Integrand size = 27, antiderivative size = 234 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=-\frac {53697 (7+8 x) \sqrt {2+5 x+3 x^2}}{5120000000 (3+2 x)^2}+\frac {17899 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{128000000 (3+2 x)^4}-\frac {17899 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{8000000 (3+2 x)^6}+\frac {7671 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{200000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {621 \left (2+5 x+3 x^2\right )^{9/2}}{2750 (3+2 x)^{10}}-\frac {3904 \left (2+5 x+3 x^2\right )^{9/2}}{20625 (3+2 x)^9}+\frac {53697 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10240000000 \sqrt {5}} \]
17899/128000000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-17899/8000000*(7+8*x )*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6+7671/200000*(7+8*x)*(3*x^2+5*x+2)^(7/2)/(3 +2*x)^8-13/55*(3*x^2+5*x+2)^(9/2)/(3+2*x)^11-621/2750*(3*x^2+5*x+2)^(9/2)/ (3+2*x)^10-3904/20625*(3*x^2+5*x+2)^(9/2)/(3+2*x)^9+53697/51200000000*arct anh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-53697/5120000000*(7+ 8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.77 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.46 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (629890144539+6058472990850 x+24817198954840 x^2+56898923222800 x^3+80329740407040 x^4+72251114756992 x^5+41485308553600 x^6+14992486229760 x^7+3387337708800 x^8+479034140160 x^9+30557343744 x^{10}\right )}{(3+2 x)^{11}}+1772001 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{844800000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(629890144539 + 6058472990850*x + 24817198954840 *x^2 + 56898923222800*x^3 + 80329740407040*x^4 + 72251114756992*x^5 + 4148 5308553600*x^6 + 14992486229760*x^7 + 3387337708800*x^8 + 479034140160*x^9 + 30557343744*x^10))/(3 + 2*x)^11 + 1772001*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/844800000000
Time = 0.49 (sec) , antiderivative size = 264, 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, 25, 1228, 1152, 1152, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{12}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{55} \int -\frac {3 (129-52 x) \left (3 x^2+5 x+2\right )^{7/2}}{2 (2 x+3)^{11}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{110} \int \frac {(129-52 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{11}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {3}{110} \left (-\frac {1}{50} \int -\frac {(5945-1242 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}}dx-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \int \frac {(5945-1242 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}}dx-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \int \frac {\left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}dx-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{110} \left (\frac {1}{50} \left (\frac {28127}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {15616 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {207 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}\) |
(-13*(2 + 5*x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) + (3*((-207*(2 + 5*x + 3*x ^2)^(9/2))/(25*(3 + 2*x)^10) + ((-15616*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (28127*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(80*(3 + 2*x)^8) - ( 7*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*(3 + 2*x)^6) + (-1/40*((7 + 8*x )*(2 + 5*x + 3*x^2)^(3/2))/(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)/24))/160))/5)/50))/110
3.25.63.3.1 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 = 0.55 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {91672031232 x^{12}+1589889139200 x^{11}+12618298514688 x^{10}+62872215513600 x^{9}+206193032227200 x^{8}+454164859498496 x^{7}+685215412113280 x^{6}+716847701217584 x^{5}+519605693792600 x^{4}+256059260192350 x^{3}+81816433297547 x^{2}+15266396704395 x +1259780289078}{168960000000 \left (3+2 x \right )^{11} \sqrt {3 x^{2}+5 x +2}}-\frac {53697 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{51200000000}\) | \(113\) |
| trager | \(\frac {\left (30557343744 x^{10}+479034140160 x^{9}+3387337708800 x^{8}+14992486229760 x^{7}+41485308553600 x^{6}+72251114756992 x^{5}+80329740407040 x^{4}+56898923222800 x^{3}+24817198954840 x^{2}+6058472990850 x +629890144539\right ) \sqrt {3 x^{2}+5 x +2}}{168960000000 \left (3+2 x \right )^{11}}-\frac {53697 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{51200000000}\) | \(122\) |
| default | \(-\frac {48583 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{32000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {237801 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{100000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {14735991 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{4000000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {14112083 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{2500000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {427819341 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{50000000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {80215647 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{12500000000}-\frac {80215647 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{6250000000 \left (x +\frac {3}{2}\right )}-\frac {31197957 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{50000000000}+\frac {519071 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{8000000000}-\frac {53697 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{6400000000}-\frac {53697 \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 )}{51200000000}-\frac {621 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{2816000 \left (x +\frac {3}{2}\right )^{10}}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{112640 \left (x +\frac {3}{2}\right )^{11}}+\frac {53697 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200000000000}+\frac {17899 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{32000000000}+\frac {7671 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{50000000000}-\frac {7671 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{8000000 \left (x +\frac {3}{2}\right )^{7}}+\frac {53697 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{51200000000}-\frac {7671 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{12800000 \left (x +\frac {3}{2}\right )^{8}}-\frac {61 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{165000 \left (x +\frac {3}{2}\right )^{9}}\) | \(411\) |
1/168960000000*(91672031232*x^12+1589889139200*x^11+12618298514688*x^10+62 872215513600*x^9+206193032227200*x^8+454164859498496*x^7+685215412113280*x ^6+716847701217584*x^5+519605693792600*x^4+256059260192350*x^3+81816433297 547*x^2+15266396704395*x+1259780289078)/(3+2*x)^11/(3*x^2+5*x+2)^(1/2)-536 97/51200000000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-1 9)^(1/2))
Time = 0.30 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=\frac {1772001 \, \sqrt {5} {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (30557343744 \, x^{10} + 479034140160 \, x^{9} + 3387337708800 \, x^{8} + 14992486229760 \, x^{7} + 41485308553600 \, x^{6} + 72251114756992 \, x^{5} + 80329740407040 \, x^{4} + 56898923222800 \, x^{3} + 24817198954840 \, x^{2} + 6058472990850 \, x + 629890144539\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{3379200000000 \, {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )}} \]
