Integrand size = 27, antiderivative size = 174 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=-\frac {4663 (7+8 x) \sqrt {2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac {4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac {433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac {4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac {4663 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1600000 \sqrt {5}} \]
4663/60000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-13/35*(3*x^2+5*x+2)^(5/2) /(3+2*x)^7-433/1050*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6-4892/13125*(3*x^2+5*x+2) ^(5/2)/(3+2*x)^5+4663/8000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^( 1/2))*5^(1/2)-4663/800000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.57 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (-6554463+15759118 x+64140640 x^2+55403520 x^3+16376240 x^4+2893088 x^5+191232 x^6\right )}{(3+2 x)^7}+97923 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{84000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(-6554463 + 15759118*x + 64140640*x^2 + 55403520 *x^3 + 16376240*x^4 + 2893088*x^5 + 191232*x^6))/(3 + 2*x)^7 + 97923*Sqrt[ 5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/84000000
Time = 0.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1237, 27, 1237, 25, 1228, 1152, 1152, 1154, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^8} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{35} \int -\frac {(199-156 x) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)^7}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{70} \int \frac {(199-156 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{70} \left (-\frac {1}{30} \int -\frac {(5887-2598 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \int \frac {(5887-2598 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \left (\frac {32641}{5} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {19568 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \left (\frac {32641}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx\right )-\frac {19568 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \left (\frac {32641}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {19568 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \left (\frac {32641}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )\right )-\frac {19568 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{70} \left (\frac {1}{30} \left (\frac {32641}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )\right )-\frac {19568 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {433 \left (3 x^2+5 x+2\right )^{5/2}}{15 (2 x+3)^6}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}\) |
(-13*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^7) + ((-433*(2 + 5*x + 3*x^2)^ (5/2))/(15*(3 + 2*x)^6) + ((-19568*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^ 5) + (32641*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(40*(3 + 2*x)^4) - (3*((( 7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[(7 + 8*x)/(2*Sq rt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80))/5)/30)/70
3.25.30.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 30.77 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {573696 x^{8}+9635424 x^{7}+63976624 x^{6}+253877936 x^{5}+502192000 x^{4}+478787594 x^{3}+187413481 x^{2}-1254079 x -13108926}{16800000 \left (3+2 x \right )^{7} \sqrt {3 x^{2}+5 x +2}}-\frac {4663 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{8000000}\) | \(93\) |
| trager | \(\frac {\left (191232 x^{6}+2893088 x^{5}+16376240 x^{4}+55403520 x^{3}+64140640 x^{2}+15759118 x -6554463\right ) \sqrt {3 x^{2}+5 x +2}}{16800000 \left (3+2 x \right )^{7}}-\frac {4663 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{8000000}\) | \(102\) |
| default | \(-\frac {433 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{67200 \left (x +\frac {3}{2}\right )^{6}}-\frac {1223 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{105000 \left (x +\frac {3}{2}\right )^{5}}-\frac {4663 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{240000 \left (x +\frac {3}{2}\right )^{4}}-\frac {4663 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{150000 \left (x +\frac {3}{2}\right )^{3}}-\frac {144553 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{3000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {135227 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1875000 \left (x +\frac {3}{2}\right )}+\frac {4663 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{15000000}-\frac {4663 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{1000000}+\frac {4663 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{8000000}-\frac {4663 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{8000000}+\frac {135227 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{3750000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}\) | \(253\) |
1/16800000*(573696*x^8+9635424*x^7+63976624*x^6+253877936*x^5+502192000*x^ 4+478787594*x^3+187413481*x^2-1254079*x-13108926)/(3+2*x)^7/(3*x^2+5*x+2)^ (1/2)-4663/8000000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16 *x-19)^(1/2))
Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {97923 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (191232 \, x^{6} + 2893088 \, x^{5} + 16376240 \, x^{4} + 55403520 \, x^{3} + 64140640 \, x^{2} + 15759118 \, x - 6554463\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{336000000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]
1/336000000*(97923*sqrt(5)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22 680*x^3 + 20412*x^2 + 10206*x + 2187)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2) *(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 20*(191232*x^6 + 2893088*x^5 + 16376240*x^4 + 55403520*x^3 + 64140640*x^2 + 15759118*x - 65 54463)*sqrt(3*x^2 + 5*x + 2))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128* x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561 ), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 161 28*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6 561), x)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (144) = 288\).
Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.94 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {144553}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {433 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{1050 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {4892 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{13125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{15000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {4663 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{18750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {144553 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{750000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {13989}{500000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {4663}{8000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {88597}{4000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {135227 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{750000 \, {\left (2 \, x + 3\right )}} \]
144553/1000000*(3*x^2 + 5*x + 2)^(3/2) - 13/35*(3*x^2 + 5*x + 2)^(5/2)/(12 8*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 433/1050*(3*x^2 + 5*x + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 4892/13125*(3*x^2 + 5*x + 2)^(5/2)/( 32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 4663/15000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 4663/18750*(3*x ^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 144553/750000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 13989/500000*sqrt(3*x^2 + 5*x + 2)*x - 4663/8000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2 /abs(2*x + 3) - 2) - 88597/4000000*sqrt(3*x^2 + 5*x + 2) - 135227/750000*( 3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (144) = 288\).
Time = 0.35 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.65 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=\frac {4663}{8000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {6267072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 122207904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 3852187808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 18344551344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 131374293680 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 134399090784 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 264419126976 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 1446858601104 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 6675760646156 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 5954681858370 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 10149146991914 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3640765552263 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 2268672558411 \, \sqrt {3} x - 208833935688 \, \sqrt {3} + 2268672558411 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16800000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]
4663/8000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt (3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/16800000*(6267072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 122207904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 3852187808*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 18344551344*sqrt(3)*(sqrt(3)*x - s qrt(3*x^2 + 5*x + 2))^10 + 131374293680*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2) )^9 + 134399090784*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 2644191 26976*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 1446858601104*sqrt(3)*(sqrt( 3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 6675760646156*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 5954681858370*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 10149146991914*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 3640765552263*sqr t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 2268672558411*sqrt(3)*x - 208 833935688*sqrt(3) + 2268672558411*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - s qrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^7
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^8} \,d x \]