\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^9} \, dx\) [233]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {48141 \sqrt {2+3 x^2}}{153664000 (3+2 x)^2}+\frac {24813 \sqrt {2+3 x^2}}{5378240000 (3+2 x)}-\frac {9 (11323+4037 x) \sqrt {2+3 x^2}}{10976000 (3+2 x)^4}-\frac {(13106-16521 x) \left (2+3 x^2\right )^{3/2}}{548800 (3+2 x)^6}-\frac {(59-173 x) \left (2+3 x^2\right )^{5/2}}{784 (3+2 x)^8}-\frac {18873 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42017500 \sqrt {35}} \] Output:

48141/153664000*(3*x^2+2)^(1/2)/(3+2*x)^2+24813*(3*x^2+2)^(1/2)/(161347200 
00+10756480000*x)-9/10976000*(11323+4037*x)*(3*x^2+2)^(1/2)/(3+2*x)^4-1/54 
8800*(13106-16521*x)*(3*x^2+2)^(3/2)/(3+2*x)^6-1/784*(59-173*x)*(3*x^2+2)^ 
(5/2)/(3+2*x)^8-18873/1470612500*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3 
*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 5.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {\frac {35 \sqrt {2+3 x^2} \left (-104577556-38788883 x-178164896 x^2+226355535 x^3+33613440 x^4+210306726 x^5+2206008 x^6+49626 x^7\right )}{(3+2 x)^8}+75492 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{2941225000} \] Input:

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

Output:

((35*Sqrt[2 + 3*x^2]*(-104577556 - 38788883*x - 178164896*x^2 + 226355535* 
x^3 + 33613440*x^4 + 210306726*x^5 + 2206008*x^6 + 49626*x^7))/(3 + 2*x)^8 
 + 75492*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sq 
rt[35]])/2941225000
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {688, 25, 679, 486, 486, 486, 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.

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

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{280} \int -\frac {(328-39 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{280} \int \frac {(328-39 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \int \frac {\left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \left (\frac {1}{7} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \left (\frac {1}{7} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{280} \left (\frac {2796}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {773 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\)

Input:

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

Output:

(-13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) + ((-773*(2 + 3*x^2)^(7/2))/(245 
*(3 + 2*x)^7) + (2796*(-1/210*((4 - 9*x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6 + 
(-1/140*((4 - 9*x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4 + (9*(-1/70*((4 - 9*x)*S 
qrt[2 + 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^ 
2])])/(35*Sqrt[35])))/70)/7))/35)/280
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 0]
 

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 679

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

rule 688

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {148878 x^{9}+6618024 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-467267808 x^{4}[phone] x^{3}-670062460 x^{2}-77577766 x -209155112}{84035000 \left (2 x +3\right )^{8} \sqrt {3 x^{2}+2}}-\frac {18873 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1470612500}\)	\(95\)
trager	\(\frac {\left (49626 x^{7}+2206008 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-178164896 x^{2}-38788883 x -104577556\right ) \sqrt {3 x^{2}+2}}{84035000 \left (2 x +3\right )^{8}}+\frac {18873 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{1470612500}\)	\(101\)
default	\(-\frac {2097 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{96040000 \left (x +\frac {3}{2}\right )^{5}}+\frac {18873 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{1470612500}+\frac {150984 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{2251875390625}+\frac {12582 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12867859375}+\frac {169857 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{2941225000}+\frac {78643791 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{9007501562500}-\frac {233 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{5488000 \left (x +\frac {3}{2}\right )^{6}}-\frac {18873 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1470612500}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{71680 \left (x +\frac {3}{2}\right )^{8}}-\frac {26214597 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9007501562500 \left (x +\frac {3}{2}\right )}-\frac {773 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{8780800 \left (x +\frac {3}{2}\right )^{7}}-\frac {20271 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1680700000 \left (x +\frac {3}{2}\right )^{4}}-\frac {207603 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{29412250000 \left (x +\frac {3}{2}\right )^{3}}-\frac {2258469 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{514714375000 \left (x +\frac {3}{2}\right )^{2}}+\frac {2208141 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{102942875000}\)	\(299\)

Input:

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

Output:

