\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^5} \, dx\) [229]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {9 (20931-6499 x) \sqrt {2+3 x^2}}{15680}+\frac {(36126+32449 x) \left (2+3 x^2\right )^{3/2}}{7840 (3+2 x)^2}+\frac {(81+145 x) \left (2+3 x^2\right )^{5/2}}{168 (3+2 x)^4}-\frac {2625}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {188379 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{896 \sqrt {35}} \] Output:

9/15680*(20931-6499*x)*(3*x^2+2)^(1/2)+1/7840*(36126+32449*x)*(3*x^2+2)^(3 
/2)/(3+2*x)^2+1/168*(81+145*x)*(3*x^2+2)^(5/2)/(3+2*x)^4-2625/128*arcsinh( 
1/2*x*6^(1/2))*3^(1/2)-188379/31360*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2) 
/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {-\frac {70 \sqrt {2+3 x^2} \left (-1421955-3335009 x-2762820 x^2-898734 x^3-57456 x^4+3024 x^5\right )}{(3+2 x)^4}+1130274 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+1929375 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{94080} \] Input:

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

Output:

((-70*Sqrt[2 + 3*x^2]*(-1421955 - 3335009*x - 2762820*x^2 - 898734*x^3 - 5 
7456*x^4 + 3024*x^5))/(3 + 2*x)^4 + 1130274*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 
2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 1929375*Sqrt[3]*Log[-(Sqrt[3] 
*x) + Sqrt[2 + 3*x^2]])/94080
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {681, 27, 680, 27, 681, 27, 719, 222, 488, 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.

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

\(\Big \downarrow \) 681

\(\displaystyle -\frac {5}{64} \int \frac {4 (8-57 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \int \frac {(8-57 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 680

\(\displaystyle -\frac {5}{16} \left (\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}-\frac {1}{560} \int -\frac {24 (374-1917 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \int \frac {(374-1917 x) \sqrt {3 x^2+2}}{(2 x+3)^2}dx+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 681

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {1}{8} \int \frac {12 (1278-6125 x)}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {3}{2} \int \frac {1278-6125 x}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 719

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {3}{2} \left (\frac {20931}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {6125}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 222

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {3}{2} \left (\frac {20931}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {6125 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {3}{2} \left (-\frac {20931}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {6125 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {5}{16} \left (\frac {3}{70} \left (-\frac {3}{2} \left (-\frac {6125 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {20931 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )-\frac {\sqrt {3 x^2+2} (1917 x+6125)}{2 (2 x+3)}\right )+\frac {(5517 x+5003) \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+2\right )^{5/2}}{16 (2 x+3)^4}\)

Input:

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

Output:

-1/16*((19 + 4*x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^4 - (5*(((5003 + 5517*x)*(2 
 + 3*x^2)^(3/2))/(210*(3 + 2*x)^3) + (3*(-1/2*((6125 + 1917*x)*Sqrt[2 + 3* 
x^2])/(3 + 2*x) - (3*((-6125*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3]) - (20931*Ar 
cTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])))/2))/70))/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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 488

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

rule 680

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

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (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 + 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 + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, 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 719

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {9072 x^{7}-172368 x^{6}-2690154 x^{5}-8403372 x^{4}-11802495 x^{3}-9791505 x^{2}-6670018 x -2843910}{1344 \left (2 x +3\right )^{4} \sqrt {3 x^{2}+2}}-\frac {2625 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}-\frac {188379 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{31360}\)	\(97\)
trager	\(-\frac {\left (3024 x^{5}-57456 x^{4}-898734 x^{3}-2762820 x^{2}-3335009 x -1421955\right ) \sqrt {3 x^{2}+2}}{1344 \left (2 x +3\right )^{4}}+\frac {188379 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{31360}+\frac {2625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}\)	\(121\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}}+\frac {23 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{117600 \left (x +\frac {3}{2}\right )^{3}}-\frac {1041 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{343000 \left (x +\frac {3}{2}\right )^{2}}+\frac {29717 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{6002500 \left (x +\frac {3}{2}\right )}+\frac {188379 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{6002500}-\frac {58629 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{274400}-\frac {58491 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{15680}-\frac {2625 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}+\frac {62793 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{137200}+\frac {188379 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{31360}-\frac {188379 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{31360}-\frac {89151 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{6002500}\)	\(227\)

Input:

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

Output:

-1/1344*(9072*x^7-172368*x^6-2690154*x^5-8403372*x^4-11802495*x^3-9791505* 
x^2-6670018*x-2843910)/(2*x+3)^4/(3*x^2+2)^(1/2)-2625/128*arcsinh(1/2*6^(1 
/2)*x)*3^(1/2)-188379/31360*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+ 
3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.40 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {1929375 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 565137 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (3024 \, x^{5} - 57456 \, x^{4} - 898734 \, x^{3} - 2762820 \, x^{2} - 3335009 \, x - 1421955\right )} \sqrt {3 \, x^{2} + 2}}{188160 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \] Input:

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

Output:

1/188160*(1929375*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqr 
t(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 565137*sqrt(35)*(16*x^4 + 96*x^3 + 2 
16*x^2 + 216*x + 81)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 3 
6*x + 43)/(4*x^2 + 12*x + 9)) - 140*(3024*x^5 - 57456*x^4 - 898734*x^3 - 2 
762820*x^2 - 3335009*x - 1421955)*sqrt(3*x^2 + 2))/(16*x^4 + 96*x^3 + 216* 
x^2 + 216*x + 81)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (100) = 200\).

