\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^8} \, dx\) [232]

Optimal result

Integrand size = 24, antiderivative size = 136 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{1200500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{34300 (3+2 x)^4}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{5/2}}{7350 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{245 (3+2 x)^7}-\frac {1107 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{600250 \sqrt {35}} \] Output:

-369/1200500*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2-41/34300*(4-9*x)*(3*x^2+2)^ 
(3/2)/(3+2*x)^4-41/7350*(4-9*x)*(3*x^2+2)^(5/2)/(3+2*x)^6-13/245*(3*x^2+2) 
^(7/2)/(3+2*x)^7-1107/21008750*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x 
^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 5.56 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (4499004-593639 x+3488490 x^2-15015225 x^3-2997810 x^4-9455994 x^5+656424 x^6\right )}{(3+2 x)^7}+13284 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{126052500} \] Input:

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

Output:

((-35*Sqrt[2 + 3*x^2]*(4499004 - 593639*x + 3488490*x^2 - 15015225*x^3 - 2 
997810*x^4 - 9455994*x^5 + 656424*x^6))/(3 + 2*x)^7 + 13284*Sqrt[35]*ArcTa 
nh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/126052500
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.

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

\(\Big \downarrow \) 679

\(\displaystyle \frac {41}{35} \int \frac {\left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {1}{7} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {1}{7} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {41}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {41}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\)

Input:

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

Output:

(-13*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) + (41*(-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)*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)/7))/35
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {1969272 x^{8}-28367982 x^{7}-7680582 x^{6}-63957663 x^{5}+4469850 x^{4}-31811367 x^{3}[phone] x^{2}-1187278 x +8998008}{3601500 \left (2 x +3\right )^{7} \sqrt {3 x^{2}+2}}-\frac {1107 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{21008750}\)	\(90\)
trager	\(-\frac {\left (656424 x^{6}-9455994 x^{5}-2997810 x^{4}-15015225 x^{3}+3488490 x^{2}-593639 x +4499004\right ) \sqrt {3 x^{2}+2}}{3601500 \left (2 x +3\right )^{7}}+\frac {1107 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{21008750}\)	\(96\)
default	\(-\frac {123 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1372000 \left (x +\frac {3}{2}\right )^{5}}+\frac {1107 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{21008750}+\frac {17712 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{64339296875}+\frac {1476 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{367653125}+\frac {9963 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{42017500}+\frac {4612869 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{128678593750}-\frac {41 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{235200 \left (x +\frac {3}{2}\right )^{6}}-\frac {1107 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{21008750}-\frac {1537623 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{128678593750 \left (x +\frac {3}{2}\right )}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{31360 \left (x +\frac {3}{2}\right )^{7}}-\frac {1189 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{24010000 \left (x +\frac {3}{2}\right )^{4}}-\frac {12177 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{420175000 \left (x +\frac {3}{2}\right )^{3}}-\frac {132471 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{7353062500 \left (x +\frac {3}{2}\right )^{2}}+\frac {129519 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1470612500}\)	\(278\)

Input:

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

Output:

-1/3601500*(1969272*x^8-28367982*x^7-7680582*x^6-63957663*x^5+4469850*x^4- 
31811367*x^3+20473992*x^2-1187278*x+8998008)/(2*x+3)^7/(3*x^2+2)^(1/2)-110 
7/21008750*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 = 0.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.21 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {3321 \, \sqrt {35} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (656424 \, x^{6} - 9455994 \, x^{5} - 2997810 \, x^{4} - 15015225 \, x^{3} + 3488490 \, x^{2} - 593639 \, x + 4499004\right )} \sqrt {3 \, x^{2} + 2}}{126052500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \] Input:

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

Output:

1/126052500*(3321*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22 
680*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*(656424*x^6 - 9455994* 
x^5 - 2997810*x^4 - 15015225*x^3 + 3488490*x^2 - 593639*x + 4499004)*sqrt( 
3*x^2 + 2))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412 
*x^2 + 10206*x + 2187)
 

Sympy [F(-1)]

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

integrate((5-x)*(3*x**2+2)**(5/2)/(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. 323 vs. \(2 (113) = 226\).

Time = 0.16 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.38 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {397413}{7353062500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{245 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{3675 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {123 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{42875 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {1189 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1500625 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {12177 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{52521875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {132471 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1838265625 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {129519}{1470612500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {1476}{367653125} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {1537623 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{7353062500 \, {\left (2 \, x + 3\right )}} + \frac {9963}{42017500} \, \sqrt {3 \, x^{2} + 2} x + \frac {1107}{21008750} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1107}{10504375} \, \sqrt {3 \, x^{2} + 2} \] Input:

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

Output:

397413/7353062500*(3*x^2 + 2)^(5/2) - 13/245*(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) 
- 41/3675*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860 
*x^2 + 2916*x + 729) - 123/42875*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720 
*x^3 + 1080*x^2 + 810*x + 243) - 1189/1500625*(3*x^2 + 2)^(7/2)/(16*x^4 + 
96*x^3 + 216*x^2 + 216*x + 81) - 12177/52521875*(3*x^2 + 2)^(7/2)/(8*x^3 + 
 36*x^2 + 54*x + 27) - 132471/1838265625*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 
 9) + 129519/1470612500*(3*x^2 + 2)^(3/2)*x + 1476/367653125*(3*x^2 + 2)^( 
3/2) - 1537623/7353062500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 9963/42017500*sqrt 
(3*x^2 + 2)*x + 1107/21008750*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) 
- 2/3*sqrt(6)/abs(2*x + 3)) + 1107/10504375*sqrt(3*x^2 + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (113) = 226\).

