\(\int \frac {(5-x) (3+2 x)^{9/2}}{(2+5 x+3 x^2)^3} \, dx\) [843]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=-\frac {32}{27} \sqrt {3+2 x}-\frac {\sqrt {3+2 x} (11597+12083 x)}{162 \left (2+5 x+3 x^2\right )^2}+\frac {\sqrt {3+2 x} (111154+137169 x)}{162 \left (2+5 x+3 x^2\right )}+1962 \text {arctanh}\left (\sqrt {3+2 x}\right )-\frac {13675}{9} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \] Output:

-32/27*(3+2*x)^(1/2)-1/162*(3+2*x)^(1/2)*(11597+12083*x)/(3*x^2+5*x+2)^2+( 
3+2*x)^(1/2)*(111154+137169*x)/(486*x^2+810*x+324)+1962*arctanh((3+2*x)^(1 
/2))-13675/27*15^(1/2)*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=1962 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {1}{54} \left (-\frac {3 \sqrt {3+2 x} \left (-23327-90465 x-112467 x^2-45083 x^3+192 x^4\right )}{\left (2+5 x+3 x^2\right )^2}-27350 \sqrt {15} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \] Input:

Integrate[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]
 

Output:

1962*ArcTanh[Sqrt[3 + 2*x]] + ((-3*Sqrt[3 + 2*x]*(-23327 - 90465*x - 11246 
7*x^2 - 45083*x^3 + 192*x^4))/(2 + 5*x + 3*x^2)^2 - 27350*Sqrt[15]*ArcTanh 
[Sqrt[3/5]*Sqrt[3 + 2*x]])/54
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1233, 25, 1233, 27, 1196, 1197, 1480, 220}

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.

\(\displaystyle \int \frac {(5-x) (2 x+3)^{9/2}}{\left (3 x^2+5 x+2\right )^3} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {1}{6} \int -\frac {(416-131 x) (2 x+3)^{5/2}}{\left (3 x^2+5 x+2\right )^2}dx-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{6} \int \frac {(416-131 x) (2 x+3)^{5/2}}{\left (3 x^2+5 x+2\right )^2}dx-\frac {(139 x+121) (2 x+3)^{7/2}}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {1}{6} \left (\frac {(2 x+3)^{3/2} (12473 x+10832)}{3 \left (3 x^2+5 x+2\right )}-\frac {1}{3} \int -\frac {3 (1903-3983 x) \sqrt {2 x+3}}{3 x^2+5 x+2}dx\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\int \frac {(1903-3983 x) \sqrt {2 x+3}}{3 x^2+5 x+2}dx+\frac {(12473 x+10832) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {1}{6} \left (\frac {1}{3} \int \frac {15401 x+33059}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )}dx+\frac {(12473 x+10832) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}-\frac {7966}{3} \sqrt {2 x+3}\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {1}{6} \left (\frac {2}{3} \int \frac {15401 (2 x+3)+19915}{3 (2 x+3)^2-8 (2 x+3)+5}d\sqrt {2 x+3}+\frac {(12473 x+10832) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}-\frac {7966}{3} \sqrt {2 x+3}\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{6} \left (\frac {2}{3} \left (68375 \int \frac {1}{3 (2 x+3)-5}d\sqrt {2 x+3}-52974 \int \frac {1}{3 (2 x+3)-3}d\sqrt {2 x+3}\right )+\frac {(12473 x+10832) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}-\frac {7966}{3} \sqrt {2 x+3}\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{6} \left (\frac {2}{3} \left (17658 \text {arctanh}\left (\sqrt {2 x+3}\right )-13675 \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )+\frac {(12473 x+10832) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}-\frac {7966}{3} \sqrt {2 x+3}\right )-\frac {(2 x+3)^{7/2} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}\)

Input:

Int[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^3,x]
 

Output:

-1/6*((3 + 2*x)^(7/2)*(121 + 139*x))/(2 + 5*x + 3*x^2)^2 + ((-7966*Sqrt[3 
+ 2*x])/3 + ((3 + 2*x)^(3/2)*(10832 + 12473*x))/(3*(2 + 5*x + 3*x^2)) + (2 
*(17658*ArcTanh[Sqrt[3 + 2*x]] - 13675*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 
+ 2*x]]))/3)/6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 1196

