Integrand size = 25, antiderivative size = 252 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{5 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {13}{3 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {292}{5 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {\sqrt {x} (6995+5241 x)}{30 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {9521 \sqrt {x} (2+3 x)}{30 \sqrt {2+5 x+3 x^2}}+\frac {\sqrt {x} (21610+28563 x)}{30 \sqrt {2+5 x+3 x^2}}+\frac {9521 \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x}}-\frac {1733 \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {2+5 x+3 x^2}} \] Output:
-2/5/x^(5/2)/(3*x^2+5*x+2)^(3/2)+13/3/x^(3/2)/(3*x^2+5*x+2)^(3/2)-292/5/x^ (1/2)/(3*x^2+5*x+2)^(3/2)-1/30*x^(1/2)*(6995+5241*x)/(3*x^2+5*x+2)^(3/2)-9 521/30*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+1/30*x^(1/2)*(21610+28563*x)/(3 *x^2+5*x+2)^(1/2)+9521/30*2^(1/2)*(3*x^2+5*x+2)^(1/2)*EllipticE(x^(1/2)/(1 +x)^(1/2),1/2*I*2^(1/2))/(1+x)^(1/2)/(2+3*x)^(1/2)-1733/4*2^(1/2)*(1+x)^(1 /2)*(2+3*x)^(1/2)*InverseJacobiAM(arctan(x^(1/2)),1/2*I*2^(1/2))/(3*x^2+5* x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.70 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-2 \left (12-130 x+39836 x^2+154195 x^3+192342 x^4+77985 x^5\right )-19042 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-6953 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{60 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:
Integrate[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
Output:
(-2*(12 - 130*x + 39836*x^2 + 154195*x^3 + 192342*x^4 + 77985*x^5) - (1904 2*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(7/2)*(2 + 5*x + 3*x^2)*EllipticE[I*Arc Sinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (6953*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(7/ 2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(60*x^( 5/2)*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.51 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {1235, 25, 1235, 27, 1237, 1237, 27, 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.
| \(\displaystyle \int \frac {2-5 x}{x^{7/2} \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}-\frac {1}{3} \int -\frac {405 x+193}{x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {405 x+193}{x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{3} \left (-\int \frac {3 (3275 x+2504)}{2 x^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {3275 x+2504}{x^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (-\frac {1}{5} \int \frac {11268 x+8665}{x^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {25995 x+19042}{2 x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \int \frac {25995 x+19042}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (-\int -\frac {3 (9521 x+8665)}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (3 \int \frac {9521 x+8665}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \int \frac {9521 x+8665}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (8665 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+9521 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (9521 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {8665 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \left (\frac {1}{5} \left (\frac {1}{6} \left (6 \left (\frac {8665 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+9521 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {19042 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {8665 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {2504 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\right )-\frac {1965 x+1541}{x^{5/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}\) |
Input:
Int[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
Output:
(2*(38 + 45*x))/(3*x^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) + (-((1541 + 1965*x)/( x^(5/2)*Sqrt[2 + 5*x + 3*x^2])) - (3*((-2504*Sqrt[2 + 5*x + 3*x^2])/(5*x^( 5/2)) + ((8665*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/2)) + ((-19042*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x] + 6*(9521*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/ 2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (8665*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ell ipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/6)/5))/2) /3
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)*(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])
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 = 1.05 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {\sqrt {3 x^{3}+5 x^{2}+2 x}}{10 x^{3}}+\frac {19 \sqrt {3 x^{3}+5 x^{2}+2 x}}{12 x^{2}}-\frac {2591 \left (3 x^{2}+5 x +2\right )}{120 \sqrt {\left (3 x^{2}+5 x +2\right ) x}}+\frac {\left (-\frac {190}{27}-\frac {68 x}{9}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {19765}{144}-\frac {8135 x}{48}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {1733 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{12 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {9521 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{60 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(282\) |
