Integrand size = 27, antiderivative size = 174 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac {1048 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac {9696 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {46108 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{625 \sqrt {5}} \]
-2/5*(37+47*x)/(3+2*x)^3/(3*x^2+5*x+2)^(3/2)+46108/3125*arctanh(1/10*(7+8* x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+12/25*(603+638*x)/(3+2*x)^3/(3*x^2 +5*x+2)^(1/2)+47552/375*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3+1048/15*(3*x^2+5*x+2 )^(1/2)/(3+2*x)^2+9696/625*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {5 \sqrt {2+5 x+3 x^2} \left (2313929+12060957 x+25105026 x^2+26717636 x^3+15334836 x^4+4495032 x^5+523584 x^6\right )}{(1+x)^2 (3+2 x)^3 (2+3 x)^2}+138324 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right )}{9375} \]
(2*((5*Sqrt[2 + 5*x + 3*x^2]*(2313929 + 12060957*x + 25105026*x^2 + 267176 36*x^3 + 15334836*x^4 + 4495032*x^5 + 523584*x^6))/((1 + x)^2*(3 + 2*x)^3* (2 + 3*x)^2) + 138324*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])) /9375
Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1235, 27, 1235, 27, 1237, 25, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{15} \int \frac {3 (470 x+491)}{(2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {470 x+491}{(2 x+3)^4 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \int \frac {2 (5742 x+5641)}{(2 x+3)^4 \sqrt {3 x^2+5 x+2}}dx-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \int \frac {5742 x+5641}{(2 x+3)^4 \sqrt {3 x^2+5 x+2}}dx-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}-\frac {1}{15} \int -\frac {35664 x+37121}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \int \frac {35664 x+37121}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \left (\frac {3275 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}-\frac {1}{10} \int -\frac {15 (6550 x+8613)}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx\right )+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \left (\frac {3}{2} \int \frac {6550 x+8613}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx+\frac {3275 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \left (\frac {3}{2} \left (\frac {11527}{5} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {2424 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )+\frac {3275 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \left (\frac {3}{2} \left (\frac {2424 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {23054}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {3275 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {4}{5} \left (\frac {1}{15} \left (\frac {3}{2} \left (\frac {11527 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}+\frac {2424 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )+\frac {3275 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )+\frac {5944 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {6 (638 x+603)}{5 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}\) |
(-2*(37 + 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-6*(603 + 638*x))/(5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - (4*((5944*Sqrt[2 + 5*x + 3 *x^2])/(15*(3 + 2*x)^3) + ((3275*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + (3*( (2424*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (11527*ArcTanh[(7 + 8*x)/(2*S qrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(5*Sqrt[5])))/2)/15))/5))/5
3.26.24.3.1 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 = 83, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {\frac {349056}{625} x^{6}+\frac {2996688}{625} x^{5}+\frac {10223224}{625} x^{4}+\frac {53435272}{1875} x^{3}+\frac {16736684}{625} x^{2}+\frac {8040638}{625} x +\frac {4627858}{1875}}{\left (3+2 x \right )^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {46108 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{3125}\) | \(83\) |
| trager | \(\frac {\frac {349056}{625} x^{6}+\frac {2996688}{625} x^{5}+\frac {10223224}{625} x^{4}+\frac {53435272}{1875} x^{3}+\frac {16736684}{625} x^{2}+\frac {8040638}{625} x +\frac {4627858}{1875}}{\left (3+2 x \right )^{3} \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {46108 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{3125}\) | \(102\) |
| default | \(-\frac {151}{200 \left (x +\frac {3}{2}\right )^{2} \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {862}{125 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {11527}{750 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {2366 \left (5+6 x \right )}{375 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {2424}{125}+\frac {14544 x}{625}}{\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {23054}{625 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {46108 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{3125}-\frac {13}{120 \left (x +\frac {3}{2}\right )^{3} \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(169\) |
