Integrand size = 24, antiderivative size = 153 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=-\frac {24201 (4-9 x) \sqrt {2+3 x^2}}{210087500 (3+2 x)^2}-\frac {2689 (4-9 x) \left (2+3 x^2\right )^{3/2}}{6002500 (3+2 x)^4}-\frac {13 \left (2+3 x^2\right )^{5/2}}{245 (3+2 x)^7}-\frac {404 \left (2+3 x^2\right )^{5/2}}{25725 (3+2 x)^6}-\frac {822 \left (2+3 x^2\right )^{5/2}}{214375 (3+2 x)^5}-\frac {72603 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{105043750 \sqrt {35}} \]
-2689/6002500*(4-9*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-13/245*(3*x^2+2)^(5/2)/(3+ 2*x)^7-404/25725*(3*x^2+2)^(5/2)/(3+2*x)^6-822/214375*(3*x^2+2)^(5/2)/(3+2 *x)^5-72603/3676531250*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^( 1/2)-24201/210087500*(4-9*x)*(3*x^2+2)^(1/2)/(3+2*x)^2
Time = 1.69 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (471103116+256388969 x+740031210 x^2-98810025 x^3+148868010 x^4+44301924 x^5+5104296 x^6\right )}{(3+2 x)^7}+871236 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{22059187500} \]
((-35*Sqrt[2 + 3*x^2]*(471103116 + 256388969*x + 740031210*x^2 - 98810025* x^3 + 148868010*x^4 + 44301924*x^5 + 5104296*x^6))/(3 + 2*x)^7 + 871236*Sq rt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/22 059187500
Time = 0.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {688, 25, 688, 27, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^8} \, dx\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {1}{245} \int -\frac {(287-78 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{245} \int \frac {(287-78 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{245} \left (-\frac {1}{210} \int -\frac {6 (2271-404 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \int \frac {(2271-404 x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{245} \left (\frac {1}{35} \left (\frac {2689}{5} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {822 \left (3 x^2+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {404 \left (3 x^2+2\right )^{5/2}}{105 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+2\right )^{5/2}}{245 (2 x+3)^7}\) |
(-13*(2 + 3*x^2)^(5/2))/(245*(3 + 2*x)^7) + ((-404*(2 + 3*x^2)^(5/2))/(105 *(3 + 2*x)^6) + ((-822*(2 + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (2689*(-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))/5)/35)/245
3.14.80.3.1 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[((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]
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[(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 = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(-\frac {15312888 x^{8}+132905772 x^{7}+456812622 x^{6}-207826227 x^{5}+2517829650 x^{4}+571546857 x^{3}+2893371768 x^{2}+512777938 x +942206232}{630262500 \left (3+2 x \right )^{7} \sqrt {3 x^{2}+2}}-\frac {72603 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3676531250}\) | \(90\) |
| trager | \(-\frac {\left (5104296 x^{6}+44301924 x^{5}+148868010 x^{4}-98810025 x^{3}+740031210 x^{2}+256388969 x +471103116\right ) \sqrt {3 x^{2}+2}}{630262500 \left (3+2 x \right )^{7}}-\frac {72603 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{3676531250}\) | \(96\) |
| default | \(-\frac {101 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{411600 \left (x +\frac {3}{2}\right )^{6}}-\frac {411 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{3430000 \left (x +\frac {3}{2}\right )^{5}}-\frac {2689 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{48020000 \left (x +\frac {3}{2}\right )^{4}}-\frac {24201 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{840350000 \left (x +\frac {3}{2}\right )^{3}}-\frac {250077 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{14706125000 \left (x +\frac {3}{2}\right )^{2}}-\frac {2831517 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{257357187500 \left (x +\frac {3}{2}\right )}+\frac {96804 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{64339296875}+\frac {653427 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{7353062500}+\frac {72603 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{3676531250}-\frac {72603 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{3676531250}+\frac {8494551 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{257357187500}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{31360 \left (x +\frac {3}{2}\right )^{7}}\) | \(245\) |
-1/630262500*(15312888*x^8+132905772*x^7+456812622*x^6-207826227*x^5+25178 29650*x^4+571546857*x^3+2893371768*x^2+512777938*x+942206232)/(3+2*x)^7/(3 *x^2+2)^(1/2)-72603/3676531250*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12* (x+3/2)^2-36*x-19)^(1/2))
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {217809 \, \sqrt {35} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (5104296 \, x^{6} + 44301924 \, x^{5} + 148868010 \, x^{4} - 98810025 \, x^{3} + 740031210 \, x^{2} + 256388969 \, x + 471103116\right )} \sqrt {3 \, x^{2} + 2}}{22059187500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
1/22059187500*(217809*sqrt(35)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*( 9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(5104296*x^6 + 443 01924*x^5 + 148868010*x^4 - 98810025*x^3 + 740031210*x^2 + 256388969*x + 4 71103116)*sqrt(3*x^2 + 2))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22 680*x^3 + 20412*x^2 + 10206*x + 2187)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (126) = 252\).