| default | \(\frac {7704 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{4}-28563 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{4}+12840 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-47605 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{3}+5136 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-19042 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+514134 x^{6}+1245870 x^{5}+959610 x^{4}+217350 x^{3}-10512 x^{2}+780 x -72}{180 x^{\frac {5}{2}} \left (x +1\right ) \left (3 x +2\right ) \sqrt {3 x^{2}+5 x +2}}\) | \(318\) |
Input:
int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
((3*x^2+5*x+2)*x)^(1/2)/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(-1/10/x^3*(3*x^3+5*x^ 2+2*x)^(1/2)+19/12/x^2*(3*x^3+5*x^2+2*x)^(1/2)-2591/120*(3*x^2+5*x+2)/((3* x^2+5*x+2)*x)^(1/2)+(-190/27-68/9*x)*(3*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/ 3)^2-2*x*(-19765/144-8135/48*x)*3^(1/2)/(x*(x^2+5/3*x+2/3))^(1/2)-1733/12* (6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^(1/2)/(3*x^3+5*x^2+2*x)^(1/2)*EllipticF (1/2*(6*x+4)^(1/2),I*2^(1/2))-9521/60*(6*x+4)^(1/2)*(3+3*x)^(1/2)*(-6*x)^( 1/2)/(3*x^3+5*x^2+2*x)^(1/2)*(1/3*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-E llipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))))
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.62 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {30380 \, \sqrt {3} {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 85689 \, \sqrt {3} {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (85689 \, x^{6} + 207645 \, x^{5} + 159935 \, x^{4} + 36225 \, x^{3} - 1752 \, x^{2} + 130 \, x - 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{270 \, {\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )}} \] Input:
integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
-1/270*(30380*sqrt(3)*(9*x^7 + 30*x^6 + 37*x^5 + 20*x^4 + 4*x^3)*weierstra ssPInverse(28/27, 80/729, x + 5/9) - 85689*sqrt(3)*(9*x^7 + 30*x^6 + 37*x^ 5 + 20*x^4 + 4*x^3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/ 27, 80/729, x + 5/9)) - 9*(85689*x^6 + 207645*x^5 + 159935*x^4 + 36225*x^3 - 1752*x^2 + 130*x - 12)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/(9*x^7 + 30*x^6 + 37*x^5 + 20*x^4 + 4*x^3)
Timed out. \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2-5*x)/x**(7/2)/(3*x**2+5*x+2)**(5/2),x)
Output:
Timed out
\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{\frac {7}{2}}} \,d x } \] Input:
integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
-integrate((5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)), x)
\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{\frac {7}{2}}} \,d x } \] Input:
integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)), x)
Timed out. \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int -\frac {5\,x-2}{x^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:
int(-(5*x - 2)/(x^(7/2)*(5*x + 3*x^2 + 2)^(5/2)),x)
Output:
int(-(5*x - 2)/(x^(7/2)*(5*x + 3*x^2 + 2)^(5/2)), x)
\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x)
Output:
(50*sqrt(3*x**2 + 5*x + 2)*x - 12*sqrt(3*x**2 + 5*x + 2) - 2160*sqrt(x)*in t(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**8 + 135*sqrt(x)*x**7 + 279*sqrt(x) *x**6 + 305*sqrt(x)*x**5 + 186*sqrt(x)*x**4 + 60*sqrt(x)*x**3 + 8*sqrt(x)* x**2),x)*x**6 - 7200*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**8 + 135*sqrt(x)*x**7 + 279*sqrt(x)*x**6 + 305*sqrt(x)*x**5 + 186*sqrt(x)*x**4 + 60*sqrt(x)*x**3 + 8*sqrt(x)*x**2),x)*x**5 - 8880*sqrt(x)*int(sqrt(3*x** 2 + 5*x + 2)/(27*sqrt(x)*x**8 + 135*sqrt(x)*x**7 + 279*sqrt(x)*x**6 + 305* sqrt(x)*x**5 + 186*sqrt(x)*x**4 + 60*sqrt(x)*x**3 + 8*sqrt(x)*x**2),x)*x** 4 - 4800*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**8 + 135*sqrt(x) *x**7 + 279*sqrt(x)*x**6 + 305*sqrt(x)*x**5 + 186*sqrt(x)*x**4 + 60*sqrt(x )*x**3 + 8*sqrt(x)*x**2),x)*x**3 - 960*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/ (27*sqrt(x)*x**8 + 135*sqrt(x)*x**7 + 279*sqrt(x)*x**6 + 305*sqrt(x)*x**5 + 186*sqrt(x)*x**4 + 60*sqrt(x)*x**3 + 8*sqrt(x)*x**2),x)*x**2 + 4968*sqrt (x)*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**7 + 135*sqrt(x)*x**6 + 279*s qrt(x)*x**5 + 305*sqrt(x)*x**4 + 186*sqrt(x)*x**3 + 60*sqrt(x)*x**2 + 8*sq rt(x)*x),x)*x**6 + 16560*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x* *7 + 135*sqrt(x)*x**6 + 279*sqrt(x)*x**5 + 305*sqrt(x)*x**4 + 186*sqrt(x)* x**3 + 60*sqrt(x)*x**2 + 8*sqrt(x)*x),x)*x**5 + 20424*sqrt(x)*int(sqrt(3*x **2 + 5*x + 2)/(27*sqrt(x)*x**7 + 135*sqrt(x)*x**6 + 279*sqrt(x)*x**5 + 30 5*sqrt(x)*x**4 + 186*sqrt(x)*x**3 + 60*sqrt(x)*x**2 + 8*sqrt(x)*x),x)*x...