2/1875*(523584*x^6+4495032*x^5+15334836*x^4+26717636*x^3+25105026*x^2+1206 0957*x+2313929)/(3+2*x)^3/(3*x^2+5*x+2)^(3/2)-46108/3125*5^(1/2)*arctanh(2 /5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.67 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (34581 \, \sqrt {5} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (523584 \, x^{6} + 4495032 \, x^{5} + 15334836 \, x^{4} + 26717636 \, x^{3} + 25105026 \, x^{2} + 12060957 \, x + 2313929\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \]
2/9375*(34581*sqrt(5)*(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2 + 756*x + 108)*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*(523584*x^6 + 4495032*x^5 + 15334836*x^4 + 26717636*x^3 + 25105026*x^2 + 12060957*x + 2313929)*sqrt(3* x^2 + 5*x + 2))/(72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223* x^2 + 756*x + 108)
\[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{144 x^{8} \sqrt {3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2484 x \sqrt {3 x^{2} + 5 x + 2} + 324 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{144 x^{8} \sqrt {3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2484 x \sqrt {3 x^{2} + 5 x + 2} + 324 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x + 2) + 5416*x**6*sqrt(3*x**2 + 5*x + 2) + 12296*x**5*sqrt(3*x**2 + 5*x + 2) + 17185*x**4*sqrt(3*x**2 + 5*x + 2) + 15126*x**3*sqrt(3*x**2 + 5*x + 2) + 8181*x**2*sqrt(3*x**2 + 5*x + 2) + 2484*x*sqrt(3*x**2 + 5*x + 2) + 324* sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(144*x**8*sqrt(3*x**2 + 5*x + 2) + 1344*x**7*sqrt(3*x**2 + 5*x + 2) + 5416*x**6*sqrt(3*x**2 + 5*x + 2) + 1 2296*x**5*sqrt(3*x**2 + 5*x + 2) + 17185*x**4*sqrt(3*x**2 + 5*x + 2) + 151 26*x**3*sqrt(3*x**2 + 5*x + 2) + 8181*x**2*sqrt(3*x**2 + 5*x + 2) + 2484*x *sqrt(3*x**2 + 5*x + 2) + 324*sqrt(3*x**2 + 5*x + 2)), x)
Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.46 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {46108}{3125} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {14544 \, x}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {35174}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {4732 \, x}{125 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{15 \, {\left (8 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + 36 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 54 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {151}{50 \, {\left (4 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1724}{125 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {12133}{750 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
-46108/3125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/a bs(2*x + 3) - 2) + 14544/625*x/sqrt(3*x^2 + 5*x + 2) + 35174/625/sqrt(3*x^ 2 + 5*x + 2) - 4732/125*x/(3*x^2 + 5*x + 2)^(3/2) - 13/15/(8*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 36*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 54*(3*x^2 + 5*x + 2)^(3/ 2)*x + 27*(3*x^2 + 5*x + 2)^(3/2)) - 151/50/(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)) - 1724/125/(2 *(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 12133/750/(3*x^2 + 5*x + 2)^(3/2)
Time = 0.32 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.64 \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {46108}{3125} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2 \, {\left ({\left (12 \, {\left (19992 \, x + 58207\right )} x + 636631\right )} x + 184301\right )}}{3125 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (296724 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2103870 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 16891990 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 21246975 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 38063715 \, \sqrt {3} x + 8723544 \, \sqrt {3} - 38063715 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]
46108/3125*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))) + 2/3125*((12*(19992*x + 58207)*x + 636631)*x + 184301)/(3*x^2 + 5*x + 2)^(3/2) - 8/9375*(296724*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2103870*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 16891990*(sqrt(3) *x - sqrt(3*x^2 + 5*x + 2))^3 + 21246975*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 38063715*sqrt(3)*x + 8723544*sqrt(3) - 38063715*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)^3
Timed out. \[ \int \frac {5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^4\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]