Integrand size = 27, antiderivative size = 149 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=-\frac {1141 (7+8 x) \sqrt {2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac {1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac {1141 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{320000 \sqrt {5}} \]
1141/12000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-13/30*(3*x^2+5*x+2)^(5/2) /(3+2*x)^6-167/375*(3*x^2+5*x+2)^(5/2)/(3+2*x)^5+1141/1600000*arctanh(1/10 *(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-1141/160000*(7+8*x)*(3*x^2+5 *x+2)^(1/2)/(3+2*x)^2
Time = 0.51 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (412679+2526920 x+4479600 x^2+3065440 x^3+799120 x^4+95616 x^5\right )}{(3+2 x)^6}+3423 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{2400000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(412679 + 2526920*x + 4479600*x^2 + 3065440*x^3 + 799120*x^4 + 95616*x^5))/(3 + 2*x)^6 + 3423*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/2400000
Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^7} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{30} \int -\frac {(217-78 x) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)^6}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \int \frac {(217-78 x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{60} \left (\frac {1141}{5} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {668 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{60} \left (\frac {1141}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx\right )-\frac {668 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{60} \left (\frac {1141}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {668 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{60} \left (\frac {1141}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )\right )-\frac {668 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{60} \left (\frac {1141}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}-\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )\right )-\frac {668 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}\) |
(-13*(2 + 5*x + 3*x^2)^(5/2))/(30*(3 + 2*x)^6) + ((-668*(2 + 5*x + 3*x^2)^ (5/2))/(25*(3 + 2*x)^5) + (1141*(((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) - Arc Tanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80))/5)/6 0
3.25.29.3.1 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 = 88, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {286848 x^{7}+2875440 x^{6}+13383152 x^{5}+30364240 x^{4}+36109640 x^{3}+22831837 x^{2}+7117235 x +825358}{480000 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+5 x +2}}-\frac {1141 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1600000}\) | \(88\) |
| trager | \(\frac {\left (95616 x^{5}+799120 x^{4}+3065440 x^{3}+4479600 x^{2}+2526920 x +412679\right ) \sqrt {3 x^{2}+5 x +2}}{480000 \left (3+2 x \right )^{6}}+\frac {1141 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1600000}\) | \(97\) |
| default | \(-\frac {167 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{12000 \left (x +\frac {3}{2}\right )^{5}}-\frac {1141 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{48000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1141 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {35371 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{600000 \left (x +\frac {3}{2}\right )^{2}}-\frac {33089 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{375000 \left (x +\frac {3}{2}\right )}+\frac {1141 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{3000000}-\frac {1141 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{200000}+\frac {1141 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1600000}-\frac {1141 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1600000}+\frac {33089 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{750000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}\) | \(232\) |
1/480000*(286848*x^7+2875440*x^6+13383152*x^5+30364240*x^4+36109640*x^3+22 831837*x^2+7117235*x+825358)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2)-1141/1600000*5^ (1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {3423 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (95616 \, x^{5} + 799120 \, x^{4} + 3065440 \, x^{3} + 4479600 \, x^{2} + 2526920 \, x + 412679\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9600000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
1/9600000*(3423*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*(95616*x^5 + 799120*x^4 + 3065440*x ^3 + 4479600*x^2 + 2526920*x + 412679)*sqrt(3*x^2 + 5*x + 2))/(64*x^6 + 57 6*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]
-Integral(-10*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(-23*x *sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 2 2680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-10*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(3*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), x)
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (123) = 246\).
Time = 0.29 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.93 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {35371}{200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {1141 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1141 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {35371 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{150000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3423}{100000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1141}{1600000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {21679}{800000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {33089 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{150000 \, {\left (2 \, x + 3\right )}} \]
35371/200000*(3*x^2 + 5*x + 2)^(3/2) - 13/30*(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) - 167/375*(3 *x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243 ) - 1141/3000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1141/3750*(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 35 371/150000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 3423/100000*sqrt(3 *x^2 + 5*x + 2)*x - 1141/1600000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2) /abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 21679/800000*sqrt(3*x^2 + 5*x + 2) - 33089/150000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (123) = 246\).
Time = 0.36 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.75 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=\frac {1141}{1600000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {109536 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6127344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 70129360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 83080800 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 3334681440 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 9802137888 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 47432214576 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 48106882440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 94851959950 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 39436262415 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 28403540997 \, \sqrt {3} x - 3009604608 \, \sqrt {3} + 28403540997 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{480000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]
1141/1600000*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))) - 1/480000*(109536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 6127344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 70129360*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^9 - 83080800*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 3334681440*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 98021378 88*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 47432214576*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 48106882440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 94851959950*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 3943626 2415*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 28403540997*sqrt(3)*x - 3009604608*sqrt(3) + 28403540997*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)^6
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^7} \,d x \]