Time = 0.15 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.00 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {1107}{21008750} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, {\left (908247 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 3755004 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} + 52905908 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 114259794 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 422075810 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 16674486 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 1093657086 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 205745364 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 1886581864 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 1023977040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 660654976 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 94952448 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 9114816 \, \sqrt {3} x - 1555968 \, \sqrt {3} + 9114816 \, \sqrt {3 \, x^{2} + 2}\right )}}{38416000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{7}} \] Input:

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

Output:

1107/21008750*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sq 
rt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 
 9/38416000*(908247*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 3755004*sqrt(3)*(sq 
rt(3)*x - sqrt(3*x^2 + 2))^12 + 52905908*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 
+ 114259794*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 422075810*(sqrt(3)* 
x - sqrt(3*x^2 + 2))^9 - 16674486*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 
- 1093657086*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 205745364*sqrt(3)*(sqrt(3)* 
x - sqrt(3*x^2 + 2))^6 + 1886581864*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 1023 
977040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 660654976*(sqrt(3)*x - sq 
rt(3*x^2 + 2))^3 - 94952448*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 9114 
816*sqrt(3)*x - 1555968*sqrt(3) + 9114816*sqrt(3*x^2 + 2))/((sqrt(3)*x - s 
qrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^7
 

Mupad [B] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.00 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {1107\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{21008750}-\frac {1107\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{21008750}+\frac {34571\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{62720\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {6213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7168\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {27351\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19208000\,\left (x+\frac {3}{2}\right )}+\frac {9095\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{12288\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {73161\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2195200\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {2275\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8192\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}-\frac {122553\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{627200\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

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

Output:

(1107*35^(1/2)*log(x + 3/2))/21008750 - (1107*35^(1/2)*log(x - (3^(1/2)*35 
^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/21008750 + (34571*3^(1/2)*(x^2 + 2/3)^ 
(1/2))/(62720*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (6213*3^(1/ 
2)*(x^2 + 2/3)^(1/2))/(7168*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x 
^4)/2 + x^5 + 243/32)) - (27351*3^(1/2)*(x^2 + 2/3)^(1/2))/(19208000*(x + 
3/2)) + (9095*3^(1/2)*(x^2 + 2/3)^(1/2))/(12288*((729*x)/16 + (1215*x^2)/1 
6 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (73161*3^(1/2)*(x 
^2 + 2/3)^(1/2))/(2195200*(3*x + x^2 + 9/4)) - (2275*3^(1/2)*(x^2 + 2/3)^( 
1/2))/(8192*((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)) - (122553*3^(1/2)*(x^2 + 2/3)^( 
1/2))/(627200*((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 = 3.02 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^8} \, dx=\frac {-22974840 \sqrt {3 x^{2}+2}\, x^{6}[phone] \sqrt {3 x^{2}+2}\, x^{5}[phone] \sqrt {3 x^{2}+2}\, x^{4}[phone] \sqrt {3 x^{2}+2}\, x^{3}-122097150 \sqrt {3 x^{2}+2}\, x^{2}[phone] \sqrt {3 x^{2}+2}\, x -157465140 \sqrt {3 x^{2}+2}+850176 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{7}+8926848 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{6}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x [phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-850176 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{7}-8926848 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{6}-40170816 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-100427040 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-150640560 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-135576504 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-67788252 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -14526054 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{16134720000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 22974840*sqrt(3*x**2 + 2)*x**6 + 330959790*sqrt(3*x**2 + 2)*x**5 + 104 
923350*sqrt(3*x**2 + 2)*x**4 + 525532875*sqrt(3*x**2 + 2)*x**3 - 122097150 
*sqrt(3*x**2 + 2)*x**2 + 20777365*sqrt(3*x**2 + 2)*x - 157465140*sqrt(3*x* 
*2 + 2) + 850176*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**7 + 
8926848*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**6 + 40170816* 
sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 100427040*sqrt(35 
)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 150640560*sqrt(35)*log(s 
qrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 135576504*sqrt(35)*log(sqrt(3*x 
**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 67788252*sqrt(35)*log(sqrt(3*x**2 + 2) 
*sqrt(35) + 9*x - 4)*x + 14526054*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 
 9*x - 4) - 850176*sqrt(35)*log(2*x + 3)*x**7 - 8926848*sqrt(35)*log(2*x + 
 3)*x**6 - 40170816*sqrt(35)*log(2*x + 3)*x**5 - 100427040*sqrt(35)*log(2* 
x + 3)*x**4 - 150640560*sqrt(35)*log(2*x + 3)*x**3 - 135576504*sqrt(35)*lo 
g(2*x + 3)*x**2 - 67788252*sqrt(35)*log(2*x + 3)*x - 14526054*sqrt(35)*log 
(2*x + 3))/(126052500*(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 226 
80*x**3 + 20412*x**2 + 10206*x + 2187))