\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^7} \, dx\) [231]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {9 (4373+5167 x) \sqrt {2+3 x^2}}{109760 (3+2 x)^2}+\frac {(202+403 x) \left (2+3 x^2\right )^{3/2}}{1568 (3+2 x)^4}+\frac {(11+159 x) \left (2+3 x^2\right )^{5/2}}{420 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {159759 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{219520 \sqrt {35}} \] Output:

9/109760*(4373+5167*x)*(3*x^2+2)^(1/2)/(3+2*x)^2+1/1568*(202+403*x)*(3*x^2 
+2)^(3/2)/(3+2*x)^4+1/420*(11+159*x)*(3*x^2+2)^(5/2)/(3+2*x)^6-9/128*arcsi 
nh(1/2*x*6^(1/2))*3^(1/2)-159759/7683200*35^(1/2)*arctanh(1/35*(4-9*x)*35^ 
(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 3.95 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {\frac {70 \sqrt {2+3 x^2} \left (10361807+39843609 x+59256588 x^2+47453802 x^3+18915336 x^4+4369608 x^5\right )}{(3+2 x)^6}+958554 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+1620675 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{23049600} \] Input:

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7,x]
 

Output:

((70*Sqrt[2 + 3*x^2]*(10361807 + 39843609*x + 59256588*x^2 + 47453802*x^3 
+ 18915336*x^4 + 4369608*x^5))/(3 + 2*x)^6 + 958554*Sqrt[35]*ArcTanh[(3*Sq 
rt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] + 1620675*Sqrt[3]*Log[- 
(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/23049600
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {680, 27, 680, 27, 680, 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.

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

\(\Big \downarrow \) 680

\(\displaystyle \frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}-\frac {\int -\frac {60 (26-21 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx}{1680}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{28} \int \frac {(26-21 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {1}{28} \left (\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}-\frac {\int -\frac {360 (23-49 x) \sqrt {3 x^2+2}}{(2 x+3)^3}dx}{1120}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \int \frac {(23-49 x) \sqrt {3 x^2+2}}{(2 x+3)^3}dx+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {(5167 x+4373) \sqrt {3 x^2+2}}{140 (2 x+3)^2}-\frac {1}{560} \int -\frac {12 (386-1715 x)}{(2 x+3) \sqrt {3 x^2+2}}dx\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {3}{140} \int \frac {386-1715 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {\sqrt {3 x^2+2} (5167 x+4373)}{140 (2 x+3)^2}\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {3}{140} \left (\frac {5917}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1715}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {\sqrt {3 x^2+2} (5167 x+4373)}{140 (2 x+3)^2}\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {3}{140} \left (\frac {5917}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1715 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (5167 x+4373)}{140 (2 x+3)^2}\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {3}{140} \left (-\frac {5917}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {1715 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (5167 x+4373)}{140 (2 x+3)^2}\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{28} \left (\frac {9}{28} \left (\frac {3}{140} \left (-\frac {1715 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {5917 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )+\frac {\sqrt {3 x^2+2} (5167 x+4373)}{140 (2 x+3)^2}\right )+\frac {(403 x+202) \left (3 x^2+2\right )^{3/2}}{56 (2 x+3)^4}\right )+\frac {(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}\)

Input:

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^7,x]
 

Output:

((11 + 159*x)*(2 + 3*x^2)^(5/2))/(420*(3 + 2*x)^6) + (((202 + 403*x)*(2 + 
3*x^2)^(3/2))/(56*(3 + 2*x)^4) + (9*(((4373 + 5167*x)*Sqrt[2 + 3*x^2])/(14 
0*(3 + 2*x)^2) + (3*((-1715*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3]) - (5917*ArcT 
anh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])))/140))/28)/28
 