1/84035000*(148878*x^9+6618024*x^8+631019430*x^7+105252336*x^6+1099680057* 
x^5-467267808*x^4+336344421*x^3-670062460*x^2-77577766*x-209155112)/(2*x+3 
)^8/(3*x^2+2)^(1/2)-18873/1470612500*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 = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {18873 \, \sqrt {35} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (49626 \, x^{7} + 2206008 \, x^{6} + 210306726 \, x^{5} + 33613440 \, x^{4} + 226355535 \, x^{3} - 178164896 \, x^{2} - 38788883 \, x - 104577556\right )} \sqrt {3 \, x^{2} + 2}}{2941225000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \] Input:

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

Output:

1/2941225000*(18873*sqrt(35)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 
 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*log(-(sqrt(35)*sqrt( 
3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(49626 
*x^7 + 2206008*x^6 + 210306726*x^5 + 33613440*x^4 + 226355535*x^3 - 178164 
896*x^2 - 38788883*x - 104577556)*sqrt(3*x^2 + 2))/(256*x^8 + 3072*x^7 + 1 
6128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (131) = 262\).

Time = 0.17 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.38 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {6775407}{514714375000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{280 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {773 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{68600 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {233 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{85750 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {2097 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3001250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {20271 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{105043750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {207603 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3676531250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2258469 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{128678593750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {2208141}{102942875000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {12582}{12867859375} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {26214597 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{514714375000 \, {\left (2 \, x + 3\right )}} + \frac {169857}{2941225000} \, \sqrt {3 \, x^{2} + 2} x + \frac {18873}{1470612500} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {18873}{735306250} \, \sqrt {3 \, x^{2} + 2} \] Input:

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

Output:

6775407/514714375000*(3*x^2 + 2)^(5/2) - 13/280*(3*x^2 + 2)^(7/2)/(256*x^8 
 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 
 34992*x + 6561) - 773/68600*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048* 
x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 233/85750*(3*x 
^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x 
+ 729) - 2097/3001250*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080 
*x^2 + 810*x + 243) - 20271/105043750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 
 216*x^2 + 216*x + 81) - 207603/3676531250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x 
^2 + 54*x + 27) - 2258469/128678593750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9 
) + 2208141/102942875000*(3*x^2 + 2)^(3/2)*x + 12582/12867859375*(3*x^2 + 
2)^(3/2) - 26214597/514714375000*(3*x^2 + 2)^(5/2)/(2*x + 3) + 169857/2941 
225000*sqrt(3*x^2 + 2)*x + 18873/1470612500*sqrt(35)*arcsinh(3/2*sqrt(6)*x 
/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 18873/735306250*sqrt(3*x^2 + 2 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (131) = 262\).

Time = 0.16 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.89 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {18873}{1470612500} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, \sqrt {3} {\left (178944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} + 138131220 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} + 30787400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 573375810 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 3328877720 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 8681082564 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 13787031160 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 1566458475 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 28541438480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 30582301680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 23140527424 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 12885596640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 1726278400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 9101541120 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 39843840 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 1411584\right )}}{10756480000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{8}} \] Input:

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

Output:

18873/1470612500*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2 
*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2)) 
) - 9/10756480000*sqrt(3)*(178944*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 
 + 138131220*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 30787400*sqrt(3)*(sqrt(3)* 
x - sqrt(3*x^2 + 2))^13 + 573375810*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 332 
8877720*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 8681082564*(sqrt(3)*x - 
 sqrt(3*x^2 + 2))^10 - 13787031160*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 
 + 1566458475*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 28541438480*sqrt(3)*(sqrt( 
3)*x - sqrt(3*x^2 + 2))^7 + 30582301680*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 
23140527424*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 12885596640*(sqrt(3) 
*x - sqrt(3*x^2 + 2))^4 + 1726278400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) 
^3 - 9101541120*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 39843840*sqrt(3)*(sqrt(3 
)*x - sqrt(3*x^2 + 2)) - 1411584)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqr 
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^8
 

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.06 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {18873\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1470612500}-\frac {18873\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1470612500}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{131072\,\left (x^8+12\,x^7+63\,x^6+189\,x^5+\frac {2835\,x^4}{8}+\frac {1701\,x^3}{4}+\frac {5103\,x^2}{16}+\frac {2187\,x}{16}+\frac {6561}{256}\right )}-\frac {4816641\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70246400\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {861381\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4014080\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}+\frac {24813\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{10756480000\,\left (x+\frac {3}{2}\right )}-\frac {81899\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{229376\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {48141\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{614656000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {20705\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{65536\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {1573857\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{175616000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

