\(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^8} \, dx\) [219]

Optimal result

Integrand size = 24, antiderivative size = 153 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {7569 \sqrt {2+3 x^2}}{6860000 (3+2 x)^3}+\frac {3897 \sqrt {2+3 x^2}}{48020000 (3+2 x)^2}-\frac {212679 \sqrt {2+3 x^2}}{1680700000 (3+2 x)}-\frac {3 (22557+7898 x) \sqrt {2+3 x^2}}{1372000 (3+2 x)^5}-\frac {(471-596 x) \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^7}-\frac {72603 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{105043750 \sqrt {35}} \] Output:

7569/6860000*(3*x^2+2)^(1/2)/(3+2*x)^3+3897/48020000*(3*x^2+2)^(1/2)/(3+2* 
x)^2-212679*(3*x^2+2)^(1/2)/(5042100000+3361400000*x)-3/1372000*(22557+789 
8*x)*(3*x^2+2)^(1/2)/(3+2*x)^5-1/2940*(471-596*x)*(3*x^2+2)^(3/2)/(3+2*x)^ 
7-72603/3676531250*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (471103116+256388969 x+740031210 x^2-98810025 x^3+148868010 x^4+44301924 x^5+5104296 x^6\right )}{(3+2 x)^7}+871236 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{22059187500} \] Input:

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]
 

Output:

((-35*Sqrt[2 + 3*x^2]*(471103116 + 256388969*x + 740031210*x^2 - 98810025* 
x^3 + 148868010*x^4 + 44301924*x^5 + 5104296*x^6))/(3 + 2*x)^7 + 871236*Sq 
rt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/22 
059187500
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {688, 25, 688, 27, 679, 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.

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

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{245} \int -\frac {(287-78 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{245} \int \frac {(287-78 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{245} \left (-\frac {1}{210} \int -\frac {6 (2271-404 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \int \frac {(2271-404 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\)

Input:

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^8,x]
 

Output:

(-13*(2 + 3*x^2)^(5/2))/(245*(3 + 2*x)^7) + ((-404*(2 + 3*x^2)^(5/2))/(105 
*(3 + 2*x)^6) + ((-822*(2 + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (2689*(-1/140 
*((4 - 9*x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^4 + (9*(-1/70*((4 - 9*x)*Sqrt[2 + 
 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/( 
35*Sqrt[35])))/70))/5)/35)/245
 

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 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 = 0.84 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.59

method	result	size
risch	\(-\frac {15312888 x^{8}[phone] x^{7}[phone] x^{6}-207826227 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{630262500 \left (2 x +3\right )^{7} \sqrt {3 x^{2}+2}}-\frac {72603 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3676531250}\)	\(90\)
trager	\(-\frac {\left (5104296 x^{6}[phone] x^{5}[phone] x^{4}-98810025 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}+2}}{630262500 \left (2 x +3\right )^{7}}+\frac {72603 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{3676531250}\)	\(96\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{31360 \left (x +\frac {3}{2}\right )^{7}}-\frac {101 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{411600 \left (x +\frac {3}{2}\right )^{6}}-\frac {411 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{3430000 \left (x +\frac {3}{2}\right )^{5}}-\frac {2689 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{48020000 \left (x +\frac {3}{2}\right )^{4}}-\frac {24201 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{840350000 \left (x +\frac {3}{2}\right )^{3}}-\frac {250077 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{14706125000 \left (x +\frac {3}{2}\right )^{2}}-\frac {2831517 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{257357187500 \left (x +\frac {3}{2}\right )}+\frac {96804 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{64339296875}+\frac {653427 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{7353062500}+\frac {72603 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{3676531250}-\frac {72603 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3676531250}+\frac {8494551 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{257357187500}\)	\(245\)

Input:

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^8,x,method=_RETURNVERBOSE)
 

Output:

-1/630262500*(15312888*x^8+132905772*x^7+456812622*x^6-207826227*x^5+25178 
29650*x^4+571546857*x^3+2893371768*x^2+512777938*x[phone])/(2*x+3)^7/(3 
*x^2+2)^(1/2)-72603/3676531250*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 = 164, normalized size of antiderivative = 1.07 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {217809 \, \sqrt {35} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (5104296 \, x^{6} + 44301924 \, x^{5} + 148868010 \, x^{4} - 98810025 \, x^{3} + 740031210 \, x^{2} + 256388969 \, x + 471103116\right )} \sqrt {3 \, x^{2} + 2}}{22059187500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="fricas")
 

Output:

1/22059187500*(217809*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 
+ 22680*x^3 + 20412*x^2 + 10206*x + 2187)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*( 
9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(5104296*x^6 + 443 
01924*x^5 + 148868010*x^4 - 98810025*x^3 + 740031210*x^2 + 256388969*x + 4 
71103116)*sqrt(3*x^2 + 2))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22 
680*x^3 + 20412*x^2 + 10206*x + 2187)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (126) = 252\).

Time = 0.12 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.96 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {750231}{14706125000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{245 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {404 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{25725 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {822 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{214375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {2689 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3001250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {24201 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{105043750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {250077 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3676531250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {653427}{7353062500} \, \sqrt {3 \, x^{2} + 2} x + \frac {72603}{3676531250} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {72603}{1838265625} \, \sqrt {3 \, x^{2} + 2} - \frac {2831517 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{14706125000 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="maxima")
 

Output:

750231/14706125000*(3*x^2 + 2)^(3/2) - 13/245*(3*x^2 + 2)^(5/2)/(128*x^7 + 
 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) 
 - 404/25725*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4 
860*x^2 + 2916*x + 729) - 822/214375*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 
 720*x^3 + 1080*x^2 + 810*x + 243) - 2689/3001250*(3*x^2 + 2)^(5/2)/(16*x^ 
4 + 96*x^3 + 216*x^2 + 216*x + 81) - 24201/105043750*(3*x^2 + 2)^(5/2)/(8* 
x^3 + 36*x^2 + 54*x + 27) - 250077/3676531250*(3*x^2 + 2)^(5/2)/(4*x^2 + 1 
2*x + 9) + 653427/7353062500*sqrt(3*x^2 + 2)*x + 72603/3676531250*sqrt(35) 
*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 72603/18 
38265625*sqrt(3*x^2 + 2) - 2831517/14706125000*(3*x^2 + 2)^(3/2)/(2*x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (126) = 252\).