Time = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.63 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {3123}{343000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac {23 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{14700 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1041 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{85750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {58629}{274400} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {62793}{137200} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {29717 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{343000 \, {\left (2 \, x + 3\right )}} - \frac {58491}{15680} \, \sqrt {3 \, x^{2} + 2} x - \frac {2625}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {188379}{31360} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {188379}{15680} \, \sqrt {3 \, x^{2} + 2} \] Input:

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

Output:

3123/343000*(3*x^2 + 2)^(5/2) - 13/140*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 
+ 216*x^2 + 216*x + 81) + 23/14700*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54* 
x + 27) - 1041/85750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 58629/274400*( 
3*x^2 + 2)^(3/2)*x + 62793/137200*(3*x^2 + 2)^(3/2) + 29717/343000*(3*x^2 
+ 2)^(5/2)/(2*x + 3) - 58491/15680*sqrt(3*x^2 + 2)*x - 2625/128*sqrt(3)*ar 
csinh(1/2*sqrt(6)*x) + 188379/31360*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x 
 + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 188379/15680*sqrt(3*x^2 + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (100) = 200\).

Time = 0.31 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.49 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=-\frac {188379}{31360} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {2625}{128} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{10752} \, {\left (\frac {7 \, {\left (\frac {35 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 2129 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 57681 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 242979 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {9 \, {\left (256 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 93 \, \sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 582 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 225 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{64 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \] Input:

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

Output:

-188379/31360*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 
 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x + 3)) + 2625/128*sqrt(3)*log(1/2 
*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/ 
(2*x + 3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/ 
(2*x + 3)))*sgn(1/(2*x + 3)) - 1/10752*(7*(35*(1365*sgn(1/(2*x + 3))/(2*x 
+ 3) - 2129*sgn(1/(2*x + 3)))/(2*x + 3) + 57681*sgn(1/(2*x + 3)))/(2*x + 3 
) - 242979*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) - 9/ 
64*(256*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^3* 
sgn(1/(2*x + 3)) - 93*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 
 sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 582*(sqrt(-18/(2*x + 3) + 35/(2* 
x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sgn(1/(2*x + 3)) + 225*sqrt(35)*sgn(1/ 
(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3 
))^2 - 3)^2
 

Mupad [B] (verification not implemented)

Time = 5.95 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.43 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {188379\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{31360}+\frac {225\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64}-\frac {2625\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {188379\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{31360}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {80993\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3584\,\left (x+\frac {3}{2}\right )}-\frac {19227\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1024\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {9\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{64}+\frac {74515\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6144\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

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

Output:

(188379*35^(1/2)*log(x + 3/2))/31360 + (225*3^(1/2)*(x^2 + 2/3)^(1/2))/64 
- (2625*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/128 - (188379*35^(1/2)*log(x 
 - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/31360 - (15925*3^(1/2)*( 
x^2 + 2/3)^(1/2))/(4096*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + ( 
80993*3^(1/2)*(x^2 + 2/3)^(1/2))/(3584*(x + 3/2)) - (19227*3^(1/2)*(x^2 + 
2/3)^(1/2))/(1024*(3*x + x^2 + 9/4)) - (9*3^(1/2)*x*(x^2 + 2/3)^(1/2))/64 
+ (74515*3^(1/2)*(x^2 + 2/3)^(1/2))/(6144*((27*x)/4 + (9*x^2)/2 + x^3 + 27 
/8))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.79 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {199073700 \sqrt {3 x^{2}+2}[phone] \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{4}-30870000 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{4}-423360 \sqrt {3 x^{2}+2}\, x^{5}+8043840 \sqrt {3 x^{2}+2}\, x^{4}[phone] \sqrt {3 x^{2}+2}\, x^{3}[phone] \sqrt {3 x^{2}+2}\, x^{2}[phone] \sqrt {3 x^{2}+2}\, x [phone] \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{3}-185220000 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{3}-244139184 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x [phone] \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x -416745000 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x [phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}-18084384 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}-108506304 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-91552194 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )[phone] \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )-156279375 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x -244139184 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}[phone] \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x^{2}-416745000 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x^{2}}{3010560 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 423360*sqrt(3*x**2 + 2)*x**5 + 8043840*sqrt(3*x**2 + 2)*x**4 + 1258227 
60*sqrt(3*x**2 + 2)*x**3 + 386794800*sqrt(3*x**2 + 2)*x**2 + 466901260*sqr 
t(3*x**2 + 2)*x + 199073700*sqrt(3*x**2 + 2) + 18084384*sqrt(35)*log(sqrt( 
3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 108506304*sqrt(35)*log(sqrt(3*x**2 
+ 2)*sqrt(35) + 9*x - 4)*x**3 + 244139184*sqrt(35)*log(sqrt(3*x**2 + 2)*sq 
rt(35) + 9*x - 4)*x**2 + 244139184*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) 
+ 9*x - 4)*x + 91552194*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) 
- 18084384*sqrt(35)*log(2*x + 3)*x**4 - 108506304*sqrt(35)*log(2*x + 3)*x* 
*3 - 244139184*sqrt(35)*log(2*x + 3)*x**2 - 244139184*sqrt(35)*log(2*x + 3 
)*x - 91552194*sqrt(35)*log(2*x + 3) + 30870000*sqrt(3)*log(sqrt(3*x**2 + 
2) - sqrt(3)*x)*x**4 + 185220000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x) 
*x**3 + 416745000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 + 4167450 
00*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x + 156279375*sqrt(3)*log(sqr 
t(3*x**2 + 2) - sqrt(3)*x) - 30870000*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt( 
3)*x)*x**4 - 185220000*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**3 - 41 
6745000*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**2 - 416745000*sqrt(3) 
*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x - 156279375*sqrt(3)*log(sqrt(3*x**2 + 
 2) + sqrt(3)*x))/(188160*(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81))