\(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^5} \, dx\) [964]

Optimal result

Integrand size = 27, antiderivative size = 173 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=\frac {7 (5009-10461 x) \sqrt {2+5 x+3 x^2}}{1024}+\frac {7 (481-10404 x) \left (2+5 x+3 x^2\right )^{3/2}}{2560}+\frac {21 (1523+1080 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^2}+\frac {(319+256 x) \left (2+5 x+3 x^2\right )^{7/2}}{80 (3+2 x)^4}-\frac {744275 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{2048 \sqrt {3}}+\frac {192171 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{2048} \] Output:

7/1024*(5009-10461*x)*(3*x^2+5*x+2)^(1/2)+7/2560*(481-10404*x)*(3*x^2+5*x+ 
2)^(3/2)+21/640*(1523+1080*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^2+1/80*(319+256* 
x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^4-744275/6144*arctanh(3^(1/2)*(1+x)/(3*x^2+ 
5*x+2)^(1/2))*3^(1/2)+192171/2048*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2+5*x 
+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.71 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=\frac {-\frac {6 \sqrt {2+5 x+3 x^2} \left (-4933171-11295211 x-9107922 x^2-2869312 x^3-253688 x^4-38288 x^5-12864 x^6+3456 x^7\right )}{(3+2 x)^4}+576513 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-744275 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{6144} \] Input:

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^5,x]
 

Output:

((-6*Sqrt[2 + 5*x + 3*x^2]*(-4933171 - 11295211*x - 9107922*x^2 - 2869312* 
x^3 - 253688*x^4 - 38288*x^5 - 12864*x^6 + 3456*x^7))/(3 + 2*x)^4 + 576513 
*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 744275*Sqrt[3]*ArcTa 
nh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/6144
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.24, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1230, 27, 1230, 27, 1230, 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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^5} \, dx\)

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {7}{128} \int -\frac {8 (43 x+36) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^4}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{16} \int \frac {(43 x+36) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^4}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{72} \int \frac {6 (343 x+293) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \int \frac {(343 x+293) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{32} \int \frac {4 (2701 x+2308) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \int \frac {(2701 x+2308) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {(2701 x+5795) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (21265 x+18171)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {(2701 x+5795) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {21265 x+18171}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {27453}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {21265}{2} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (2701 x+5795)}{2 (2 x+3)}\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {27453}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-21265 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (2701 x+5795)}{2 (2 x+3)}\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {27453}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {21265 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (2701 x+5795)}{2 (2 x+3)}\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (-27453 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {21265 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (2701 x+5795)}{2 (2 x+3)}\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{16} \left (\frac {(43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}-\frac {5}{12} \left (\frac {(343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {27453 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {5}}-\frac {21265 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (2701 x+5795)}{2 (2 x+3)}\right )\right )\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}\)

Input:

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^5,x]
 

Output:

-1/8*((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4 + (7*(((93 + 43*x)*(2 + 
 5*x + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) - (5*(((736 + 343*x)*(2 + 5*x + 3*x^2 
)^(3/2))/(4*(3 + 2*x)^2) - (3*(((5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(2* 
(3 + 2*x)) + ((-21265*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])] 
)/(2*Sqrt[3]) + (27453*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]) 
])/(2*Sqrt[5]))/4))/8))/12))/16
 

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 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 = 1.82 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {10368 x^{9}-21312 x^{8}-172272 x^{7}-978232 x^{6}-9952952 x^{5}-42177702 x^{4}-85163867 x^{3}-89491412 x^{2}-47256277 x -9866342}{1024 \left (2 x +3\right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {744275 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{12288}-\frac {192171 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{4096}\)	\(127\)
trager	\(-\frac {\left (3456 x^{7}-12864 x^{6}-38288 x^{5}-253688 x^{4}-2869312 x^{3}-9107922 x^{2}-11295211 x -4933171\right ) \sqrt {3 x^{2}+5 x +2}}{1024 \left (2 x +3\right )^{4}}-\frac {744275 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{12288}+\frac {192171 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{2 x +3}\right )}{4096}\)	\(148\)
default	\(\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{100 \left (x +\frac {3}{2}\right )^{3}}+\frac {27453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{4000}+\frac {64057 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{2560}+\frac {192171 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{16000}+\frac {192171 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{4096}-\frac {10101 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4000}+\frac {1479 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{500 \left (x +\frac {3}{2}\right )}-\frac {192171 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{4096}-\frac {6069 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1280}-\frac {1263 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{4000 \left (x +\frac {3}{2}\right )^{2}}-\frac {744275 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{12288}-\frac {1479 \left (6 x +5\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1000}-\frac {24409 \left (6 x +5\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{2048}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}}\)	\(295\)

Input:

int((5-x)*(3*x^2+5*x+2)^(7/2)/(2*x+3)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/1024*(10368*x^9-21312*x^8-172272*x^7-978232*x^6-9952952*x^5-42177702*x^ 
4-85163867*x^3-89491412*x^2-47256277*x-9866342)/(2*x+3)^4/(3*x^2+5*x+2)^(1 
/2)-744275/12288*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-192 
171/4096*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 = 203, normalized size of antiderivative = 1.17 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=\frac {744275 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 576513 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 24 \, {\left (3456 \, x^{7} - 12864 \, x^{6} - 38288 \, x^{5} - 253688 \, x^{4} - 2869312 \, x^{3} - 9107922 \, x^{2} - 11295211 \, x - 4933171\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{24576 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^5,x, algorithm="fricas")
 

