Integrand size = 24, antiderivative size = 153 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {26+41 x}{210 (3+2 x)^3 \left (2+3 x^2\right )^{3/2}}-\frac {114-3331 x}{7350 (3+2 x)^3 \sqrt {2+3 x^2}}+\frac {1471 \sqrt {2+3 x^2}}{18375 (3+2 x)^3}+\frac {541 \sqrt {2+3 x^2}}{42875 (3+2 x)^2}-\frac {5987 \sqrt {2+3 x^2}}{1500625 (3+2 x)}-\frac {55344 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1500625 \sqrt {35}} \]
1/210*(26+41*x)/(3+2*x)^3/(3*x^2+2)^(3/2)-55344/52521875*arctanh(1/35*(4-9 *x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/7350*(-114+3331*x)/(3+2*x)^3/(3*x ^2+2)^(1/2)+1471/18375*(3*x^2+2)^(1/2)/(3+2*x)^3+541/42875*(3*x^2+2)^(1/2) /(3+2*x)^2-5987/1500625*(3*x^2+2)^(1/2)/(3+2*x)
Time = 0.77 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\frac {35 \left (3788738+9103449 x+7866162 x^2+9795297 x^3+4920642 x^4-1834596 x^5-1293192 x^6\right )}{(3+2 x)^3 \left (2+3 x^2\right )^{3/2}}+664128 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{315131250} \]
((35*(3788738 + 9103449*x + 7866162*x^2 + 9795297*x^3 + 4920642*x^4 - 1834 596*x^5 - 1293192*x^6))/((3 + 2*x)^3*(2 + 3*x^2)^(3/2)) + 664128*Sqrt[35]* ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/315131250
Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {686, 27, 686, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^4 \left (3 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}-\frac {1}{630} \int -\frac {6 (205 x+279)}{(2 x+3)^4 \left (3 x^2+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \int \frac {205 x+279}{(2 x+3)^4 \left (3 x^2+2\right )^{3/2}}dx+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {1}{105} \left (-\frac {1}{210} \int \frac {18 (152-3331 x)}{(2 x+3)^4 \sqrt {3 x^2+2}}dx-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \int \frac {152-3331 x}{(2 x+3)^4 \sqrt {3 x^2+2}}dx-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {1}{105} \int \frac {42 (1471 x+854)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \int \frac {1471 x+854}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \left (\frac {541 \sqrt {3 x^2+2}}{14 (2 x+3)^2}-\frac {1}{70} \int -\frac {5 (1623 x+5428)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx\right )-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \left (\frac {1}{14} \int \frac {1623 x+5428}{(2 x+3)^2 \sqrt {3 x^2+2}}dx+\frac {541 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \left (\frac {1}{14} \left (\frac {55344}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {5987 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {541 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \left (\frac {1}{14} \left (-\frac {55344}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {5987 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {541 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{105} \left (-\frac {3}{35} \left (-\frac {2}{5} \left (\frac {1}{14} \left (-\frac {55344 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {5987 \sqrt {3 x^2+2}}{35 (2 x+3)}\right )+\frac {541 \sqrt {3 x^2+2}}{14 (2 x+3)^2}\right )-\frac {1471 \sqrt {3 x^2+2}}{15 (2 x+3)^3}\right )-\frac {114-3331 x}{70 (2 x+3)^3 \sqrt {3 x^2+2}}\right )+\frac {41 x+26}{210 (2 x+3)^3 \left (3 x^2+2\right )^{3/2}}\) |
(26 + 41*x)/(210*(3 + 2*x)^3*(2 + 3*x^2)^(3/2)) + (-1/70*(114 - 3331*x)/(( 3 + 2*x)^3*Sqrt[2 + 3*x^2]) - (3*((-1471*Sqrt[2 + 3*x^2])/(15*(3 + 2*x)^3) - (2*((541*Sqrt[2 + 3*x^2])/(14*(3 + 2*x)^2) + ((-5987*Sqrt[2 + 3*x^2])/( 35*(3 + 2*x)) - (55344*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35* Sqrt[35]))/14))/5))/35)/105
3.15.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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])
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])
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {1293192 x^{6}+1834596 x^{5}-4920642 x^{4}-9795297 x^{3}-7866162 x^{2}-9103449 x -3788738}{9003750 \left (3+2 x \right )^{3} \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {55344 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{52521875}\) | \(80\) |
| trager | \(-\frac {1293192 x^{6}+1834596 x^{5}-4920642 x^{4}-9795297 x^{3}-7866162 x^{2}-9103449 x -3788738}{9003750 \left (3+2 x \right )^{3} \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {55344 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{52521875}\) | \(96\) |
| default | \(-\frac {79}{2450 \left (x +\frac {3}{2}\right )^{2} \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {516}{6125 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {2306}{42875 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {4071 x}{85750 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {17961 x}{3001250 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {27672}{1500625 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {55344 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{52521875}-\frac {13}{840 \left (x +\frac {3}{2}\right )^{3} \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(161\) |
-1/9003750*(1293192*x^6+1834596*x^5-4920642*x^4-9795297*x^3-7866162*x^2-91 03449*x-3788738)/(3+2*x)^3/(3*x^2+2)^(3/2)-55344/52521875*35^(1/2)*arctanh (2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.36 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {166032 \, \sqrt {35} {\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (1293192 \, x^{6} + 1834596 \, x^{5} - 4920642 \, x^{4} - 9795297 \, x^{3} - 7866162 \, x^{2} - 9103449 \, x - 3788738\right )} \sqrt {3 \, x^{2} + 2}}{315131250 \, {\left (72 \, x^{7} + 324 \, x^{6} + 582 \, x^{5} + 675 \, x^{4} + 680 \, x^{3} + 468 \, x^{2} + 216 \, x + 108\right )}} \]
