Integrand size = 27, antiderivative size = 197 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=\frac {63 (44365+20678 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {8547 \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024}+\frac {6620481 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{204800 \sqrt {5}} \]
-7/25600*(40201+31174*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3-7/1920*(1301+1046*x )*(3*x^2+5*x+2)^(5/2)/(3+2*x)^5-1/12*(11+3*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^ 6-8547/1024*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+66204 81/1024000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+63/10 2400*(44365+20678*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.75 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (-1835461379-6648875480 x-9799959120 x^2-7425343520 x^3-2968126160 x^4-550079616 x^5-23155200 x^6+2073600 x^7\right )}{(3+2 x)^6}+19861443 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-25641000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1536000} \]
((-5*Sqrt[2 + 5*x + 3*x^2]*(-1835461379 - 6648875480*x - 9799959120*x^2 - 7425343520*x^3 - 2968126160*x^4 - 550079616*x^5 - 23155200*x^6 + 2073600*x ^7))/(3 + 2*x)^6 + 19861443*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 25641000*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/153600 0
Time = 0.46 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1229, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^7} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {7}{96} \int -\frac {4 (51 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}dx-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{24} \int \frac {(51 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}dx-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {7}{24} \left (-\frac {1}{160} \int -\frac {3 (3086 x+2661) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \int \frac {(3086 x+2661) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}dx-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (-\frac {1}{80} \int -\frac {18 (20678 x+17669) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \int \frac {(20678 x+17669) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {(20678 x+44365) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (162800 x+139113)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {(20678 x+44365) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {162800 x+139113}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {1}{4} \left (105087 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-81400 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (20678 x+44365)}{2 (2 x+3)}\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {1}{4} \left (105087 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-162800 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (20678 x+44365)}{2 (2 x+3)}\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {1}{4} \left (105087 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {81400 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (20678 x+44365)}{2 (2 x+3)}\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {1}{4} \left (-210174 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {81400 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (20678 x+44365)}{2 (2 x+3)}\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{24} \left (\frac {3}{160} \left (\frac {9}{40} \left (\frac {1}{4} \left (\frac {105087 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {5}}-\frac {81400 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (20678 x+44365)}{2 (2 x+3)}\right )-\frac {(31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^3}\right )-\frac {(1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}\right )-\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}\) |
-1/12*((11 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6 + (7*(-1/80*((1301 + 1046*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^5 + (3*(-1/20*((40201 + 31174 *x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^3 + (9*(((44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(2*(3 + 2*x)) + ((-81400*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[ 2 + 5*x + 3*x^2])])/Sqrt[3] + (105087*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[5])/4))/40))/160))/24
3.25.58.3.1 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 = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {6220800 x^{9}-59097600 x^{8}-1761867648 x^{7}-11701086960 x^{6}-38216820592 x^{5}-72462847280 x^{4}-83797109080 x^{3}-58350679777 x^{2}-22475057855 x -3670922758}{307200 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+5 x +2}}-\frac {8547 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{1024}-\frac {6620481 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024000}\) | \(127\) |
| trager | \(-\frac {\left (2073600 x^{7}-23155200 x^{6}-550079616 x^{5}-2968126160 x^{4}-7425343520 x^{3}-9799959120 x^{2}-6648875480 x -1835461379\right ) \sqrt {3 x^{2}+5 x +2}}{307200 \left (3+2 x \right )^{6}}-\frac {6620481 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1024000}+\frac {8547 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{1024}\) | \(148\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}-\frac {21 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{4000 \left (x +\frac {3}{2}\right )^{5}}-\frac {1143 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {459 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{50000 \left (x +\frac {3}{2}\right )^{3}}-\frac {63693 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {47169 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{250000}+\frac {47169 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {349461 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000000}-\frac {104517 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{160000}-\frac {210231 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{128000}-\frac {8547 \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}}{1024}-\frac {6620481 \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 )}{1024000}+\frac {6620481 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4000000}+\frac {2206827 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{640000}+\frac {945783 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1000000}+\frac {6620481 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024000}\) | \(337\) |
