\(\int \frac {(5-x) (2+3 x^2)^{3/2}}{(3+2 x)^6} \, dx\) [217]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=-\frac {369 (4-9 x) \sqrt {2+3 x^2}}{171500 (3+2 x)^2}-\frac {41 (4-9 x) \left (2+3 x^2\right )^{3/2}}{4900 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{175 (3+2 x)^5}-\frac {1107 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{85750 \sqrt {35}} \] Output:

-369/171500*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2-41/4900*(4-9*x)*(3*x^2+2)^(3 
/2)/(3+2*x)^4-13/175*(3*x^2+2)^(5/2)/(3+2*x)^5-1107/3001250*35^(1/2)*arcta 
nh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 2.65 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (125252-64493 x+26682 x^2-189543 x^3+10602 x^4\right )}{(3+2 x)^5}+4428 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{6002500} \] Input:

Integrate[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]
 

Output:

((-35*Sqrt[2 + 3*x^2]*(125252 - 64493*x + 26682*x^2 - 189543*x^3 + 10602*x 
^4))/(3 + 2*x)^5 + 4428*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt 
[2 + 3*x^2])/Sqrt[35]])/6002500
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {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.

\(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6} \, dx\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {41}{35} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {41}{35} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {41}{35} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {41}{35} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{175 (2 x+3)^5}\)

Input:

Int[((5 - x)*(2 + 3*x^2)^(3/2))/(3 + 2*x)^6,x]
 

Output:

(-13*(2 + 3*x^2)^(5/2))/(175*(3 + 2*x)^5) + (41*(-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)) 
/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 = 0.80 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {31806 x^{6}-568629 x^{5}+101250 x^{4}-572565 x^{3}+429120 x^{2}-128986 x +250504}{171500 \left (2 x +3\right )^{5} \sqrt {3 x^{2}+2}}-\frac {1107 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3001250}\)	\(80\)
trager	\(-\frac {\left (10602 x^{4}-189543 x^{3}+26682 x^{2}-64493 x +125252\right ) \sqrt {3 x^{2}+2}}{171500 \left (2 x +3\right )^{5}}-\frac {1107 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{3001250}\)	\(87\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {41 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{39200 \left (x +\frac {3}{2}\right )^{4}}-\frac {369 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{686000 \left (x +\frac {3}{2}\right )^{3}}-\frac {3813 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{12005000 \left (x +\frac {3}{2}\right )^{2}}-\frac {43173 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{210087500 \left (x +\frac {3}{2}\right )}+\frac {1476 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{52521875}+\frac {9963 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{6002500}+\frac {1107 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{3001250}-\frac {1107 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3001250}+\frac {129519 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{210087500}\)	\(203\)

Input:

int((5-x)*(3*x^2+2)^(3/2)/(2*x+3)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/171500*(31806*x^6-568629*x^5+101250*x^4-572565*x^3+429120*x^2-128986*x+ 
250504)/(2*x+3)^5/(3*x^2+2)^(1/2)-1107/3001250*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 = 134, normalized size of antiderivative = 1.23 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {1107 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (10602 \, x^{4} - 189543 \, x^{3} + 26682 \, x^{2} - 64493 \, x + 125252\right )} \sqrt {3 \, x^{2} + 2}}{6002500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="fricas")
 

Output:

1/6002500*(1107*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 
243)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 
 + 12*x + 9)) - 35*(10602*x^4 - 189543*x^3 + 26682*x^2 - 64493*x + 125252) 
*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(3/2)/(3+2*x)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (90) = 180\).

Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {11439}{12005000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {41 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{2450 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {369 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{85750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {3813 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3001250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {9963}{6002500} \, \sqrt {3 \, x^{2} + 2} x + \frac {1107}{3001250} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1107}{1500625} \, \sqrt {3 \, x^{2} + 2} - \frac {43173 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{12005000 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="maxima")
 

Output:

11439/12005000*(3*x^2 + 2)^(3/2) - 13/175*(3*x^2 + 2)^(5/2)/(32*x^5 + 240* 
x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 41/2450*(3*x^2 + 2)^(5/2)/(16*x^ 
4 + 96*x^3 + 216*x^2 + 216*x + 81) - 369/85750*(3*x^2 + 2)^(5/2)/(8*x^3 + 
36*x^2 + 54*x + 27) - 3813/3001250*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) + 
9963/6002500*sqrt(3*x^2 + 2)*x + 1107/3001250*sqrt(35)*arcsinh(3/2*sqrt(6) 
*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 1107/1500625*sqrt(3*x^2 + 2) 
 - 43173/12005000*(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. 318 vs. \(2 (90) = 180\).

