Integrand size = 27, antiderivative size = 199 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=-\frac {153927 (7+8 x) \sqrt {2+5 x+3 x^2}}{32000000 (3+2 x)^2}+\frac {51309 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{800000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{40 (3+2 x)^8}-\frac {19 \left (2+5 x+3 x^2\right )^{5/2}}{50 (3+2 x)^7}-\frac {717 \left (2+5 x+3 x^2\right )^{5/2}}{2000 (3+2 x)^6}-\frac {3879 \left (2+5 x+3 x^2\right )^{5/2}}{12500 (3+2 x)^5}+\frac {153927 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{64000000 \sqrt {5}} \]
51309/800000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-13/40*(3*x^2+5*x+2)^(5/ 2)/(3+2*x)^8-19/50*(3*x^2+5*x+2)^(5/2)/(3+2*x)^7-717/2000*(3*x^2+5*x+2)^(5 /2)/(3+2*x)^6-3879/12500*(3*x^2+5*x+2)^(5/2)/(3+2*x)^5+153927/320000000*ar ctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-153927/32000000*(7 +8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.62 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.47 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (131091161+512781828 x+924451956 x^2+1007243840 x^3+682163760 x^4+272314944 x^5+60161472 x^6+5681664 x^7\right )}{(3+2 x)^8}+153927 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{160000000} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(131091161 + 512781828*x + 924451956*x^2 + 1007 243840*x^3 + 682163760*x^4 + 272314944*x^5 + 60161472*x^6 + 5681664*x^7))/ (3 + 2*x)^8 + 153927*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/1 60000000
Time = 0.43 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1237, 27, 1237, 27, 1237, 27, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^9} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{40} \int -\frac {(181-234 x) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)^8}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \int \frac {(181-234 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^8}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{80} \left (-\frac {1}{35} \int -\frac {21 (261-304 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \int \frac {(261-304 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^7}dx-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (-\frac {1}{30} \int -\frac {9 (1007-478 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \int \frac {(1007-478 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \left (\frac {5701}{5} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {3448 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \left (\frac {5701}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx\right )-\frac {3448 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \left (\frac {5701}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {3448 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \left (\frac {5701}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )\right )-\frac {3448 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{80} \left (\frac {3}{5} \left (\frac {3}{10} \left (\frac {5701}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )\right )-\frac {3448 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {239 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^6}\right )-\frac {152 \left (3 x^2+5 x+2\right )^{5/2}}{5 (2 x+3)^7}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{40 (2 x+3)^8}\) |
(-13*(2 + 5*x + 3*x^2)^(5/2))/(40*(3 + 2*x)^8) + ((-152*(2 + 5*x + 3*x^2)^ (5/2))/(5*(3 + 2*x)^7) + (3*((-239*(2 + 5*x + 3*x^2)^(5/2))/(5*(3 + 2*x)^6 ) + (3*((-3448*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (5701*(((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*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80))/5))/10))/5)/80
3.25.31.3.1 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 = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {17044992 x^{9}+208892736 x^{8}+1129115520 x^{7}+3528388944 x^{6}+6977180208 x^{5}+9173902588 x^{4}+8175092944 x^{3}+4806086535 x^{2}+1681019461 x +262182322}{32000000 \left (3+2 x \right )^{8} \sqrt {3 x^{2}+5 x +2}}-\frac {153927 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{320000000}\) | \(98\) |
| trager | \(-\frac {\left (5681664 x^{7}+60161472 x^{6}+272314944 x^{5}+682163760 x^{4}+1007243840 x^{3}+924451956 x^{2}+512781828 x +131091161\right ) \sqrt {3 x^{2}+5 x +2}}{32000000 \left (3+2 x \right )^{8}}+\frac {153927 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{320000000}\) | \(107\) |
| default | \(-\frac {19 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{6400 \left (x +\frac {3}{2}\right )^{7}}-\frac {717 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{128000 \left (x +\frac {3}{2}\right )^{6}}-\frac {3879 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{400000 \left (x +\frac {3}{2}\right )^{5}}-\frac {51309 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{3200000 \left (x +\frac {3}{2}\right )^{4}}-\frac {51309 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{2000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1590579 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{40000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {1487961 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{50000000}-\frac {1487961 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{25000000 \left (x +\frac {3}{2}\right )}-\frac {153927 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{40000000}-\frac {153927 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{320000000}+\frac {51309 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{200000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{10240 \left (x +\frac {3}{2}\right )^{8}}+\frac {153927 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{320000000}\) | \(274\) |
-1/32000000*(17044992*x^9+208892736*x^8+1129115520*x^7+3528388944*x^6+6977 180208*x^5+9173902588*x^4+8175092944*x^3+4806086535*x^2+1681019461*x+26218 2322)/(3+2*x)^8/(3*x^2+5*x+2)^(1/2)-153927/320000000*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 = 185, normalized size of antiderivative = 0.93 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=\frac {153927 \, \sqrt {5} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (5681664 \, x^{7} + 60161472 \, x^{6} + 272314944 \, x^{5} + 682163760 \, x^{4} + 1007243840 \, x^{3} + 924451956 \, x^{2} + 512781828 \, x + 131091161\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{640000000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]