Time = 0.15 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.67 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {72603}{3676531250} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, {\left (258144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 5033808 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} + 225898166 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 26360013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 555459995 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 2679767547 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 4252091247 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 6029804778 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 11677158028 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 7324195080 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 2245361152 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 675266496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 174039168 \, \sqrt {3} x - 6049536 \, \sqrt {3} - 174039168 \, \sqrt {3 \, x^{2} + 2}\right )}}{3361400000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{7}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x, algorithm="giac")
 

Output:

72603/3676531250*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)) 
) - 9/3361400000*(258144*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 5033808*sqrt(3 
)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 225898166*(sqrt(3)*x - sqrt(3*x^2 + 2 
))^11 + 26360013*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 555459995*(sqr 
t(3)*x - sqrt(3*x^2 + 2))^9 - 2679767547*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 
 2))^8 - 4252091247*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 6029804778*sqrt(3)*( 
sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 11677158028*(sqrt(3)*x - sqrt(3*x^2 + 2)) 
^5 - 7324195080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 2245361152*(sqrt 
(3)*x - sqrt(3*x^2 + 2))^3 - 675266496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
))^2 + 174039168*sqrt(3)*x - 6049536*sqrt(3) - 174039168*sqrt(3*x^2 + 2))/ 
((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) 
 - 2)^7
 

Mupad [B] (verification not implemented)

Time = 5.97 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.78 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {72603\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{3676531250}-\frac {72603\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{3676531250}+\frac {92453\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{21952000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {507\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {212679\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3361400000\,\left (x+\frac {3}{2}\right )}+\frac {125\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2688\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {3897\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{192080000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {65\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2048\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {7569\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{54880000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^8,x)
 

Output:

(72603*35^(1/2)*log(x + 3/2))/3676531250 - (72603*35^(1/2)*log(x - (3^(1/2 
)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/3676531250 + (92453*3^(1/2)*(x^2 + 
 2/3)^(1/2))/(21952000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (5 
07*3^(1/2)*(x^2 + 2/3)^(1/2))/(19600*((405*x)/16 + (135*x^2)/4 + (45*x^3)/ 
2 + (15*x^4)/2 + x^5 + 243/32)) - (212679*3^(1/2)*(x^2 + 2/3)^(1/2))/(3361 
400000*(x + 3/2)) + (125*3^(1/2)*(x^2 + 2/3)^(1/2))/(2688*((729*x)/16 + (1 
215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (3897*3 
^(1/2)*(x^2 + 2/3)^(1/2))/(192080000*(3*x + x^2 + 9/4)) - (65*3^(1/2)*(x^2 
 + 2/3)^(1/2))/(2048*((5103*x)/64 + (5103*x^2)/32 + (2835*x^3)/16 + (945*x 
^4)/8 + (189*x^5)/4 + (21*x^6)/2 + x^7 + 2187/128)) + (7569*3^(1/2)*(x^2 + 
 2/3)^(1/2))/(54880000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.69 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {-178650360 \sqrt {3 x^{2}+2}\, x^{6}-1550567340 \sqrt {3 x^{2}+2}\, x^{5}-5210380350 \sqrt {3 x^{2}+2}\, x^{4}[phone] \sqrt {3 x^{2}+2}\, x^{3}-25901092350 \sqrt {3 x^{2}+2}\, x^{2}-8973613915 \sqrt {3 x^{2}+2}\, x -16488609060 \sqrt {3 x^{2}+2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{7}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{6}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x [phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-55759104 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{7}-585470592 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{6}-2634617664 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-6586544160 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-9879816240 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-8891834616 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-4445917308 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -952696566 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{2823576000000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

int((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^8,x)
 

Output:

( - 178650360*sqrt(3*x**2 + 2)*x**6 - 1550567340*sqrt(3*x**2 + 2)*x**5 - 5 
210380350*sqrt(3*x**2 + 2)*x**4 + 3458350875*sqrt(3*x**2 + 2)*x**3 - 25901 
092350*sqrt(3*x**2 + 2)*x**2 - 8973613915*sqrt(3*x**2 + 2)*x - 16488609060 
*sqrt(3*x**2 + 2) + 55759104*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x 
- 4)*x**7 + 585470592*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x* 
*6 + 2634617664*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 6 
586544160*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 9879816 
240*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 8891834616*sq 
rt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 4445917308*sqrt(35) 
*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 952696566*sqrt(35)*log(sqrt( 
3*x**2 + 2)*sqrt(35) + 9*x - 4) - 55759104*sqrt(35)*log(2*x + 3)*x**7 - 58 
5470592*sqrt(35)*log(2*x + 3)*x**6 - 2634617664*sqrt(35)*log(2*x + 3)*x**5 
 - 6586544160*sqrt(35)*log(2*x + 3)*x**4 - 9879816240*sqrt(35)*log(2*x + 3 
)*x**3 - 8891834616*sqrt(35)*log(2*x + 3)*x**2 - 4445917308*sqrt(35)*log(2 
*x + 3)*x - 952696566*sqrt(35)*log(2*x + 3))/(22059187500*(128*x**7 + 1344 
*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187) 
)