Time = 0.15 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.92 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {1107}{3001250} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, {\left (89686 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 138886 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 1224478 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 245133 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 1224531 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 4374874 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 4855928 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1339152 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 586816 \, \sqrt {3} x - 37696 \, \sqrt {3} + 586816 \, \sqrt {3 \, x^{2} + 2}\right )}}{2744000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \] Input:

integrate((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x, algorithm="giac")
 

Output:

1107/3001250*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqr 
t(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 
9/2744000*(89686*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 138886*sqrt(3)*(sqrt(3) 
*x - sqrt(3*x^2 + 2))^8 + 1224478*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 245133 
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 1224531*(sqrt(3)*x - sqrt(3*x^2 
 + 2))^5 - 4374874*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 4855928*(sqrt 
(3)*x - sqrt(3*x^2 + 2))^3 - 1339152*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) 
^2 - 586816*sqrt(3)*x - 37696*sqrt(3) + 586816*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)^5
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.64 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {1107\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{3001250}-\frac {1107\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{3001250}+\frac {731\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2560\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {91\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {5301\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2744000\,\left (x+\frac {3}{2}\right )}+\frac {7233\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{156800\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {8349\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{44800\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(3/2)*(x - 5))/(2*x + 3)^6,x)
 

Output:

(1107*35^(1/2)*log(x + 3/2))/3001250 - (1107*35^(1/2)*log(x - (3^(1/2)*35^ 
(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/3001250 + (731*3^(1/2)*(x^2 + 2/3)^(1/2 
))/(2560*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (91*3^(1/2)*(x^2 
 + 2/3)^(1/2))/(512*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + 
x^5 + 243/32)) - (5301*3^(1/2)*(x^2 + 2/3)^(1/2))/(2744000*(x + 3/2)) + (7 
233*3^(1/2)*(x^2 + 2/3)^(1/2))/(156800*(3*x + x^2 + 9/4)) - (8349*3^(1/2)* 
(x^2 + 2/3)^(1/2))/(44800*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.76 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx=\frac {-371070 \sqrt {3 x^{2}+2}\, x^{4}+6634005 \sqrt {3 x^{2}+2}\, x^{3}-933870 \sqrt {3 x^{2}+2}\, x^{2}+2257255 \sqrt {3 x^{2}+2}\, x -4383820 \sqrt {3 x^{2}+2}+70848 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}+531360 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}+1594080 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+2391120 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+1793340 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +538002 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-70848 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-531360 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-1594080 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-2391120 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-1793340 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -538002 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{192080000 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

int((5-x)*(3*x^2+2)^(3/2)/(3+2*x)^6,x)
 

Output:

( - 371070*sqrt(3*x**2 + 2)*x**4 + 6634005*sqrt(3*x**2 + 2)*x**3 - 933870* 
sqrt(3*x**2 + 2)*x**2 + 2257255*sqrt(3*x**2 + 2)*x - 4383820*sqrt(3*x**2 + 
 2) + 70848*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 53136 
0*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 1594080*sqrt(35 
)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 2391120*sqrt(35)*log(sqr 
t(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 1793340*sqrt(35)*log(sqrt(3*x**2 
+ 2)*sqrt(35) + 9*x - 4)*x + 538002*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) 
 + 9*x - 4) - 70848*sqrt(35)*log(2*x + 3)*x**5 - 531360*sqrt(35)*log(2*x + 
 3)*x**4 - 1594080*sqrt(35)*log(2*x + 3)*x**3 - 2391120*sqrt(35)*log(2*x + 
 3)*x**2 - 1793340*sqrt(35)*log(2*x + 3)*x - 538002*sqrt(35)*log(2*x + 3)) 
/(6002500*(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243))