\(\int \frac {5-x}{(3+2 x)^2 (2+3 x^2)^{5/2}} \, dx\) [260]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {220-51 x}{1470 \left (2+3 x^2\right )^{3/2}}-\frac {13}{35 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {176+277 x}{3430 \sqrt {2+3 x^2}}-\frac {176 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1715 \sqrt {35}} \] Output:

1/1470*(220-51*x)/(3*x^2+2)^(3/2)-13/35/(3+2*x)/(3*x^2+2)^(3/2)+1/3430*(17 
6+277*x)/(3*x^2+2)^(1/2)-176/60025*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/ 
(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {35 \left (3966+9107 x+7362 x^2+10647 x^3+4986 x^4\right )-1056 \sqrt {35} \sqrt {2+3 x^2} \left (6+4 x+9 x^2+6 x^3\right ) \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{360150 (3+2 x) \left (2+3 x^2\right )^{3/2}} \] Input:

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

Output:

(35*(3966 + 9107*x + 7362*x^2 + 10647*x^3 + 4986*x^4) - 1056*Sqrt[35]*Sqrt 
[2 + 3*x^2]*(6 + 4*x + 9*x^2 + 6*x^3)*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 
 3*x^2])])/(360150*(3 + 2*x)*(2 + 3*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {686, 27, 686, 27, 679, 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.

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}-\frac {1}{630} \int -\frac {6 (123 x+227)}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{105} \int \frac {123 x+227}{(2 x+3)^2 \left (3 x^2+2\right )^{3/2}}dx+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {1}{105} \left (\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {30 (507 x+68)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{105} \left (\frac {1}{7} \int \frac {507 x+68}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {528}{7} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}-\frac {528}{7} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{105} \left (\frac {1}{7} \left (\frac {277 \sqrt {3 x^2+2}}{7 (2 x+3)}-\frac {528 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7 \sqrt {35}}\right )+\frac {507 x+34}{14 (2 x+3) \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}\)

Input:

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

Output:

(26 + 41*x)/(210*(3 + 2*x)*(2 + 3*x^2)^(3/2)) + ((34 + 507*x)/(14*(3 + 2*x 
)*Sqrt[2 + 3*x^2]) + ((277*Sqrt[2 + 3*x^2])/(7*(3 + 2*x)) - (528*ArcTanh[( 
4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(7*Sqrt[35]))/7)/105
 

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 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {4986 x^{4}+10647 x^{3}+7362 x^{2}+9107 x +3966}{10290 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +3\right )}-\frac {176 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}\)	\(70\)
trager	\(\frac {4986 x^{4}+10647 x^{3}+7362 x^{2}+9107 x +3966}{10290 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (2 x +3\right )}+\frac {176 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{60025}\)	\(86\)
default	\(-\frac {13}{70 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {22}{147 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {17 x}{490 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {277 x}{3430 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {88}{1715 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {176 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}\)	\(119\)

Input:

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

Output:

1/10290*(4986*x^4+10647*x^3+7362*x^2+9107*x+3966)/(3*x^2+2)^(3/2)/(2*x+3)- 
176/60025*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 = 134, normalized size of antiderivative = 1.41 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {528 \, \sqrt {35} {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (4986 \, x^{4} + 10647 \, x^{3} + 7362 \, x^{2} + 9107 \, x + 3966\right )} \sqrt {3 \, x^{2} + 2}}{360150 \, {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )}} \] Input:

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

Output:

1/360150*(528*sqrt(35)*(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)*log( 
-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 
 9)) + 35*(4986*x^4 + 10647*x^3 + 7362*x^2 + 9107*x + 3966)*sqrt(3*x^2 + 2 
))/(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {176}{60025} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {277 \, x}{3430 \, \sqrt {3 \, x^{2} + 2}} + \frac {88}{1715 \, \sqrt {3 \, x^{2} + 2}} - \frac {17 \, x}{490 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{35 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {22}{147 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

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

Output:

176/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2* 
x + 3)) + 277/3430*x/sqrt(3*x^2 + 2) + 88/1715/sqrt(3*x^2 + 2) - 17/490*x/ 
(3*x^2 + 2)^(3/2) - 13/35/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) + 
22/147/(3*x^2 + 2)^(3/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (76) = 152\).