-1/307200*(6220800*x^9-59097600*x^8-1761867648*x^7-11701086960*x^6-3821682 0592*x^5-72462847280*x^4-83797109080*x^3-58350679777*x^2-22475057855*x-367 0922758)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2)-8547/1024*ln(1/3*(5/2+3*x)*3^(1/2)+ (3*x^2+5*x+2)^(1/2))*3^(1/2)-6620481/1024000*5^(1/2)*arctanh(2/5*(-7/2-4*x )*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=\frac {25641000 \, \sqrt {3} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 19861443 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (2073600 \, x^{7} - 23155200 \, x^{6} - 550079616 \, x^{5} - 2968126160 \, x^{4} - 7425343520 \, x^{3} - 9799959120 \, x^{2} - 6648875480 \, x - 1835461379\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{6144000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
1/6144000*(25641000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860 *x^2 + 2916*x + 729)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x ^2 + 120*x + 49) + 19861443*sqrt(5)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^ 3 + 4860*x^2 + 2916*x + 729)*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*(2073600*x^7 - 23155200 *x^6 - 550079616*x^5 - 2968126160*x^4 - 7425343520*x^3 - 9799959120*x^2 - 6648875480*x - 1835461379)*sqrt(3*x^2 + 5*x + 2))/(64*x^6 + 576*x^5 + 2160 *x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 1 5120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-292* x*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), x) - Integral(-870*x**2*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), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5 *x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 2041 2*x**2 + 10206*x + 2187), x) - Integral(-1090*x**4*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), x) - Integral(-396*x**5*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), x) - Integral(27*x**7*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), x)
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (161) = 322\).
Time = 0.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.89 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=\frac {191079}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {21 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {1143 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {459 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{6250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {63693 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1048383}{500000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {368739}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {47169 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{50000 \, {\left (2 \, x + 3\right )}} - \frac {313551}{80000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {116487}{640000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {630693}{64000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {8547}{1024} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {6620481}{1024000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {2415861}{512000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
191079/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/30*(3*x^2 + 5*x + 2)^(9/2)/(64 *x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 21/125*( 3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 24 3) - 1143/5000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 459/6250*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 63 693/250000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 1048383/500000*(3* x^2 + 5*x + 2)^(5/2)*x - 368739/4000000*(3*x^2 + 5*x + 2)^(5/2) + 47169/50 000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 313551/80000*(3*x^2 + 5*x + 2)^(3/ 2)*x + 116487/640000*(3*x^2 + 5*x + 2)^(3/2) - 630693/64000*sqrt(3*x^2 + 5 *x + 2)*x - 8547/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/ 2) - 6620481/1024000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3 ) + 5/2/abs(2*x + 3) - 2) + 2415861/512000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (161) = 322\).
Time = 0.35 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.37 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=-\frac {9}{512} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 121\right )} + \frac {6620481}{1024000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {8547}{1024} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {\sqrt {3} {\left (1761054624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 78359519088 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 522182992240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 6180007168800 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 16013156565600 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 85756996584864 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 107556795368496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 284279833881720 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 172447244925750 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 205883289380025 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 48408731804817 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 15295619190024\right )}}{921600 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]
-9/512*sqrt(3*x^2 + 5*x + 2)*(6*x - 121) + 6620481/1024000*sqrt(5)*log(abs (-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sq rt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 8547/1024*sq rt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 1/921 600*sqrt(3)*(1761054624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 7 8359519088*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 522182992240*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 6180007168800*(sqrt(3)*x - sqrt(3*x ^2 + 5*x + 2))^8 + 16013156565600*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 85756996584864*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 10755679536 8496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 284279833881720*(sqrt (3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 172447244925750*sqrt(3)*(sqrt(3)*x - sq rt(3*x^2 + 5*x + 2))^3 + 205883289380025*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2 ))^2 + 48408731804817*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 152956 19190024)/(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)^6
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^7} \,d x \]