Time = 0.28 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.96 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {750231}{14706125000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{245 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {404 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{25725 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {822 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{214375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {2689 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3001250 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {24201 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{105043750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {250077 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3676531250 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {653427}{7353062500} \, \sqrt {3 \, x^{2} + 2} x + \frac {72603}{3676531250} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {72603}{1838265625} \, \sqrt {3 \, x^{2} + 2} - \frac {2831517 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{14706125000 \, {\left (2 \, x + 3\right )}} \]
750231/14706125000*(3*x^2 + 2)^(3/2) - 13/245*(3*x^2 + 2)^(5/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 404/25725*(3*x^2 + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4 860*x^2 + 2916*x + 729) - 822/214375*(3*x^2 + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 2689/3001250*(3*x^2 + 2)^(5/2)/(16*x^ 4 + 96*x^3 + 216*x^2 + 216*x + 81) - 24201/105043750*(3*x^2 + 2)^(5/2)/(8* x^3 + 36*x^2 + 54*x + 27) - 250077/3676531250*(3*x^2 + 2)^(5/2)/(4*x^2 + 1 2*x + 9) + 653427/7353062500*sqrt(3*x^2 + 2)*x + 72603/3676531250*sqrt(35) *arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 72603/18 38265625*sqrt(3*x^2 + 2) - 2831517/14706125000*(3*x^2 + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (126) = 252\).
Time = 0.30 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.67 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {72603}{3676531250} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, {\left (258144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} + 5033808 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} + 225898166 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 26360013 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 555459995 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 2679767547 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 4252091247 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 6029804778 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 11677158028 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 7324195080 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 2245361152 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 675266496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 174039168 \, \sqrt {3} x - 6049536 \, \sqrt {3} - 174039168 \, \sqrt {3 \, x^{2} + 2}\right )}}{3361400000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{7}} \]
72603/3676531250*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)) ) - 9/3361400000*(258144*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 5033808*sqrt(3 )*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 + 225898166*(sqrt(3)*x - sqrt(3*x^2 + 2 ))^11 + 26360013*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 555459995*(sqr t(3)*x - sqrt(3*x^2 + 2))^9 - 2679767547*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 4252091247*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 6029804778*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 11677158028*(sqrt(3)*x - sqrt(3*x^2 + 2)) ^5 - 7324195080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 2245361152*(sqrt (3)*x - sqrt(3*x^2 + 2))^3 - 675266496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 ))^2 + 174039168*sqrt(3)*x - 6049536*sqrt(3) - 174039168*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)^7
Time = 0.14 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.78 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {72603\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{3676531250}-\frac {72603\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{3676531250}+\frac {92453\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{21952000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {507\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {212679\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3361400000\,\left (x+\frac {3}{2}\right )}+\frac {125\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2688\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {3897\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{192080000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {65\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2048\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {7569\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{54880000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(72603*35^(1/2)*log(x + 3/2))/3676531250 - (72603*35^(1/2)*log(x - (3^(1/2 )*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/3676531250 + (92453*3^(1/2)*(x^2 + 2/3)^(1/2))/(21952000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (5 07*3^(1/2)*(x^2 + 2/3)^(1/2))/(19600*((405*x)/16 + (135*x^2)/4 + (45*x^3)/ 2 + (15*x^4)/2 + x^5 + 243/32)) - (212679*3^(1/2)*(x^2 + 2/3)^(1/2))/(3361 400000*(x + 3/2)) + (125*3^(1/2)*(x^2 + 2/3)^(1/2))/(2688*((729*x)/16 + (1 215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (3897*3 ^(1/2)*(x^2 + 2/3)^(1/2))/(192080000*(3*x + x^2 + 9/4)) - (65*3^(1/2)*(x^2 + 2/3)^(1/2))/(2048*((5103*x)/64 + (5103*x^2)/32 + (2835*x^3)/16 + (945*x ^4)/8 + (189*x^5)/4 + (21*x^6)/2 + x^7 + 2187/128)) + (7569*3^(1/2)*(x^2 + 2/3)^(1/2))/(54880000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))