Integrand size = 25, antiderivative size = 233 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\frac {3229 \sqrt {x} (2+3 x)}{1386 \sqrt {2+5 x+3 x^2}}+\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac {3229 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{693 \sqrt {2} \sqrt {2+5 x+3 x^2}}+\frac {1357 (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{231 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]
-4/99*(9-20*x)*(3*x^2+5*x+2)^(3/2)/x^(11/2)+3229/1386*(2+3*x)*x^(1/2)/(3*x ^2+5*x+2)^(1/2)-3229/1386*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticE(x^(1/2)/(1 +x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2) +1357/462*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I* 2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)+1357/693*(3*x^2 +5*x+2)^(1/2)/x^(3/2)+1/231*(634+1367*x)*(3*x^2+5*x+2)^(1/2)/x^(7/2)-3229/ 1386*(3*x^2+5*x+2)^(1/2)/x^(1/2)
Result contains complex when optimal does not.
Time = 21.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.71 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\frac {-2016-5600 x+11360 x^2+61744 x^3+86914 x^4+48256 x^5+8142 x^6+3229 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{13/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+842 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{13/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{1386 x^{11/2} \sqrt {2+5 x+3 x^2}} \]
(-2016 - 5600*x + 11360*x^2 + 61744*x^3 + 86914*x^4 + 48256*x^5 + 8142*x^6 + (3229*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(13/2)*EllipticE[I*Ar cSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (842*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(13/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(1386*x^(11/2 )*Sqrt[2 + 5*x + 3*x^2])
Time = 0.46 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1229, 1229, 27, 1237, 1237, 27, 1240, 1503, 1413, 1456}
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.
| \(\displaystyle \int \frac {(2-5 x) \left (3 x^2+5 x+2\right )^{3/2}}{x^{13/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {1}{33} \int \frac {(375 x+317) \sqrt {3 x^2+5 x+2}}{x^{9/2}}dx-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{35} \int -\frac {5 (3447 x+2714)}{2 x^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {(1367 x+634) \sqrt {3 x^2+5 x+2}}{7 x^{7/2}}-\frac {1}{14} \int \frac {3447 x+2714}{x^{5/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \int \frac {4071 x+3229}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (-\int -\frac {3 (3229 x+2714)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {3229 x+2714}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (3 \int \frac {3229 x+2714}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (3 \left (2714 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+3229 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (3 \left (3229 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {1357 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {1}{33} \left (\frac {1}{14} \left (\frac {1}{3} \left (3 \left (\frac {1357 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+3229 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {3229 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {2714 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1367 x+634)}{7 x^{7/2}}\right )-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}\) |
(-4*(9 - 20*x)*(2 + 5*x + 3*x^2)^(3/2))/(99*x^(11/2)) + (((634 + 1367*x)*S qrt[2 + 5*x + 3*x^2])/(7*x^(7/2)) + ((2714*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/ 2)) + ((-3229*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x] + 3*(3229*((Sqrt[x]*(2 + 3*x) )/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ell ipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (1357*Sqrt[2]* (1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/3)/14)/33
3.11.53.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[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])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(-\frac {1545 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{5}-3229 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{5}+58122 x^{7}+48018 x^{6}-250788 x^{5}-521484 x^{4}-370464 x^{3}-68160 x^{2}+33600 x +12096}{8316 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {11}{2}}}\) | \(139\) |
| risch | \(-\frac {9687 x^{7}+8003 x^{6}-41798 x^{5}-86914 x^{4}-61744 x^{3}-11360 x^{2}+5600 x +2016}{1386 x^{\frac {11}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {1357 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1386 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {3229 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2772 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(213\) |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{11 x^{6}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{99 x^{5}}+\frac {3946 \sqrt {3 x^{3}+5 x^{2}+2 x}}{693 x^{4}}+\frac {1927 \sqrt {3 x^{3}+5 x^{2}+2 x}}{231 x^{3}}+\frac {1357 \sqrt {3 x^{3}+5 x^{2}+2 x}}{693 x^{2}}-\frac {3229 \left (3 x^{2}+5 x +2\right )}{1386 \sqrt {x \left (3 x^{2}+5 x +2\right )}}+\frac {1357 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1386 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {3229 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2772 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(290\) |
-1/8316*(1545*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticF(1/2 *(6*x+4)^(1/2),I*2^(1/2))*x^5-3229*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x )^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*x^5+58122*x^7+48018*x^6-250 788*x^5-521484*x^4-370464*x^3-68160*x^2+33600*x+12096)/(3*x^2+5*x+2)^(1/2) /x^(11/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\frac {8281 \, \sqrt {3} x^{6} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 29061 \, \sqrt {3} x^{6} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (3229 \, x^{5} - 2714 \, x^{4} - 11562 \, x^{3} - 7892 \, x^{2} + 280 \, x + 1008\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{12474 \, x^{6}} \]
1/12474*(8281*sqrt(3)*x^6*weierstrassPInverse(28/27, 80/729, x + 5/9) - 29 061*sqrt(3)*x^6*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(3229*x^5 - 2714*x^4 - 11562*x^3 - 7892*x^2 + 280*x + 1008)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/x^6
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{13/2}} \,d x \]