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + 
(c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c   Int 
[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + 
 b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & 
& GtQ[m, 0]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (192 x^{4}-45083 x^{3}-112467 x^{2}-90465 x -23327\right ) \sqrt {2 x +3}}{18 \left (3 x^{2}+5 x +2\right )^{2}}+981 \ln \left (\sqrt {2 x +3}+1\right )-981 \ln \left (\sqrt {2 x +3}-1\right )-\frac {13675 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}\)	\(85\)
trager	\(-\frac {\left (192 x^{4}-45083 x^{3}-112467 x^{2}-90465 x -23327\right ) \sqrt {2 x +3}}{18 \left (3 x^{2}+5 x +2\right )^{2}}-\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4488135\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4488135\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-4488135\right )+8205 \sqrt {2 x +3}}{3 x +2}\right )}{54}+981 \ln \left (\frac {\sqrt {2 x +3}+2+x}{x +1}\right )\)	\(106\)
pseudoelliptic	\(\frac {-\frac {54700 \sqrt {15}\, \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}-35316 \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \ln \left (\sqrt {2 x +3}-1\right )+35316 \left (x +1\right )^{2} \left (x +\frac {2}{3}\right )^{2} \ln \left (\sqrt {2 x +3}+1\right )-\frac {128 \left (x^{4}-\frac {45083}{192} x^{3}-\frac {37489}{64} x^{2}-\frac {30155}{64} x -\frac {23327}{192}\right ) \sqrt {2 x +3}}{3}}{\left (\sqrt {2 x +3}+1\right )^{2} \left (\sqrt {2 x +3}-1\right )^{2} \left (3 x +2\right )^{2}}\)	\(132\)
derivativedivides	\(-\frac {32 \sqrt {2 x +3}}{27}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {104}{\sqrt {2 x +3}+1}+981 \ln \left (\sqrt {2 x +3}+1\right )+\frac {3}{\left (\sqrt {2 x +3}-1\right )^{2}}+\frac {104}{\sqrt {2 x +3}-1}-981 \ln \left (\sqrt {2 x +3}-1\right )+\frac {\frac {9625 \left (2 x +3\right )^{\frac {3}{2}}}{3}-\frac {165625 \sqrt {2 x +3}}{27}}{\left (6 x +4\right )^{2}}-\frac {13675 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}\)	\(133\)
default	\(-\frac {32 \sqrt {2 x +3}}{27}-\frac {3}{\left (\sqrt {2 x +3}+1\right )^{2}}+\frac {104}{\sqrt {2 x +3}+1}+981 \ln \left (\sqrt {2 x +3}+1\right )+\frac {3}{\left (\sqrt {2 x +3}-1\right )^{2}}+\frac {104}{\sqrt {2 x +3}-1}-981 \ln \left (\sqrt {2 x +3}-1\right )+\frac {\frac {9625 \left (2 x +3\right )^{\frac {3}{2}}}{3}-\frac {165625 \sqrt {2 x +3}}{27}}{\left (6 x +4\right )^{2}}-\frac {13675 \sqrt {15}\, \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{27}\)	\(133\)

Input:

int((5-x)*(2*x+3)^(9/2)/(3*x^2+5*x+2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/18*(192*x^4-45083*x^3-112467*x^2-90465*x-23327)/(3*x^2+5*x+2)^2*(2*x+3) 
^(1/2)+981*ln((2*x+3)^(1/2)+1)-981*ln((2*x+3)^(1/2)-1)-13675/27*15^(1/2)*a 
rctanh(1/5*15^(1/2)*(2*x+3)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.48 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {13675 \, \sqrt {\frac {5}{3}} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac {3 \, \sqrt {\frac {5}{3}} \sqrt {2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 17658 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) - 17658 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - {\left (192 \, x^{4} - 45083 \, x^{3} - 112467 \, x^{2} - 90465 \, x - 23327\right )} \sqrt {2 \, x + 3}}{18 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="fricas")
 

Output:

