Integrand size = 27, antiderivative size = 147 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt {2+5 x+3 x^2}}+\frac {152 \sqrt {2+5 x+3 x^2}}{(3+2 x)^2}+\frac {11808 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)}+\frac {4884 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{125 \sqrt {5}} \]
-2/5*(37+47*x)/(3+2*x)^2/(3*x^2+5*x+2)^(3/2)+4884/625*arctanh(1/10*(7+8*x) *5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+4/25*(1907+2112*x)/(3+2*x)^2/(3*x^2+ 5*x+2)^(1/2)+152*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2+11808/125*(3*x^2+5*x+2)^(1/ 2)/(3+2*x)
Time = 0.44 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.65 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2}{625} \left (\frac {5 \sqrt {2+5 x+3 x^2} \left (146063+727887 x+1405814 x^2+1316616 x^3+599148 x^4+106272 x^5\right )}{(1+x)^2 (3+2 x)^2 (2+3 x)^2}+4884 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right ) \]
(2*((5*Sqrt[2 + 5*x + 3*x^2]*(146063 + 727887*x + 1405814*x^2 + 1316616*x^ 3 + 599148*x^4 + 106272*x^5))/((1 + x)^2*(3 + 2*x)^2*(2 + 3*x)^2) + 4884*S qrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)]))/625
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1235, 27, 1235, 27, 1237, 27, 1228, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{15} \int \frac {3 (376 x+417)}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {376 x+417}{(2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \int \frac {2 (4224 x+3961)}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \int \frac {4224 x+3961}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {475 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}-\frac {1}{10} \int -\frac {15 (950 x+933)}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {3}{2} \int \frac {950 x+933}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx+\frac {475 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {3}{2} \left (\frac {407}{5} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {984 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )+\frac {475 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {3}{2} \left (\frac {984 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {814}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {475 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {3}{2} \left (\frac {407 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}+\frac {984 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )+\frac {475 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {2 (2112 x+1907)}{5 (2 x+3)^2 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}\) |
(-2*(37 + 47*x))/(5*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(1907 + 2112*x))/(5*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]) - (4*((475*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + (3*((984*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (407 *ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(5*Sqrt[5])))/2))/5 ))/5
3.26.23.3.1 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/(((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[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (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[(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 = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {\frac {212544}{125} x^{5}+\frac {1198296}{125} x^{4}+\frac {2633232}{125} x^{3}+\frac {2811628}{125} x^{2}+\frac {1455774}{125} x +\frac {292126}{125}}{\left (3+2 x \right )^{2} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {4884 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{625}\) | \(78\) |
| trager | \(\frac {2 \left (106272 x^{5}+599148 x^{4}+1316616 x^{3}+1405814 x^{2}+727887 x +146063\right ) \sqrt {3 x^{2}+5 x +2}}{125 \left (6 x^{3}+19 x^{2}+19 x +6\right )^{2}}-\frac {4884 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{625}\) | \(107\) |
| default | \(-\frac {177}{50 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {407}{50 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {106 \left (5+6 x \right )}{25 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {2952}{25}+\frac {17712 x}{125}}{\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {2442}{125 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {4884 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{625}-\frac {13}{40 \left (x +\frac {3}{2}\right )^{2} \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(148\) |
2/125*(106272*x^5+599148*x^4+1316616*x^3+1405814*x^2+727887*x+146063)/(3+2 *x)^2/(3*x^2+5*x+2)^(3/2)-4884/625*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/ (12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.36 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (1221 \, \sqrt {5} {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (106272 \, x^{5} + 599148 \, x^{4} + 1316616 \, x^{3} + 1405814 \, x^{2} + 727887 \, x + 146063\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{625 \, {\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )}} \]
2/625*(1221*sqrt(5)*(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228* x + 36)*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)) + 5*(106272*x^5 + 599148*x^4 + 1316616*x^3 + 1405 814*x^2 + 727887*x + 146063)*sqrt(3*x^2 + 5*x + 2))/(36*x^6 + 228*x^5 + 58 9*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)
\[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{72 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{72 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 756 x \sqrt {3 x^{2} + 5 x + 2} + 108 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(72*x**7*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt(3*x**2 + 5*x + 2) + 1862*x**5*sqrt(3*x**2 + 5*x + 2) + 3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*x**3*sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(3*x**2 + 5*x + 2) + 75 6*x*sqrt(3*x**2 + 5*x + 2) + 108*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5 /(72*x**7*sqrt(3*x**2 + 5*x + 2) + 564*x**6*sqrt(3*x**2 + 5*x + 2) + 1862* x**5*sqrt(3*x**2 + 5*x + 2) + 3355*x**4*sqrt(3*x**2 + 5*x + 2) + 3560*x**3 *sqrt(3*x**2 + 5*x + 2) + 2223*x**2*sqrt(3*x**2 + 5*x + 2) + 756*x*sqrt(3* x**2 + 5*x + 2) + 108*sqrt(3*x**2 + 5*x + 2)), x)
Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.27 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {4884}{625} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {17712 \, x}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {17202}{125 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {636 \, x}{25 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{10 \, {\left (4 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {177}{25 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {653}{50 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
-4884/625*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs (2*x + 3) - 2) + 17712/125*x/sqrt(3*x^2 + 5*x + 2) + 17202/125/sqrt(3*x^2 + 5*x + 2) - 636/25*x/(3*x^2 + 5*x + 2)^(3/2) - 13/10/(4*(3*x^2 + 5*x + 2) ^(3/2)*x^2 + 12*(3*x^2 + 5*x + 2)^(3/2)*x + 9*(3*x^2 + 5*x + 2)^(3/2)) - 1 77/25/(2*(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 653/50/( 3*x^2 + 5*x + 2)^(3/2)
Time = 0.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.59 \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {4884}{625} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2 \, {\left ({\left (6 \, {\left (23826 \, x + 61591\right )} x + 309599\right )} x + 84259\right )}}{625 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (4106 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 16447 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 57729 \, \sqrt {3} x + 20987 \, \sqrt {3} - 57729 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{625 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]
4884/625*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))) + 2/625*((6*(23826*x + 61591)*x + 309599)*x + 84259)/(3*x^2 + 5* x + 2)^(3/2) - 8/625*(4106*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 16447*s qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 57729*sqrt(3)*x + 20987*sqr t(3) - 57729*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)^2
Timed out. \[ \int \frac {5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^3\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]