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 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 = 1.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {13108824 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{329280 \left (2 x +3\right )^{6} \sqrt {3 x^{2}+2}}-\frac {9 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}-\frac {159759 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7683200}\)	\(97\)
trager	\(\frac {\left (4369608 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}+2}}{329280 \left (2 x +3\right )^{6}}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{128}-\frac {159759 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{7683200}\)	\(122\)
default	\(-\frac {\left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{3136 \left (x +\frac {3}{2}\right )^{5}}+\frac {159759 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{7683200}+\frac {159759 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1470612500}+\frac {53253 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{33614000}-\frac {45711 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{3841600}+\frac {123129 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1470612500}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {159759 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7683200}-\frac {9 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{128}-\frac {41043 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1470612500 \left (x +\frac {3}{2}\right )}-\frac {113 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{548800 \left (x +\frac {3}{2}\right )^{4}}-\frac {1039 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9604000 \left (x +\frac {3}{2}\right )^{3}}-\frac {6561 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{84035000 \left (x +\frac {3}{2}\right )^{2}}-\frac {27009 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{67228000}\)	\(269\)

Input:

int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^7,x,method=_RETURNVERBOSE)
 

Output:

1/329280*(13108824*x^7+56746008*x^6+151100622*x^5+215600436*x^4+214438431* 
x^3+149598597*x^2+79687218*x[phone])/(2*x+3)^6/(3*x^2+2)^(1/2)-9/128*arc 
sinh(1/2*6^(1/2)*x)*3^(1/2)-159759/7683200*35^(1/2)*arctanh(2/35*(4-9*x)*3 
5^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {1620675 \, \sqrt {3} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 479277 \, \sqrt {35} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 140 \, {\left (4369608 \, x^{5} + 18915336 \, x^{4} + 47453802 \, x^{3} + 59256588 \, x^{2} + 39843609 \, x + 10361807\right )} \sqrt {3 \, x^{2} + 2}}{46099200 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="fricas")
 

Output:

1/46099200*(1620675*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860 
*x^2 + 2916*x + 729)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 479277*s 
qrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) 
*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 1 
2*x + 9)) + 140*(4369608*x^5 + 18915336*x^4 + 47453802*x^3 + 59256588*x^2 
+ 39843609*x + 10361807)*sqrt(3*x^2 + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4 
320*x^3 + 4860*x^2 + 2916*x + 729)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (107) = 214\).

Time = 0.15 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.16 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {19683}{84035000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {{\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{98 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {113 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{34300 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1039 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1200500 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {6561 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{21008750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {27009}{67228000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {53253}{33614000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {41043 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{84035000 \, {\left (2 \, x + 3\right )}} - \frac {45711}{3841600} \, \sqrt {3 \, x^{2} + 2} x - \frac {9}{128} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {159759}{7683200} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {159759}{3841600} \, \sqrt {3 \, x^{2} + 2} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="maxima")
 

Output:

19683/84035000*(3*x^2 + 2)^(5/2) - 13/210*(3*x^2 + 2)^(7/2)/(64*x^6 + 576* 
x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1/98*(3*x^2 + 2)^(7 
/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 113/34300*(3*x 
^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1039/1200500*(3*x 
^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 6561/21008750*(3*x^2 + 2)^(7/ 
2)/(4*x^2 + 12*x + 9) - 27009/67228000*(3*x^2 + 2)^(3/2)*x + 53253/3361400 
0*(3*x^2 + 2)^(3/2) - 41043/84035000*(3*x^2 + 2)^(5/2)/(2*x + 3) - 45711/3 
841600*sqrt(3*x^2 + 2)*x - 9/128*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 159759/7 
683200*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 
 3)) + 159759/3841600*sqrt(3*x^2 + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (107) = 214\).

Time = 0.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {9}{128} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {159759}{7683200} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, \sqrt {3} {\left (566976 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 16427322 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 70792520 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 421378065 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 244013814 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 879808433 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 512612604 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 2079633300 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 831934400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 500387712 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 51770496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 7768192\right )}}{878080 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \] Input:

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x, algorithm="giac")
 

Output:

9/128*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 159759/7683200*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))) + 3/878080*sqrt(3)*(5669 
76*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 16427322*(sqrt(3)*x - sqrt(3 
*x^2 + 2))^10 + 70792520*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 4213780 
65*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 244013814*sqrt(3)*(sqrt(3)*x - sqrt(3 
*x^2 + 2))^7 - 879808433*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 512612604*sqrt( 
3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 2079633300*(sqrt(3)*x - sqrt(3*x^2 + 
2))^4 - 831934400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 500387712*(sqr 
t(3)*x - sqrt(3*x^2 + 2))^2 - 51770496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
)) + 7768192)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sq 
rt(3*x^2 + 2)) - 2)^6
 

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.79 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx=\frac {159759\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{7683200}-\frac {9\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{128}-\frac {159759\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7683200}-\frac {9019\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {7315\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4096\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}+\frac {182067\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{878080\,\left (x+\frac {3}{2}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{24576\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}-\frac {164961\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{250880\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {109789\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{71680\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(5/2)*(x - 5))/(2*x + 3)^7,x)
 

Output:

(159759*35^(1/2)*log(x + 3/2))/7683200 - (9*3^(1/2)*asinh((2^(1/2)*3^(1/2) 
*x)/2))/128 - (159759*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2) 
)/9 - 4/9))/7683200 - (9019*3^(1/2)*(x^2 + 2/3)^(1/2))/(4096*((27*x)/2 + ( 
27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (7315*3^(1/2)*(x^2 + 2/3)^(1/2))/(4096 
*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) + (1 
82067*3^(1/2)*(x^2 + 2/3)^(1/2))/(878080*(x + 3/2)) - (15925*3^(1/2)*(x^2 
+ 2/3)^(1/2))/(24576*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4) 
/4 + 9*x^5 + x^6 + 729/64)) - (164961*3^(1/2)*(x^2 + 2/3)^(1/2))/(250880*( 
3*x + x^2 + 9/4)) + (109789*3^(1/2)*(x^2 + 2/3)^(1/2))/(71680*((27*x)/4 + 
(9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 647, normalized size of antiderivative = 4.86 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx =\text {Too large to display} \] Input:

int((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^7,x)
 

Output:

(611745120*sqrt(3*x**2 + 2)*x**5 + 2648147040*sqrt(3*x**2 + 2)*x**4 + 6643 
532280*sqrt(3*x**2 + 2)*x**3 + 8295922320*sqrt(3*x**2 + 2)*x**2 + 55781052 
60*sqrt(3*x**2 + 2)*x + 1450652980*sqrt(3*x**2 + 2) + 61347456*sqrt(35)*lo 
g(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**6 + 552127104*sqrt(35)*log(sqrt( 
3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 2070476640*sqrt(35)*log(sqrt(3*x**2 
 + 2)*sqrt(35) + 9*x - 4)*x**4 + 4140953280*sqrt(35)*log(sqrt(3*x**2 + 2)* 
sqrt(35) + 9*x - 4)*x**3 + 4658572440*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(3 
5) + 9*x - 4)*x**2 + 2795143464*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9 
*x - 4)*x + 698785866*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 
61347456*sqrt(35)*log(2*x + 3)*x**6 - 552127104*sqrt(35)*log(2*x + 3)*x**5 
 - 2070476640*sqrt(35)*log(2*x + 3)*x**4 - 4140953280*sqrt(35)*log(2*x + 3 
)*x**3 - 4658572440*sqrt(35)*log(2*x + 3)*x**2 - 2795143464*sqrt(35)*log(2 
*x + 3)*x - 698785866*sqrt(35)*log(2*x + 3) + 103723200*sqrt(3)*log(sqrt(3 
*x**2 + 2) - sqrt(3)*x)*x**6 + 933508800*sqrt(3)*log(sqrt(3*x**2 + 2) - sq 
rt(3)*x)*x**5 + 3500658000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**4 
+ 7001316000*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**3 + 7876480500*s 
qrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x)*x**2 + 4725888300*sqrt(3)*log(sqr 
t(3*x**2 + 2) - sqrt(3)*x)*x + 1181472075*sqrt(3)*log(sqrt(3*x**2 + 2) - s 
qrt(3)*x) - 103723200*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**6 - 933 
508800*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x**5 - 3500658000*sqrt...