Integrand size = 25, antiderivative size = 183 \[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {x} (74+95 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {1450 \sqrt {x} (2+3 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2 \sqrt {x} (1831+2175 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {1450 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {1+x} \sqrt {2+3 x}}+\frac {598 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}} \] Output:
2/9*x^(1/2)*(74+95*x)/(3*x^2+5*x+2)^(3/2)+1450/9*x^(1/2)*(2+3*x)/(3*x^2+5* x+2)^(1/2)-2/9*x^(1/2)*(1831+2175*x)/(3*x^2+5*x+2)^(1/2)-1450/9*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)+598/3*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJacobiAM(arc tan(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 = 21.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {5800+21824 x+26830 x^2+10764 x^3+1450 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+344 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{9 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:
Integrate[((2 - 5*x)*x^(3/2))/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(5800 + 21824*x + 26830*x^2 + 10764*x^3 + (1450*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2 ] + (344*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*Elliptic F[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(9*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.38 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1233, 25, 1235, 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.
| \(\displaystyle \int \frac {(2-5 x) x^{3/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int -\frac {37-135 x}{\sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \int \frac {37-135 x}{\sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-\int \frac {3 (725 x+598)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-\frac {3}{2} \int \frac {725 x+598}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-3 \int \frac {725 x+598}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-3 \left (598 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+725 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-3 \left (725 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {299 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2 \sqrt {x} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {\sqrt {x} (2175 x+1831)}{\sqrt {3 x^2+5 x+2}}-3 \left (\frac {299 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+725 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )\) |
Input:
Int[((2 - 5*x)*x^(3/2))/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(2*Sqrt[x]*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((Sqrt[x]*(1831 + 2175*x))/Sqrt[2 + 5*x + 3*x^2] - 3*(725*((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])) + (299*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2])) )/9
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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[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[((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 = 0.98 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.17
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (\frac {\left (\frac {148}{81}+\frac {190 x}{81}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (\frac {1831}{27}+\frac {725 x}{9}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}+\frac {598 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {725 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{9 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(214\) |
| default | \(-\frac {\left (1143 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-2175 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+1905 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -3625 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +762 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1450 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+39150 x^{4}+98208 x^{3}+80460 x^{2}+21528 x \right ) \sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, \left (3 x +2\right )^{2} \left (x +1\right )^{2}}\) | \(297\) |
Input:
int((2-5*x)*x^(3/2)/(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)*((148/81+190/81*x)*(3* x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(1831/27+725/9*x)*3^(1/2)/(x*(x ^2+5/3*x+2/3))^(1/2)+598/9*(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))+725/9*(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))-EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))))
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (1757 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6525 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (2175 \, x^{3} + 5456 \, x^{2} + 4470 \, x + 1196\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:
integrate((2-5*x)*x^(3/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
2/81*(1757*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInvers e(28/27, 80/729, x + 5/9) - 6525*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 27*(2175*x^3 + 5456*x^2 + 4470*x + 1196)*sqrt(3*x^2 + 5*x + 2)*sqr t(x))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
\[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {2 x^{\frac {3}{2}}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {5 x^{\frac {5}{2}}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx \] Input:
integrate((2-5*x)*x**(3/2)/(3*x**2+5*x+2)**(5/2),x)
Output:
-Integral(-2*x**(3/2)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(5*x**(5/2)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {3}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2-5*x)*x^(3/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
-integrate((5*x - 2)*x^(3/2)/(3*x^2 + 5*x + 2)^(5/2), x)
\[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {3}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2-5*x)*x^(3/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
integrate(-(5*x - 2)*x^(3/2)/(3*x^2 + 5*x + 2)^(5/2), x)
Timed out. \[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{3/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:
int(-(x^(3/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2),x)
Output:
-int((x^(3/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2), x)
\[ \int \frac {(2-5 x) x^{3/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x +6 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}-54 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{4}-180 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{3}-222 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{2}-120 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x -24 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right )+567 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{4}+1890 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{3}+2331 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{2}+1260 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x +252 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right )}{81 x^{4}+270 x^{3}+333 x^{2}+180 x +36} \] Input:
int((2-5*x)*x^(3/2)/(3*x^2+5*x+2)^(5/2),x)
Output:
(10*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x + 6*sqrt(x)*sqrt(3*x**2 + 5*x + 2) - 54*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**6 + 135*sqrt(x)*x**5 + 279*sq rt(x)*x**4 + 305*sqrt(x)*x**3 + 186*sqrt(x)*x**2 + 60*sqrt(x)*x + 8*sqrt(x )),x)*x**4 - 180*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**6 + 135*sqrt(x) *x**5 + 279*sqrt(x)*x**4 + 305*sqrt(x)*x**3 + 186*sqrt(x)*x**2 + 60*sqrt(x )*x + 8*sqrt(x)),x)*x**3 - 222*int(sqrt(3*x**2 + 5*x + 2)/(27*sqrt(x)*x**6 + 135*sqrt(x)*x**5 + 279*sqrt(x)*x**4 + 305*sqrt(x)*x**3 + 186*sqrt(x)*x* *2 + 60*sqrt(x)*x + 8*sqrt(x)),x)*x**2 - 120*int(sqrt(3*x**2 + 5*x + 2)/(2 7*sqrt(x)*x**6 + 135*sqrt(x)*x**5 + 279*sqrt(x)*x**4 + 305*sqrt(x)*x**3 + 186*sqrt(x)*x**2 + 60*sqrt(x)*x + 8*sqrt(x)),x)*x - 24*int(sqrt(3*x**2 + 5 *x + 2)/(27*sqrt(x)*x**6 + 135*sqrt(x)*x**5 + 279*sqrt(x)*x**4 + 305*sqrt( x)*x**3 + 186*sqrt(x)*x**2 + 60*sqrt(x)*x + 8*sqrt(x)),x) + 567*int((sqrt( x)*sqrt(3*x**2 + 5*x + 2)*x)/(27*x**6 + 135*x**5 + 279*x**4 + 305*x**3 + 1 86*x**2 + 60*x + 8),x)*x**4 + 1890*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/ (27*x**6 + 135*x**5 + 279*x**4 + 305*x**3 + 186*x**2 + 60*x + 8),x)*x**3 + 2331*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(27*x**6 + 135*x**5 + 279*x** 4 + 305*x**3 + 186*x**2 + 60*x + 8),x)*x**2 + 1260*int((sqrt(x)*sqrt(3*x** 2 + 5*x + 2)*x)/(27*x**6 + 135*x**5 + 279*x**4 + 305*x**3 + 186*x**2 + 60* x + 8),x)*x + 252*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(27*x**6 + 135*x* *5 + 279*x**4 + 305*x**3 + 186*x**2 + 60*x + 8),x))/(9*(9*x**4 + 30*x**...