\(\int \frac {5-x}{(3+2 x)^3 (2+3 x^2)^{3/2}} \, dx\) [251]

Optimal result

Integrand size = 24, antiderivative size = 97 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {3 (654-331 x)}{17150 \sqrt {2+3 x^2}}-\frac {13}{70 (3+2 x)^2 \sqrt {2+3 x^2}}-\frac {103}{490 (3+2 x) \sqrt {2+3 x^2}}-\frac {1962 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8575 \sqrt {35}} \] Output:

3/17150*(654-331*x)/(3*x^2+2)^(1/2)-13/70/(3+2*x)^2/(3*x^2+2)^(1/2)-103/49 
0/(3+2*x)/(3*x^2+2)^(1/2)-1962/300125*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/ 
2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\frac {35 \left (3658+7397 x-4068 x^2-3972 x^3\right )}{(3+2 x)^2 \sqrt {2+3 x^2}}+7848 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{600250} \] Input:

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

Output:

((35*(3658 + 7397*x - 4068*x^2 - 3972*x^3))/((3 + 2*x)^2*Sqrt[2 + 3*x^2]) 
+ 7848*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt 
[35]])/600250
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {686, 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.

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}-\frac {1}{210} \int -\frac {12 (41 x+39)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{35} \int \frac {41 x+39}{(2 x+3)^3 \sqrt {3 x^2+2}}dx+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {2}{35} \left (\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}-\frac {1}{70} \int -\frac {5 (27 x+206)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \int \frac {27 x+206}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (\frac {1962}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (-\frac {1962}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2}{35} \left (\frac {1}{14} \left (-\frac {1962 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {331 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {9 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )+\frac {41 x+26}{70 (2 x+3)^2 \sqrt {3 x^2+2}}\)

Input:

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

Output:

(26 + 41*x)/(70*(3 + 2*x)^2*Sqrt[2 + 3*x^2]) + (2*((9*Sqrt[2 + 3*x^2])/(14 
*(3 + 2*x)^2) + ((-331*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (1962*ArcTanh[(4 
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35]))/14))/35
 

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])
 

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])
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {3972 x^{3}+4068 x^{2}-7397 x -3658}{17150 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {1962 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}\)	\(65\)
trager	\(-\frac {3972 x^{3}+4068 x^{2}-7397 x -3658}{17150 \left (2 x +3\right )^{2} \sqrt {3 x^{2}+2}}-\frac {1962 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{300125}\)	\(82\)
default	\(-\frac {13}{280 \left (x +\frac {3}{2}\right )^{2} \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {103}{980 \left (x +\frac {3}{2}\right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {981}{8575 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {993 x}{17150 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {1962 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{300125}\)	\(107\)

Input:

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

Output:

-1/17150*(3972*x^3+4068*x^2-7397*x-3658)/(2*x+3)^2/(3*x^2+2)^(1/2)-1962/30 
0125*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 = 119, normalized size of antiderivative = 1.23 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962 \, \sqrt {35} {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (3972 \, x^{3} + 4068 \, x^{2} - 7397 \, x - 3658\right )} \sqrt {3 \, x^{2} + 2}}{600250 \, {\left (12 \, x^{4} + 36 \, x^{3} + 35 \, x^{2} + 24 \, x + 18\right )}} \] Input:

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

Output:

1/600250*(1962*sqrt(35)*(12*x^4 + 36*x^3 + 35*x^2 + 24*x + 18)*log(-(sqrt( 
35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 
35*(3972*x^3 + 4068*x^2 - 7397*x - 3658)*sqrt(3*x^2 + 2))/(12*x^4 + 36*x^3 
 + 35*x^2 + 24*x + 18)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.32 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962}{300125} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {993 \, x}{17150 \, \sqrt {3 \, x^{2} + 2}} + \frac {981}{8575 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{70 \, {\left (4 \, \sqrt {3 \, x^{2} + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 2} x + 9 \, \sqrt {3 \, x^{2} + 2}\right )}} - \frac {103}{490 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\right )}} \] Input:

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

Output:

1962/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs( 
2*x + 3)) - 993/17150*x/sqrt(3*x^2 + 2) + 981/8575/sqrt(3*x^2 + 2) - 13/70 
/(4*sqrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 103/ 
490/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (78) = 156\).

Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.05 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962}{300125} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, {\left (157 \, x - 1478\right )}}{85750 \, \sqrt {3 \, x^{2} + 2}} - \frac {768 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 2461 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 6168 \, \sqrt {3} x + 856 \, \sqrt {3} + 6168 \, \sqrt {3 \, x^{2} + 2}}{6125 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \] Input:

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

Output:

1962/300125*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 
/85750*(157*x - 1478)/sqrt(3*x^2 + 2) - 1/6125*(768*(sqrt(3)*x - sqrt(3*x^ 
2 + 2))^3 + 2461*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 6168*sqrt(3)*x 
+ 856*sqrt(3) + 6168*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)^2
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.87 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1962\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{300125}-\frac {1962\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{300125}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {157\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {107\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2450\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,739{}\mathrm {i}}{171500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(1962*35^(1/2)*log(x + 3/2))/300125 - (1962*35^(1/2)*log(x - (3^(1/2)*35^( 
1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/300125 - (157*3^(1/2)*(x^2 + 2/3)^(1/2)) 
/(171500*(x - (6^(1/2)*1i)/3)) - (157*3^(1/2)*(x^2 + 2/3)^(1/2))/(171500*( 
x + (6^(1/2)*1i)/3)) - (107*3^(1/2)*(x^2 + 2/3)^(1/2))/(6125*(x + 3/2)) - 
(13*3^(1/2)*(x^2 + 2/3)^(1/2))/(2450*(3*x + x^2 + 9/4)) - (3^(1/2)*6^(1/2) 
*(x^2 + 2/3)^(1/2)*739i)/(171500*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)* 
(x^2 + 2/3)^(1/2)*739i)/(171500*(x + (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.54 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-139020 \sqrt {3 x^{2}+2}\, x^{3}-142380 \sqrt {3 x^{2}+2}\, x^{2}+258895 \sqrt {3 x^{2}+2}\, x +128030 \sqrt {3 x^{2}+2}+47088 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}+141264 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+137340 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+94176 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +70632 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-47088 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-141264 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-137340 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-94176 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -70632 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{7203000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 139020*sqrt(3*x**2 + 2)*x**3 - 142380*sqrt(3*x**2 + 2)*x**2 + 258895*s 
qrt(3*x**2 + 2)*x + 128030*sqrt(3*x**2 + 2) + 47088*sqrt(35)*log(sqrt(3*x* 
*2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 141264*sqrt(35)*log(sqrt(3*x**2 + 2)*sq 
rt(35) + 9*x - 4)*x**3 + 137340*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9 
*x - 4)*x**2 + 94176*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 
 70632*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 47088*sqrt(35)* 
log(2*x + 3)*x**4 - 141264*sqrt(35)*log(2*x + 3)*x**3 - 137340*sqrt(35)*lo 
g(2*x + 3)*x**2 - 94176*sqrt(35)*log(2*x + 3)*x - 70632*sqrt(35)*log(2*x + 
 3))/(600250*(12*x**4 + 36*x**3 + 35*x**2 + 24*x + 18))