1/18*(13675*sqrt(5/3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(-(3*sqrt(5/ 
3)*sqrt(2*x + 3) - 3*x - 7)/(3*x + 2)) + 17658*(9*x^4 + 30*x^3 + 37*x^2 + 
20*x + 4)*log(sqrt(2*x + 3) + 1) - 17658*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 
 4)*log(sqrt(2*x + 3) - 1) - (192*x^4 - 45083*x^3 - 112467*x^2 - 90465*x - 
 23327)*sqrt(2*x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
 

Sympy [A] (verification not implemented)

Time = 106.57 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.65 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=- \frac {32 \sqrt {2 x + 3}}{27} + \frac {5875 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{27} - \frac {25000 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{3} + \frac {425000 \left (\begin {cases} \frac {\sqrt {15} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )^{2}}\right )}{375} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{27} - 981 \log {\left (\sqrt {2 x + 3} - 1 \right )} + 981 \log {\left (\sqrt {2 x + 3} + 1 \right )} + \frac {104}{\sqrt {2 x + 3} + 1} - \frac {3}{\left (\sqrt {2 x + 3} + 1\right )^{2}} + \frac {104}{\sqrt {2 x + 3} - 1} + \frac {3}{\left (\sqrt {2 x + 3} - 1\right )^{2}} \] Input:

integrate((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**3,x)
 

Output:

-32*sqrt(2*x + 3)/27 + 5875*sqrt(15)*(log(sqrt(2*x + 3) - sqrt(15)/3) - lo 
g(sqrt(2*x + 3) + sqrt(15)/3))/27 - 25000*Piecewise((sqrt(15)*(-log(sqrt(1 
5)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqrt(2*x + 3)/5 + 1)/4 - 1/(4*(sq 
rt(15)*sqrt(2*x + 3)/5 + 1)) - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 - 1)))/75, ( 
sqrt(2*x + 3) > -sqrt(15)/3) & (sqrt(2*x + 3) < sqrt(15)/3)))/3 + 425000*P 
iecewise((sqrt(15)*(3*log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/16 - 3*log(sqrt(15 
)*sqrt(2*x + 3)/5 + 1)/16 + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) + 1/(16* 
(sqrt(15)*sqrt(2*x + 3)/5 + 1)**2) + 3/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1)) 
 - 1/(16*(sqrt(15)*sqrt(2*x + 3)/5 - 1)**2))/375, (sqrt(2*x + 3) > -sqrt(1 
5)/3) & (sqrt(2*x + 3) < sqrt(15)/3)))/27 - 981*log(sqrt(2*x + 3) - 1) + 9 
81*log(sqrt(2*x + 3) + 1) + 104/(sqrt(2*x + 3) + 1) - 3/(sqrt(2*x + 3) + 1 
)**2 + 104/(sqrt(2*x + 3) - 1) + 3/(sqrt(2*x + 3) - 1)**2
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {13675}{54} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (9 \, {\left (2 \, x + 3\right )}^{4} - 48 \, {\left (2 \, x + 3\right )}^{3} + 94 \, {\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \] Input:

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="maxima")
 

Output:

13675/54*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x 
 + 3))) - 32/27*sqrt(2*x + 3) + 1/27*(137169*(2*x + 3)^(7/2) - 554983*(2*x 
 + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(9*(2*x + 3)^ 
4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160*x - 215) + 981*log(sqrt(2*x + 3) 
 + 1) - 981*log(sqrt(2*x + 3) - 1)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {13675}{54} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {32}{27} \, \sqrt {2 \, x + 3} + \frac {137169 \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} - 554983 \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + 717035 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 297925 \, \sqrt {2 \, x + 3}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 981 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) - 981 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \] Input:

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x, algorithm="giac")
 

Output:

13675/54*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3 
*sqrt(2*x + 3))) - 32/27*sqrt(2*x + 3) + 1/27*(137169*(2*x + 3)^(7/2) - 55 
4983*(2*x + 3)^(5/2) + 717035*(2*x + 3)^(3/2) - 297925*sqrt(2*x + 3))/(3*( 
2*x + 3)^2 - 16*x - 19)^2 + 981*log(sqrt(2*x + 3) + 1) - 981*log(abs(sqrt( 
2*x + 3) - 1))
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {\frac {297925\,\sqrt {2\,x+3}}{243}-\frac {717035\,{\left (2\,x+3\right )}^{3/2}}{243}+\frac {554983\,{\left (2\,x+3\right )}^{5/2}}{243}-\frac {15241\,{\left (2\,x+3\right )}^{7/2}}{27}}{\frac {160\,x}{9}-\frac {94\,{\left (2\,x+3\right )}^2}{9}+\frac {16\,{\left (2\,x+3\right )}^3}{3}-{\left (2\,x+3\right )}^4+\frac {215}{9}}-\frac {32\,\sqrt {2\,x+3}}{27}-\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,1962{}\mathrm {i}+\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,13675{}\mathrm {i}}{27} \] Input:

int(-((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^3,x)
 