1/315131250*(166032*sqrt(35)*(72*x^7 + 324*x^6 + 582*x^5 + 675*x^4 + 680*x ^3 + 468*x^2 + 216*x + 108)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93* x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(1293192*x^6 + 1834596*x^5 - 492 0642*x^4 - 9795297*x^3 - 7866162*x^2 - 9103449*x - 3788738)*sqrt(3*x^2 + 2 ))/(72*x^7 + 324*x^6 + 582*x^5 + 675*x^4 + 680*x^3 + 468*x^2 + 216*x + 108 )
Timed out. \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.35 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {55344}{52521875} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {17961 \, x}{3001250 \, \sqrt {3 \, x^{2} + 2}} + \frac {27672}{1500625 \, \sqrt {3 \, x^{2} + 2}} - \frac {4071 \, x}{85750 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{105 \, {\left (8 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + 36 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {158}{1225 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1032}{6125 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {2306}{42875 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]
55344/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/a bs(2*x + 3)) - 17961/3001250*x/sqrt(3*x^2 + 2) + 27672/1500625/sqrt(3*x^2 + 2) - 4071/85750*x/(3*x^2 + 2)^(3/2) - 13/105/(8*(3*x^2 + 2)^(3/2)*x^3 + 36*(3*x^2 + 2)^(3/2)*x^2 + 54*(3*x^2 + 2)^(3/2)*x + 27*(3*x^2 + 2)^(3/2)) - 158/1225/(4*(3*x^2 + 2)^(3/2)*x^2 + 12*(3*x^2 + 2)^(3/2)*x + 9*(3*x^2 + 2)^(3/2)) - 1032/6125/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2)) + 2306 /42875/(3*x^2 + 2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.68 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {55344}{52521875} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {9 \, {\left ({\left (49879 \, x + 344464\right )} x - 6729\right )} x + 2510374}{105043750 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {8 \, \sqrt {3} {\left (37652 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 695865 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 729630 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 3472470 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 1016800 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 259424\right )}}{52521875 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \]
55344/52521875*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*s qrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/105043750*(9*((49879*x + 344464)*x - 6729)*x + 2510374)/(3*x^2 + 2)^(3 /2) - 8/52521875*sqrt(3)*(37652*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 695865*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 729630*sqrt(3)*(sqrt(3)*x - sqrt( 3*x^2 + 2))^3 - 3472470*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 1016800*sqrt(3)* (sqrt(3)*x - sqrt(3*x^2 + 2)) - 259424)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^3
Time = 10.54 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.16 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {55344\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{52521875}-\frac {55344\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{52521875}-\frac {6337\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{36015000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {49879\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {49879\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{210087500\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {6337\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{36015000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {129712\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{52521875\,\left (x+\frac {3}{2}\right )}-\frac {1256\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1500625\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128625\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,3427{}\mathrm {i}}{72030000\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,2288579{}\mathrm {i}}{2521050000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,2288579{}\mathrm {i}}{2521050000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,3427{}\mathrm {i}}{72030000\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \]
(55344*35^(1/2)*log(x + 3/2))/52521875 - (55344*35^(1/2)*log(x - (3^(1/2)* 35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/52521875 - (6337*3^(1/2)*(x^2 + 2/3) ^(1/2))/(36015000*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (49879*3^(1/2)*(x^2 + 2/3)^(1/2))/(210087500*(x - (6^(1/2)*1i)/3)) + (49879*3^(1/2)*(x^2 + 2/3)^ (1/2))/(210087500*(x + (6^(1/2)*1i)/3)) + (6337*3^(1/2)*(x^2 + 2/3)^(1/2)) /(36015000*((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - (129712*3^(1/2)*(x^2 + 2/3)^( 1/2))/(52521875*(x + 3/2)) - (1256*3^(1/2)*(x^2 + 2/3)^(1/2))/(1500625*(3* x + x^2 + 9/4)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(128625*((27*x)/4 + (9*x^ 2)/2 + x^3 + 27/8)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*3427i)/(72030000* ((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*22885 79i)/(2521050000*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2 )*2288579i)/(2521050000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(x^2 + 2/ 3)^(1/2)*3427i)/(72030000*((6^(1/2)*x*2i)/3 - x^2 + 2/3))