Output:

1/24576*(744275*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-4*sq 
rt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 576513*sqrt 
(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((4*sqrt(5)*sqrt(3*x^2 + 5 
*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 24*(3456*x 
^7 - 12864*x^6 - 38288*x^5 - 253688*x^4 - 2869312*x^3 - 9107922*x^2 - 1129 
5211*x - 4933171)*sqrt(3*x^2 + 5*x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216* 
x + 81)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \] Input:

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**5,x)
 

Output:

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080 
*x**2 + 810*x + 243), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 
 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-870*x**2 
*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x 
 + 243), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x* 
*4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-1090*x**4*sqrt(3* 
x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), 
 x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720* 
x**3 + 1080*x**2 + 810*x + 243), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 
 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (141) = 282\).

Time = 0.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.65 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=\frac {3789}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac {6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1263 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {30303}{2000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {9849}{16000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{200 \, {\left (2 \, x + 3\right )}} - \frac {18207}{640} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {3367}{2560} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {73227}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {744275}{12288} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {192171}{4096} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {35063}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^5,x, algorithm="maxima")
 

Output:

3789/4000*(3*x^2 + 5*x + 2)^(7/2) - 13/20*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 
+ 96*x^3 + 216*x^2 + 216*x + 81) + 6/25*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 3 
6*x^2 + 54*x + 27) - 1263/1000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) 
- 30303/2000*(3*x^2 + 5*x + 2)^(5/2)*x - 9849/16000*(3*x^2 + 5*x + 2)^(5/2 
) + 1479/200*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 18207/640*(3*x^2 + 5*x + 
2)^(3/2)*x + 3367/2560*(3*x^2 + 5*x + 2)^(3/2) - 73227/1024*sqrt(3*x^2 + 5 
*x + 2)*x - 744275/12288*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 
 5/2) - 192171/4096*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) 
 + 5/2/abs(2*x + 3) - 2) + 35063/1024*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. 636 vs. \(2 (141) = 282\).

Time = 0.74 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx =\text {Too large to display} \] Input:

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^5,x, algorithm="giac")
 

Output:

744275/12288*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)) - 192171/4096*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)) - 1/4096*(5*(50*(13*sgn(1/(2*x + 3))/(2*x + 
3) - 88*sgn(1/(2*x + 3)))/(2*x + 3) + 14343*sgn(1/(2*x + 3)))/(2*x + 3) - 
181996*sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 1/2048*( 
479709*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn( 
1/(2*x + 3)) - 499296*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sq 
rt(5)/(2*x + 3))^6*sgn(1/(2*x + 3)) - 3133183*(sqrt(-8/(2*x + 3) + 5/(2*x 
+ 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) + 3365712*sqrt(5)*(sqr 
t(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3) 
) + 7550211*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3 
*sgn(1/(2*x + 3)) - 8139744*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3 
) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 6574257*(sqrt(-8/(2*x + 3) + 5 
/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3)) + 6966000*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)^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^5} \,d x \] Input:

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^5,x)
 

Output:

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^5, x)
 

Reduce [B] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 612, normalized size of antiderivative = 3.54 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx =\text {Too large to display} \] Input:

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^5,x)
 

Output:

( - 5142528*sqrt(3*x**2 + 5*x + 2)*x**7 + 19141632*sqrt(3*x**2 + 5*x + 2)* 
x**6 + 56972544*sqrt(3*x**2 + 5*x + 2)*x**5 + 377487744*sqrt(3*x**2 + 5*x 
+ 2)*x**4 + 4269536256*sqrt(3*x**2 + 5*x + 2)*x**3 + 13552587936*sqrt(3*x* 
*2 + 5*x + 2)*x**2 + 16807273968*sqrt(3*x**2 + 5*x + 2)*x + 7340558448*sqr 
t(3*x**2 + 5*x + 2) + 1143801792*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt 
(3) - sqrt(15) + 6*x + 9)*x**4 + 6862810752*sqrt(5)*log(2*sqrt(3*x**2 + 5* 
x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x**3 + 15441324192*sqrt(5)*log(2*sqrt 
(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x**2 + 15441324192*sqrt(5 
)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x + 579049657 
2*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9) - 114 
3801792*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9) 
*x**4 - 6862810752*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) 
 + 6*x + 9)*x**3 - 15441324192*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3 
) + sqrt(15) + 6*x + 9)*x**2 - 15441324192*sqrt(5)*log(2*sqrt(3*x**2 + 5*x 
 + 2)*sqrt(3) + sqrt(15) + 6*x + 9)*x - 5790496572*sqrt(5)*log(2*sqrt(3*x* 
*2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9) - 1476641600*sqrt(3)*log(2*sqr 
t(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**4 - 8859849600*sqrt(3)*log(2*sqr 
t(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**3 - 19934661600*sqrt(3)*log(2*sq 
rt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**2 - 19934661600*sqrt(3)*log(2*s 
qrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x - 7475498100*sqrt(3)*log(2*s...