Output:

((297925*(2*x + 3)^(1/2))/243 - (717035*(2*x + 3)^(3/2))/243 + (554983*(2* 
x + 3)^(5/2))/243 - (15241*(2*x + 3)^(7/2))/27)/((160*x)/9 - (94*(2*x + 3) 
^2)/9 + (16*(2*x + 3)^3)/3 - (2*x + 3)^4 + 215/9) - atan((2*x + 3)^(1/2)*1 
i)*1962i + (15^(1/2)*atan((15^(1/2)*(2*x + 3)^(1/2)*1i)/5)*13675i)/27 - (3 
2*(2*x + 3)^(1/2))/27
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.43 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^3} \, dx=\frac {505975 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{2}+273500 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x -505975 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{2}-273500 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x -211896 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right )+211896 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right )-123075 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{4}-410250 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right ) x^{3}-1960038 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{2}-1059480 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x +1960038 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{2}+1059480 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x +135249 \sqrt {2 x +3}\, x^{3}+337401 \sqrt {2 x +3}\, x^{2}+271395 \sqrt {2 x +3}\, x +54700 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right )-54700 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}+\sqrt {15}\right )+123075 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{4}+410250 \sqrt {15}\, \mathrm {log}\left (3 \sqrt {2 x +3}-\sqrt {15}\right ) x^{3}-576 \sqrt {2 x +3}\, x^{4}-476766 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{4}-1589220 \,\mathrm {log}\left (\sqrt {2 x +3}-1\right ) x^{3}+476766 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{4}+1589220 \,\mathrm {log}\left (\sqrt {2 x +3}+1\right ) x^{3}+69981 \sqrt {2 x +3}}{486 x^{4}+1620 x^{3}+1998 x^{2}+1080 x +216} \] Input:

int((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^3,x)
 

Output:

( - 576*sqrt(2*x + 3)*x**4 + 135249*sqrt(2*x + 3)*x**3 + 337401*sqrt(2*x + 
 3)*x**2 + 271395*sqrt(2*x + 3)*x + 69981*sqrt(2*x + 3) + 123075*sqrt(15)* 
log(3*sqrt(2*x + 3) - sqrt(15))*x**4 + 410250*sqrt(15)*log(3*sqrt(2*x + 3) 
 - sqrt(15))*x**3 + 505975*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x**2 + 
 273500*sqrt(15)*log(3*sqrt(2*x + 3) - sqrt(15))*x + 54700*sqrt(15)*log(3* 
sqrt(2*x + 3) - sqrt(15)) - 123075*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15) 
)*x**4 - 410250*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x**3 - 505975*sqr 
t(15)*log(3*sqrt(2*x + 3) + sqrt(15))*x**2 - 273500*sqrt(15)*log(3*sqrt(2* 
x + 3) + sqrt(15))*x - 54700*sqrt(15)*log(3*sqrt(2*x + 3) + sqrt(15)) - 47 
6766*log(sqrt(2*x + 3) - 1)*x**4 - 1589220*log(sqrt(2*x + 3) - 1)*x**3 - 1 
960038*log(sqrt(2*x + 3) - 1)*x**2 - 1059480*log(sqrt(2*x + 3) - 1)*x - 21 
1896*log(sqrt(2*x + 3) - 1) + 476766*log(sqrt(2*x + 3) + 1)*x**4 + 1589220 
*log(sqrt(2*x + 3) + 1)*x**3 + 1960038*log(sqrt(2*x + 3) + 1)*x**2 + 10594 
80*log(sqrt(2*x + 3) + 1)*x + 211896*log(sqrt(2*x + 3) + 1))/(54*(9*x**4 + 
 30*x**3 + 37*x**2 + 20*x + 4))