1/3379200000000*(1772001*sqrt(5)*(2048*x^11 + 33792*x^10 + 253440*x^9 + 11 40480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660 520*x^3 + 4330260*x^2 + 1299078*x + 177147)*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*(3055734 3744*x^10 + 479034140160*x^9 + 3387337708800*x^8 + 14992486229760*x^7 + 41 485308553600*x^6 + 72251114756992*x^5 + 80329740407040*x^4 + 5689892322280 0*x^3 + 24817198954840*x^2 + 6058472990850*x + 629890144539)*sqrt(3*x^2 + 5*x + 2))/(2048*x^11 + 33792*x^10 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x + 177147)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x* *10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427 328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531 441), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**1 1 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144 *x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 42 51528*x + 531441), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(4096*x* *12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368 *x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15 588936*x**2 + 4251528*x + 531441), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 3041280*x**9 + 1026432 0*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 3 4642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(-1090*x* *4*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 608256*x**10 + 30412 80*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 73728*x**11 + 6082 56*x**10 + 3041280*x**9 + 10264320*x**8 + 24634368*x**7 + 43110144*x**6 + 55427328*x**5 + 51963120*x**4 + 34642080*x**3 + 15588936*x**2 + 4251528*x + 531441), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(4096*x**12 + 7...
Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (196) = 392\).
Time = 0.29 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.78 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=\frac {1283458023}{50000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{55 \, {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )}} - \frac {621 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2750 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} - \frac {3904 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{20625 \, {\left (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 )}} - \frac {7671 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000 \, {\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 {7671 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{62500 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {48583 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{500000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {237801 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{3125000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {14735991 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {14112083 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{312500000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {427819341 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {93593871}{25000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {623905443}{200000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {80215647 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{2500000000 \, {\left (2 \, x + 3\right )}} + \frac {1557213}{4000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {10399319}{32000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {161091}{3200000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {53697}{51200000000} \, \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 {1020243}{25600000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
1283458023/50000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/55*(3*x^2 + 5*x + 2)^( 9/2)/(2048*x^11 + 33792*x^10 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 71 85024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299 078*x + 177147) - 621/2750*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099 520*x^3 + 1180980*x^2 + 393660*x + 59049) - 3904/20625*(3*x^2 + 5*x + 2)^( 9/2)/(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) - 7671/50000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 10886 4*x^3 + 81648*x^2 + 34992*x + 6561) - 7671/62500*(3*x^2 + 5*x + 2)^(9/2)/( 128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206* x + 2187) - 48583/500000*(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) - 237801/3125000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 14735991/ 250000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 14112083/312500000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 427819341/12500000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 9359 3871/25000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 623905443/200000000000*(3*x^2 + 5*x + 2)^(5/2) - 80215647/2500000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 1557213/4000000000*(3*x^2 + 5*x + 2)^(3/2)*x + 10399319/32000000000*(...
Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (196) = 392\).
Time = 0.33 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.84 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=\frac {53697}{51200000000} \, \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 {1814529024 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{21} + 57157664256 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{20} + 57290941171200 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 557490020440320 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} + 3116590396465920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} - 40571342658595584 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 1098653419392131328 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 4929229513296950400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 44860439685628251520 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 101067124429527527040 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 530008429621517017088 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 735944911884403670592 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 2465807894359584887200 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 2226326899649908579920 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 4870616002552398497520 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 2849658548882889760632 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 3959763769847021107884 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 1420163541040959876150 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 1141537424727199856070 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 215130617786249721765 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 76323347715579462729 \, \sqrt {3} x - 4261520459402725896 \, \sqrt {3} + 76323347715579462729 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{168960000000 \, {\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 )}^{11}} \]
53697/51200000000*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))) - 1/168960000000*(1814529024*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^21 + 57157664256*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 57290941171200*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 557490020440320* sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 + 3116590396465920*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^17 - 40571342658595584*sqrt(3)*(sqrt(3)*x - sq rt(3*x^2 + 5*x + 2))^16 - 1098653419392131328*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^15 - 4929229513296950400*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2 ))^14 - 44860439685628251520*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 1010 67124429527527040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 5300084 29621517017088*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 735944911884403670 592*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 246580789435958488720 0*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 2226326899649908579920*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 4870616002552398497520*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 2849658548882889760632*sqrt(3)*(sqrt(3)*x - sq rt(3*x^2 + 5*x + 2))^6 - 3959763769847021107884*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 1420163541040959876150*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 1141537424727199856070*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 215130617786249721765*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 7...
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{12}} \,d x \]