1/640000000*(153927*sqrt(5)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*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 *(5681664*x^7 + 60161472*x^6 + 272314944*x^5 + 682163760*x^4 + 1007243840* x^3 + 924451956*x^2 + 512781828*x + 131091161)*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)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{512 x^{9} + 6912 x^{8} + 41472 x^{7} + 145152 x^{6} + 326592 x^{5} + 489888 x^{4} + 489888 x^{3} + 314928 x^{2} + 118098 x + 19683}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 1180 98*x + 19683), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912 *x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x) - Integral(-10*x**2*sqrt(3*x**2 + 5 *x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152*x**6 + 326592*x**5 + 4 89888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x) - Integral( 3*x**3*sqrt(3*x**2 + 5*x + 2)/(512*x**9 + 6912*x**8 + 41472*x**7 + 145152* x**6 + 326592*x**5 + 489888*x**4 + 489888*x**3 + 314928*x**2 + 118098*x + 19683), x)
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (165) = 330\).
Time = 0.29 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=\frac {4771737}{40000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{40 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {19 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {717 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{2000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {3879 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{12500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {51309 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{200000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {51309 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{250000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1590579 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {461781}{20000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {153927}{320000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {2924613}{160000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1487961 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{10000000 \, {\left (2 \, x + 3\right )}} \]
4771737/40000000*(3*x^2 + 5*x + 2)^(3/2) - 13/40*(3*x^2 + 5*x + 2)^(5/2)/( 256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 8164 8*x^2 + 34992*x + 6561) - 19/50*(3*x^2 + 5*x + 2)^(5/2)/(128*x^7 + 1344*x^ 6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 717/2 000*(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) - 3879/12500*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^ 4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 51309/200000*(3*x^2 + 5*x + 2)^(5/ 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 51309/250000*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1590579/10000000*(3*x^2 + 5*x + 2) ^(5/2)/(4*x^2 + 12*x + 9) - 461781/20000000*sqrt(3*x^2 + 5*x + 2)*x - 1539 27/320000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/ abs(2*x + 3) - 2) - 2924613/160000000*sqrt(3*x^2 + 5*x + 2) - 1487961/1000 0000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (165) = 330\).
Time = 0.32 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.57 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=\frac {153927}{320000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {19702656 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 443309760 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 13775440320 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 88813739520 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 1135723030560 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 3326100961968 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 20795205897360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 31719485197440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 108381222834920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 93303707056820 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 182905948708404 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 90199904722080 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 98616726439110 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 25302796273485 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 12323187970155 \, \sqrt {3} x + 954490882968 \, \sqrt {3} - 12323187970155 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{32000000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{8}} \]
153927/320000000*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))) - 1/32000000*(19702656*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2 ))^15 + 443309760*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 1377544 0320*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 88813739520*sqrt(3)*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^12 + 1135723030560*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2))^11 + 3326100961968*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 20795205897360*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 31719485197440*s qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 108381222834920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 93303707056820*sqrt(3)*(sqrt(3)*x - sqrt(3*x^ 2 + 5*x + 2))^6 + 182905948708404*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 90199904722080*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 98616726439 110*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 25302796273485*sqrt(3)*(sqrt(3 )*x - sqrt(3*x^2 + 5*x + 2))^2 + 12323187970155*sqrt(3)*x + 954490882968*s qrt(3) - 12323187970155*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^8
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^9} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^9} \,d x \]