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

Output:

(18873*35^(1/2)*log(x + 3/2))/1470612500 - (18873*35^(1/2)*log(x - (3^(1/2 
)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1470612500 - (15925*3^(1/2)*(x^2 + 
 2/3)^(1/2))/(131072*((2187*x)/16 + (5103*x^2)/16 + (1701*x^3)/4 + (2835*x 
^4)/8 + 189*x^5 + 63*x^6 + 12*x^7 + x^8 + 6561/256)) - (4816641*3^(1/2)*(x 
^2 + 2/3)^(1/2))/(70246400*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) 
+ (861381*3^(1/2)*(x^2 + 2/3)^(1/2))/(4014080*((405*x)/16 + (135*x^2)/4 + 
(45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) + (24813*3^(1/2)*(x^2 + 2/3)^(1/2 
))/(10756480000*(x + 3/2)) - (81899*3^(1/2)*(x^2 + 2/3)^(1/2))/(229376*((7 
29*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/6 
4)) + (48141*3^(1/2)*(x^2 + 2/3)^(1/2))/(614656000*(3*x + x^2 + 9/4)) + (2 
0705*3^(1/2)*(x^2 + 2/3)^(1/2))/(65536*((5103*x)/64 + (5103*x^2)/32 + (283 
5*x^3)/16 + (945*x^4)/8 + (189*x^5)/4 + (21*x^6)/2 + x^7 + 2187/128)) + (1 
573857*3^(1/2)*(x^2 + 2/3)^(1/2))/(175616000*((27*x)/4 + (9*x^2)/2 + x^3 + 
 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.95 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx=\frac {-9662976 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{8}+9662976 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{8}-3660214460 \sqrt {3 x^{2}+2}[phone] \sqrt {3 x^{2}+2}\, x^{6}[phone] \sqrt {3 x^{2}+2}\, x^{5}[phone] \sqrt {3 x^{2}+2}\, x^{4}[phone] \sqrt {3 x^{2}+2}\, x^{3}-6235771360 \sqrt {3 x^{2}+2}\, x^{2}-1357610905 \sqrt {3 x^{2}+2}\, x +1736910 \sqrt {3 x^{2}+2}\, x^{7}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{6}-608767488 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{6}-1320808032 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x [phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}-1826302464 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}-3424317120 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{7}-115955712 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{7}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}-4109180544 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-247651506 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x -3081885408 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}}{752953600000 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(1736910*sqrt(3*x**2 + 2)*x**7 + 77210280*sqrt(3*x**2 + 2)*x**6 + 73607354 
10*sqrt(3*x**2 + 2)*x**5 + 1176470400*sqrt(3*x**2 + 2)*x**4 + 7922443725*s 
qrt(3*x**2 + 2)*x**3 - 6235771360*sqrt(3*x**2 + 2)*x**2 - 1357610905*sqrt( 
3*x**2 + 2)*x - 3660214460*sqrt(3*x**2 + 2) + 9662976*sqrt(35)*log(sqrt(3* 
x**2 + 2)*sqrt(35) + 9*x - 4)*x**8 + 115955712*sqrt(35)*log(sqrt(3*x**2 + 
2)*sqrt(35) + 9*x - 4)*x**7 + 608767488*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt 
(35) + 9*x - 4)*x**6 + 1826302464*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 
 9*x - 4)*x**5 + 3424317120*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 
 4)*x**4 + 4109180544*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x* 
*3 + 3081885408*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 1 
320808032*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 247651506* 
sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 9662976*sqrt(35)*log(2 
*x + 3)*x**8 - 115955712*sqrt(35)*log(2*x + 3)*x**7 - 608767488*sqrt(35)*l 
og(2*x + 3)*x**6 - 1826302464*sqrt(35)*log(2*x + 3)*x**5 - 3424317120*sqrt 
(35)*log(2*x + 3)*x**4 - 4109180544*sqrt(35)*log(2*x + 3)*x**3 - 308188540 
8*sqrt(35)*log(2*x + 3)*x**2 - 1320808032*sqrt(35)*log(2*x + 3)*x - 247651 
506*sqrt(35)*log(2*x + 3))/(2941225000*(256*x**8 + 3072*x**7 + 16128*x**6 
+ 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561))