3.15.7 \(\int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx\) [1407]

3.15.7.1 Optimal result

Integrand size = 24, antiderivative size = 143 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=-\frac {13 \sqrt {2+3 x^2}}{175 (3+2 x)^5}-\frac {439 \sqrt {2+3 x^2}}{12250 (3+2 x)^4}-\frac {797 \sqrt {2+3 x^2}}{61250 (3+2 x)^3}-\frac {1611 \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {10023 \sqrt {2+3 x^2}}{15006250 (3+2 x)}+\frac {19737 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{7503125 \sqrt {35}} \]

output

19737/262609375*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-13 
/175*(3*x^2+2)^(1/2)/(3+2*x)^5-439/12250*(3*x^2+2)^(1/2)/(3+2*x)^4-797/612 
50*(3*x^2+2)^(1/2)/(3+2*x)^3-1611/428750*(3*x^2+2)^(1/2)/(3+2*x)^2-10023/1 
5006250*(3*x^2+2)^(1/2)/(3+2*x)
 

3.15.7.2 Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (3409859+4314244 x+2487944 x^2+706644 x^3+80184 x^4\right )}{(3+2 x)^5}-39474 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{262609375} \]

input

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

output

((-35*Sqrt[2 + 3*x^2]*(3409859 + 4314244*x + 2487944*x^2 + 706644*x^3 + 80 
184*x^4))/(3 + 2*x)^5 - 39474*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 
2*Sqrt[2 + 3*x^2])/Sqrt[35]])/262609375
 

3.15.7.3 Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {688, 25, 688, 27, 688, 27, 688, 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.

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

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{175} \int -\frac {205-156 x}{(2 x+3)^5 \sqrt {3 x^2+2}}dx-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{175} \int \frac {205-156 x}{(2 x+3)^5 \sqrt {3 x^2+2}}dx-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{175} \left (-\frac {1}{140} \int -\frac {6 (814-1317 x)}{(2 x+3)^4 \sqrt {3 x^2+2}}dx-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \int \frac {814-1317 x}{(2 x+3)^4 \sqrt {3 x^2+2}}dx-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (-\frac {1}{105} \int -\frac {42 (147-797 x)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \int \frac {147-797 x}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \left (-\frac {1}{70} \int \frac {5 (1611 x+746)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {537 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \left (-\frac {1}{14} \int \frac {1611 x+746}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {537 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \left (\frac {1}{14} \left (-\frac {13158}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {3341 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {537 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {13158}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {3341 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {537 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{175} \left (\frac {3}{70} \left (\frac {2}{5} \left (\frac {1}{14} \left (\frac {13158 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {3341 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )-\frac {537 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {797 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {439 \sqrt {3 x^2+2}}{70 (2 x+3)^4}\right )-\frac {13 \sqrt {3 x^2+2}}{175 (2 x+3)^5}\)

input

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

output

(-13*Sqrt[2 + 3*x^2])/(175*(3 + 2*x)^5) + ((-439*Sqrt[2 + 3*x^2])/(70*(3 + 
 2*x)^4) + (3*((-797*Sqrt[2 + 3*x^2])/(15*(3 + 2*x)^3) + (2*((-537*Sqrt[2 
+ 3*x^2])/(14*(3 + 2*x)^2) + ((-3341*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) + (13 
158*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35]))/14))/5)) 
/70)/175
 

3.15.7.3.1 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 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])
 

3.15.7.4 Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56

method	result	size
risch	\(-\frac {240552 x^{6}+2119932 x^{5}+7624200 x^{4}[phone] x^{3}[phone] x^{2}+8628488 x +6819718}{7503125 \left (3+2 x \right )^{5} \sqrt {3 x^{2}+2}}+\frac {19737 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{262609375}\)	\(80\)
trager	\(-\frac {\left (80184 x^{4}+706644 x^{3}+2487944 x^{2}+4314244 x +3409859\right ) \sqrt {3 x^{2}+2}}{7503125 \left (3+2 x \right )^{5}}+\frac {19737 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{262609375}\)	\(86\)
default	\(-\frac {439 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{196000 \left (x +\frac {3}{2}\right )^{4}}-\frac {797 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{490000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1611 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{1715000 \left (x +\frac {3}{2}\right )^{2}}-\frac {10023 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{30012500 \left (x +\frac {3}{2}\right )}+\frac {19737 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{262609375}-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{5600 \left (x +\frac {3}{2}\right )^{5}}\)	\(137\)

input

int((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/7503125*(240552*x^6+2119932*x^5+7624200*x^4+14356020*x^3+15205465*x^2+8 
628488*x+6819718)/(3+2*x)^5/(3*x^2+2)^(1/2)+19737/262609375*35^(1/2)*arcta 
nh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

