Integrand size = 27, antiderivative size = 167 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {(12265+5718 x) \sqrt {2+5 x+3 x^2}}{512 (3+2 x)}-\frac {(3727+2898 x) \left (2+5 x+3 x^2\right )^{3/2}}{384 (3+2 x)^3}-\frac {(19+4 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^4}-\frac {1875}{256} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {29047 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024 \sqrt {5}} \]
-1/384*(3727+2898*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3-1/16*(19+4*x)*(3*x^2+5* x+2)^(5/2)/(3+2*x)^4-1875/256*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1 /2))*3^(1/2)+29047/5120*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))* 5^(1/2)+1/512*(12265+5718*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.65 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (-896721-2059268 x-1672268 x^2-533280 x^3-39744 x^4+3456 x^5\right )}{(3+2 x)^4}+87141 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-112500 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{7680} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(-896721 - 2059268*x - 1672268*x^2 - 533280*x^3 - 39744*x^4 + 3456*x^5))/(3 + 2*x)^4 + 87141*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 112500*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/ (1 + x)])/7680
Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1230, 27, 1229, 27, 1230, 27, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^5} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {5}{64} \int -\frac {2 (94 x+79) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \int \frac {(94 x+79) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5}{32} \left (-\frac {1}{80} \int -\frac {2 (5718 x+4889) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \int \frac {(5718 x+4889) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {(5718 x+12265) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (45000 x+38453)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {(5718 x+12265) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {45000 x+38453}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {1}{4} \left (29047 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-22500 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (5718 x+12265)}{2 (2 x+3)}\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {1}{4} \left (29047 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-45000 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (5718 x+12265)}{2 (2 x+3)}\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {1}{4} \left (29047 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-7500 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (5718 x+12265)}{2 (2 x+3)}\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {1}{4} \left (-58094 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-7500 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (5718 x+12265)}{2 (2 x+3)}\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{32} \left (\frac {1}{40} \left (\frac {1}{4} \left (\frac {29047 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}-7500 \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {\sqrt {3 x^2+5 x+2} (5718 x+12265)}{2 (2 x+3)}\right )-\frac {(2898 x+3727) \left (3 x^2+5 x+2\right )^{3/2}}{60 (2 x+3)^3}\right )-\frac {(4 x+19) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^4}\) |
-1/16*((19 + 4*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^4 + (5*(-1/60*((3727 + 2898*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^3 + (((12265 + 5718*x)*Sqrt[2 + 5*x + 3*x^2])/(2*(3 + 2*x)) + (-7500*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[ 3]*Sqrt[2 + 5*x + 3*x^2])] + (29047*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[5])/4)/40))/32
3.25.41.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {10368 x^{7}-101952 x^{6}-1791648 x^{5}-7762692 x^{4}-15605704 x^{3}-16331039 x^{2}-8602141 x -1793442}{1536 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {1875 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{256}-\frac {29047 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{5120}\) | \(117\) |
| trager | \(-\frac {\left (3456 x^{5}-39744 x^{4}-533280 x^{3}-1672268 x^{2}-2059268 x -896721\right ) \sqrt {3 x^{2}+5 x +2}}{1536 \left (3+2 x \right )^{4}}+\frac {29047 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{5120}+\frac {1875 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{256}\) | \(138\) |
| default | \(-\frac {\left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{75 \left (x +\frac {3}{2}\right )^{3}}-\frac {1627 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{12000 \left (x +\frac {3}{2}\right )^{2}}+\frac {1307 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{2500 \left (x +\frac {3}{2}\right )}+\frac {29047 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{20000}-\frac {1387 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{2400}-\frac {461 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{320}-\frac {1875 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{256}+\frac {29047 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{9600}+\frac {29047 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{5120}-\frac {29047 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{5120}-\frac {1307 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{5000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}}\) | \(258\) |
-1/1536*(10368*x^7-101952*x^6-1791648*x^5-7762692*x^4-15605704*x^3-1633103 9*x^2-8602141*x-1793442)/(3+2*x)^4/(3*x^2+5*x+2)^(1/2)-1875/256*ln(1/3*(5/ 2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-29047/5120*5^(1/2)*arctanh(2/5 *(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {112500 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 87141 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (3456 \, x^{5} - 39744 \, x^{4} - 533280 \, x^{3} - 1672268 \, x^{2} - 2059268 \, x - 896721\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{30720 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
1/30720*(112500*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-4*sq rt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 87141*sqrt( 5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(3456*x^ 5 - 39744*x^4 - 533280*x^3 - 1672268*x^2 - 2059268*x - 896721)*sqrt(3*x^2 + 5*x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080 *x**2 + 810*x + 243), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-165*x**2* sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)
Time = 0.28 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {1627}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {8 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1627 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1387}{400} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {1307}{9600} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {1307 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{1000 \, {\left (2 \, x + 3\right )}} - \frac {1383}{160} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1875}{256} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {29047}{5120} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {10607}{2560} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
1627/4000*(3*x^2 + 5*x + 2)^(5/2) - 13/20*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 8/75*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 3 6*x^2 + 54*x + 27) - 1627/3000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1387/400*(3*x^2 + 5*x + 2)^(3/2)*x + 1307/9600*(3*x^2 + 5*x + 2)^(3/2) + 1307/1000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 1383/160*sqrt(3*x^2 + 5*x + 2)*x - 1875/256*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 29047/5120*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/ab s(2*x + 3) - 2) + 10607/2560*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (135) = 270\).
Time = 0.66 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=\frac {1875}{256} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {29047}{5120} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{3072} \, {\left (\frac {\frac {10 \, {\left (\frac {195 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 904 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 18577 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 27132 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {9 \, {\left (157 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 126 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 409 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 330 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{128 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{2}} \]
1875/256*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) - 29047/5120*sqrt(5)*l og(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3) ) - 4))*sgn(1/(2*x + 3)) - 1/3072*((10*(195*sgn(1/(2*x + 3))/(2*x + 3) - 9 04*sgn(1/(2*x + 3)))/(2*x + 3) + 18577*sgn(1/(2*x + 3)))/(2*x + 3) - 27132 *sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 9/128*(157*(sq rt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3 )) - 126*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 409*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqr t(5)/(2*x + 3))*sgn(1/(2*x + 3)) + 330*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8 /(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2 - 3)^2
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^5} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^5} \,d x \]