Time = 0.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.45 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1}{360150} \, \sqrt {35} {\left (277 \, \sqrt {35} \sqrt {3} - 1056 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {176 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{60025 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {\frac {\frac {\frac {7 \, {\left (\frac {4813}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {4368}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {53523}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {19269}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2493}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{10290 \, {\left (\frac {18}{2 \, x + 3} - \frac {35}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} \] Input:

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

Output:

-1/360150*sqrt(35)*(277*sqrt(35)*sqrt(3) - 1056*log(sqrt(35)*sqrt(3) - 9)) 
*sgn(1/(2*x + 3)) - 176/60025*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 
35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3)) + 1/10290* 
(((7*(4813/sgn(1/(2*x + 3)) + 4368/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) 
 - 53523/sgn(1/(2*x + 3)))/(2*x + 3) + 19269/sgn(1/(2*x + 3)))/(2*x + 3) - 
 2493/sgn(1/(2*x + 3)))/((18/(2*x + 3) - 35/(2*x + 3)^2 - 3)*sqrt(-18/(2*x 
 + 3) + 35/(2*x + 3)^2 + 3))
 

Mupad [B] (verification not implemented)

Time = 6.62 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.84 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {35}\,\left (3464\,\ln \left (x+\frac {3}{2}\right )-3464\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{1500625}+\frac {\sqrt {35}\,\left (\frac {1872\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {1872\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{70}-\frac {104\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(35^(1/2)*(3464*log(x + 3/2) - 3464*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^ 
(1/2))/9 - 4/9)))/1500625 + (35^(1/2)*((1872*log(x + 3/2))/42875 - (1872*l 
og(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875))/70 - (104*3^ 
(1/2)*(x^2 + 2/3)^(1/2))/(42875*(x + 3/2)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(( 
(6^(1/2)*597i)/19600 - 639/19600)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2 
)*199i)/9800 - 213/9800)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x 
^2 + 2/3)^(1/2)*(((6^(1/2)*597i)/19600 + 639/19600)/(x + (6^(1/2)*1i)/3) + 
 (6^(1/2)*((6^(1/2)*199i)/9800 + 213/9800)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)) 
)/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i - 41568)*(x^2 + 2/3)^(1/2)*1i)/(12 
348000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i + 41568)*( 
x^2 + 2/3)^(1/2)*1i)/(12348000*(x - (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.17 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {174510 \sqrt {3 x^{2}+2}\, x^{4}+372645 \sqrt {3 x^{2}+2}\, x^{3}+257670 \sqrt {3 x^{2}+2}\, x^{2}+318745 \sqrt {3 x^{2}+2}\, x +138810 \sqrt {3 x^{2}+2}+19008 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}+28512 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}+25344 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+38016 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+8448 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +12672 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-19008 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-28512 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-25344 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-38016 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-8448 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -12672 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{6482700 x^{5}+9724050 x^{4}+8643600 x^{3}[phone] x^{2}+2881200 x +4321800} \] Input:

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

Output:

(174510*sqrt(3*x**2 + 2)*x**4 + 372645*sqrt(3*x**2 + 2)*x**3 + 257670*sqrt 
(3*x**2 + 2)*x**2 + 318745*sqrt(3*x**2 + 2)*x + 138810*sqrt(3*x**2 + 2) + 
19008*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 28512*sqrt( 
35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 25344*sqrt(35)*log(sqr 
t(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 38016*sqrt(35)*log(sqrt(3*x**2 + 
2)*sqrt(35) + 9*x - 4)*x**2 + 8448*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) 
+ 9*x - 4)*x + 12672*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 1 
9008*sqrt(35)*log(2*x + 3)*x**5 - 28512*sqrt(35)*log(2*x + 3)*x**4 - 25344 
*sqrt(35)*log(2*x + 3)*x**3 - 38016*sqrt(35)*log(2*x + 3)*x**2 - 8448*sqrt 
(35)*log(2*x + 3)*x - 12672*sqrt(35)*log(2*x + 3))/(360150*(18*x**5 + 27*x 
**4 + 24*x**3 + 36*x**2 + 8*x + 12))