3.15.7.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=\frac {19737 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 70 \, {\left (80184 \, x^{4} + 706644 \, x^{3} + 2487944 \, x^{2} + 4314244 \, x + 3409859\right )} \sqrt {3 \, x^{2} + 2}}{525218750 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

input

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="fricas")
 

output

1/525218750*(19737*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)) - 70*(80184*x^4 + 706644*x^3 + 2487944*x^2 + 4314244*x + 3 
409859)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 
243)
 

3.15.7.6 Sympy [F(-1)]

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

input

integrate((5-x)/(3+2*x)**6/(3*x**2+2)**(1/2),x)
 

output

Timed out
 

3.15.7.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.22 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=-\frac {19737}{262609375} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {439 \, \sqrt {3 \, x^{2} + 2}}{12250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {797 \, \sqrt {3 \, x^{2} + 2}}{61250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1611 \, \sqrt {3 \, x^{2} + 2}}{428750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {10023 \, \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (2 \, x + 3\right )}} \]

input

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="maxima")
 

output

-19737/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6) 
/abs(2*x + 3)) - 13/175*sqrt(3*x^2 + 2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080 
*x^2 + 810*x + 243) - 439/12250*sqrt(3*x^2 + 2)/(16*x^4 + 96*x^3 + 216*x^2 
 + 216*x + 81) - 797/61250*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 
1611/428750*sqrt(3*x^2 + 2)/(4*x^2 + 12*x + 9) - 10023/15006250*sqrt(3*x^2 
 + 2)/(2*x + 3)
 

3.15.7.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (116) = 232\).

Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.25 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=-\frac {19737}{262609375} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, \sqrt {3} {\left (8772 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 355266 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 1773406 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 11098773 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 2315313 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 49794206 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 25535944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 16740688 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 1744032 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 213824\right )}}{30012500 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]

input

integrate((5-x)/(3+2*x)^6/(3*x^2+2)^(1/2),x, algorithm="giac")
 

output

-19737/262609375*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)) 
) + 3/30012500*sqrt(3)*(8772*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 355 
266*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 1773406*sqrt(3)*(sqrt(3)*x - sqrt(3* 
x^2 + 2))^7 + 11098773*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 2315313*sqrt(3)*( 
sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 49794206*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 
- 25535944*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 16740688*(sqrt(3)*x - 
 sqrt(3*x^2 + 2))^2 - 1744032*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 2138 
24)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 
 2)) - 2)^5
 

3.15.7.9 Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12 \[ \int \frac {5-x}{(3+2 x)^6 \sqrt {2+3 x^2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {1404}{42875\,\left (x+\frac {3}{2}\right )}+\frac {54}{1225\,{\left (x+\frac {3}{2}\right )}^2}+\frac {9}{175\,{\left (x+\frac {3}{2}\right )}^3}+\frac {3}{70\,{\left (x+\frac {3}{2}\right )}^4}\right )}{96}-\frac {\sqrt {35}\,\left (\frac {555984\,\ln \left (x+\frac {3}{2}\right )}{7503125}-\frac {555984\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7503125}\right )}{1120}-\frac {\sqrt {35}\,\left (\frac {216\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {216\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{560}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {972504}{7503125\,\left (x+\frac {3}{2}\right )}+\frac {57564}{214375\,{\left (x+\frac {3}{2}\right )}^2}+\frac {12714}{30625\,{\left (x+\frac {3}{2}\right )}^3}+\frac {3159}{6125\,{\left (x+\frac {3}{2}\right )}^4}+\frac {78}{175\,{\left (x+\frac {3}{2}\right )}^5}\right )}{192} \]

input

int(-(x - 5)/((2*x + 3)^6*(3*x^2 + 2)^(1/2)),x)
 

output

(3^(1/2)*(x^2 + 2/3)^(1/2)*(1404/(42875*(x + 3/2)) + 54/(1225*(x + 3/2)^2) 
 + 9/(175*(x + 3/2)^3) + 3/(70*(x + 3/2)^4)))/96 - (35^(1/2)*((555984*log( 
x + 3/2))/7503125 - (555984*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 
 - 4/9))/7503125))/1120 - (35^(1/2)*((216*log(x + 3/2))/42875 - (216*log(x 
 - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875))/560 - (3^(1/2)*( 
x^2 + 2/3)^(1/2)*(972504/(7503125*(x + 3/2)) + 57564/(214375*(x + 3/2)^2) 
+ 12714/(30625*(x + 3/2)^3) + 3159/(6125*(x + 3/2)^4) + 78/(175*(x + 3/2)^ 
5)))/192