (sec) , antiderivative size = 671, normalized size of antiderivative = 4.76 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-41053/3072*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3) 
^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2* 
x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 1325/128*sqrt(5)* 
log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3 
)) - 4))*sgn(1/(2*x + 3)) - 325/128*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) 
*sgn(1/(2*x + 3)) + 1/7680*(1304805*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3 
) + sqrt(5)/(2*x + 3))^9*sgn(1/(2*x + 3)) - 2064120*sqrt(5)*(sqrt(-8/(2*x 
+ 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^8*sgn(1/(2*x + 3)) - 438295 
0*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2* 
x + 3)) + 10490640*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt( 
5)/(2*x + 3))^6*sgn(1/(2*x + 3)) + 19083456*(sqrt(-8/(2*x + 3) + 5/(2*x + 
3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 33372000*sqrt(5)*(sqrt 
(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) 
 - 42760170*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3 
*sgn(1/(2*x + 3)) + 60102000*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 
3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 21448395*(sqrt(-8/(2*x + 3) + 
 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3)) - 36498600*sqrt( 
5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2 
*x + 3))^2 - 3)^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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