Integrand size = 27, antiderivative size = 169 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {26453 (7+8 x) \sqrt {2+5 x+3 x^2}}{200000 (3+2 x)^2}-\frac {13 \left (2+5 x+3 x^2\right )^{3/2}}{30 (3+2 x)^6}-\frac {73 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^5}-\frac {3113 \left (2+5 x+3 x^2\right )^{3/2}}{5000 (3+2 x)^4}-\frac {2237 \left (2+5 x+3 x^2\right )^{3/2}}{3750 (3+2 x)^3}-\frac {26453 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{400000 \sqrt {5}} \]
-13/30*(3*x^2+5*x+2)^(3/2)/(3+2*x)^6-73/125*(3*x^2+5*x+2)^(3/2)/(3+2*x)^5- 3113/5000*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-2237/3750*(3*x^2+5*x+2)^(3/2)/(3+2 *x)^3-26453/2000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1 /2)+26453/200000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.49 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (16322393+50707640 x+62797200 x^2+39304480 x^3+12381040 x^4+1567872 x^5\right )}{(3+2 x)^6}-79359 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{3000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(16322393 + 50707640*x + 62797200*x^2 + 39304480 *x^3 + 12381040*x^4 + 1567872*x^5))/(3 + 2*x)^6 - 79359*Sqrt[5]*ArcTanh[Sq rt[2/5 + x + (3*x^2)/5]/(1 + x)])/3000000
Time = 0.38 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1237, 27, 1237, 25, 1237, 25, 1228, 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.
| \(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^7} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{30} \int -\frac {3 (29-78 x) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)^6}dx-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{20} \int \frac {(29-78 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{20} \left (-\frac {1}{25} \int -\frac {(485-1752 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^5}dx-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \int \frac {(485-1752 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^5}dx-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (-\frac {1}{20} \int -\frac {(16723-18678 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^4}dx-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (\frac {1}{20} \int \frac {(16723-18678 x) \sqrt {3 x^2+5 x+2}}{(2 x+3)^4}dx-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (\frac {1}{20} \left (26453 \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {17896 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (\frac {1}{20} \left (26453 \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {17896 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (\frac {1}{20} \left (26453 \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {17896 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{20} \left (\frac {1}{25} \left (\frac {1}{20} \left (26453 \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {17896 \left (3 x^2+5 x+2\right )^{3/2}}{3 (2 x+3)^3}\right )-\frac {3113 \left (3 x^2+5 x+2\right )^{3/2}}{10 (2 x+3)^4}\right )-\frac {292 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{3/2}}{30 (2 x+3)^6}\) |
(-13*(2 + 5*x + 3*x^2)^(3/2))/(30*(3 + 2*x)^6) + ((-292*(2 + 5*x + 3*x^2)^ (3/2))/(25*(3 + 2*x)^5) + ((-3113*(2 + 5*x + 3*x^2)^(3/2))/(10*(3 + 2*x)^4 ) + ((-17896*(2 + 5*x + 3*x^2)^(3/2))/(3*(3 + 2*x)^3) + 26453*(((7 + 8*x)* Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqr t[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/20)/25)/20
3.25.17.3.1 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 = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {4703616 x^{7}+44982480 x^{6}+182954384 x^{5}+409676080 x^{4}+544717880 x^{3}+428099779 x^{2}+183027245 x +32644786}{600000 \left (3+2 x \right )^{6} \sqrt {3 x^{2}+5 x +2}}+\frac {26453 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{2000000}\) | \(88\) |
| trager | \(\frac {\left (1567872 x^{5}+12381040 x^{4}+39304480 x^{3}+62797200 x^{2}+50707640 x +16322393\right ) \sqrt {3 x^{2}+5 x +2}}{600000 \left (3+2 x \right )^{6}}-\frac {26453 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{2000000}\) | \(97\) |
| default | \(-\frac {73 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{4000 \left (x +\frac {3}{2}\right )^{5}}-\frac {3113 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {2237 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {26453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{200000 \left (x +\frac {3}{2}\right )^{2}}-\frac {26453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {26453 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{2000000}+\frac {26453 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{2000000}+\frac {26453 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{250000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}\) | \(195\) |
1/600000*(4703616*x^7+44982480*x^6+182954384*x^5+409676080*x^4+544717880*x ^3+428099779*x^2+183027245*x+32644786)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2)+26453 /2000000*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 = 156, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {79359 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1567872 \, x^{5} + 12381040 \, x^{4} + 39304480 \, x^{3} + 62797200 \, x^{2} + 50707640 \, x + 16322393\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{12000000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
1/12000000*(79359*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*(1567872*x^5 + 12381040*x^4 + 39 304480*x^3 + 62797200*x^2 + 50707640*x + 16322393)*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)
\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]
-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15 120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(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)
Time = 0.29 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.53 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=\frac {26453}{2000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {79359}{200000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {73 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {3113 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {2237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {26453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{50000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {26453 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{50000 \, {\left (2 \, x + 3\right )}} \]
26453/2000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2 /abs(2*x + 3) - 2) + 79359/200000*sqrt(3*x^2 + 5*x + 2) - 13/30*(3*x^2 + 5 *x + 2)^(3/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 73/125*(3*x^2 + 5*x + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080 *x^2 + 810*x + 243) - 3113/5000*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 2237/3750*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 26453/50000*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 2 6453/50000*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (139) = 278\).
Time = 0.33 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.43 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=-\frac {26453}{2000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2539488 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 41901552 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 924796880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 3988893600 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 33933192480 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 66530947296 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 275158218192 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 265623867480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 526452161650 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 226453420305 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 171288605499 \, \sqrt {3} x + 19197814536 \, \sqrt {3} - 171288605499 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{600000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]
-26453/2000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sq rt(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))) + 1/600000*(2539488*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 41901552*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 924796880*(sq rt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 3988893600*sqrt(3)*(sqrt(3)*x - sqrt( 3*x^2 + 5*x + 2))^8 + 33933192480*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 66530947296*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 275158218192*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 265623867480*sqrt(3)*(sqrt(3)*x - s qrt(3*x^2 + 5*x + 2))^4 + 526452161650*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) ^3 + 226453420305*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 17128860 5499*sqrt(3)*x + 19197814536*sqrt(3) - 171288605499*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. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^7} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{{\left (2\,x+3\right )}^7} \,d x \]