\(\int \frac {(2-5 x) x^{5/2}}{(2+5 x+3 x^2)^{5/2}} \, dx\) [221]

Optimal result

Integrand size = 25, antiderivative size = 183 \[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {x} (190+253 x)}{27 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {3464 \sqrt {x} (2+3 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {2 \sqrt {x} (4385+5196 x)}{27 \sqrt {2+5 x+3 x^2}}+\frac {3464 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {1+x} \sqrt {2+3 x}}-\frac {1430 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}} \] Output:

-2/27*x^(1/2)*(190+253*x)/(3*x^2+5*x+2)^(3/2)-3464/27*x^(1/2)*(2+3*x)/(3*x 
^2+5*x+2)^(1/2)+2/27*x^(1/2)*(4385+5196*x)/(3*x^2+5*x+2)^(1/2)+3464/27*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)-1430/9*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJaco 
biAM(arctan(x^(1/2)),1/2*I*2^(1/2))/(3*x^2+5*x+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-2 \left (6928+26060 x+32020 x^2+12825 x^3\right )-3464 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-826 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{27 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:

Integrate[((2 - 5*x)*x^(5/2))/(2 + 5*x + 3*x^2)^(5/2),x]
 

Output:

(-2*(6928 + 26060*x + 32020*x^2 + 12825*x^3) - (3464*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] - (826*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*Ell 
ipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(27*Sqrt[x]*(2 + 5*x + 3*x^2)^( 
3/2))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1233, 25, 1234, 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.

\(\displaystyle \int \frac {(2-5 x) x^{5/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2}{9} \int -\frac {(111-40 x) \sqrt {x}}{\left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (95 x+74) x^{3/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \int \frac {(111-40 x) \sqrt {x}}{\left (3 x^2+5 x+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 1234

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (-2 \int -\frac {866 x+715}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\int \frac {866 x+715}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (2 \int \frac {866 x+715}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (2 \left (715 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+866 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (2 \left (866 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {715 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2 x^{3/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (2 \left (\frac {715 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+866 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 \sqrt {x} (866 x+715)}{\sqrt {3 x^2+5 x+2}}\right )\)

Input:

Int[((2 - 5*x)*x^(5/2))/(2 + 5*x + 3*x^2)^(5/2),x]
 

Output:

(2*x^(3/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*Sqrt[x]*(715 
 + 866*x))/Sqrt[2 + 5*x + 3*x^2] + 2*(866*((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])) + (715*(1 + x)*Sqrt[(2 + 3*x) 
/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2] 
))))/9
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1233

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])
 

rule 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1240

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]
 

rule 1413

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]
 

rule 1456

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]
 

rule 1503

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]
 

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.17

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (\frac {\left (-\frac {380}{243}-\frac {506 x}{243}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {4385}{81}-\frac {1732 x}{27}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {1430 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1732 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(214\)
default	\(\frac {2 \left (1359 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-2598 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+2265 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -4330 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +906 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1732 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+46764 x^{4}+117405 x^{3}+96192 x^{2}+25740 x \right ) \sqrt {3 x^{2}+5 x +2}}{81 \sqrt {x}\, \left (3 x +2\right )^{2} \left (x +1\right )^{2}}\)	\(297\)

Input:

int((2-5*x)*x^(5/2)/(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)*((-380/243-506/243*x)* 
(3*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(-4385/81-1732/27*x)*3^(1/2) 
/(x*(x^2+5/3*x+2/3))^(1/2)-1430/27*(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))-1732/27*( 
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*Ellip 
ticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67 \[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (4210 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 15588 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (5196 \, x^{3} + 13045 \, x^{2} + 10688 \, x + 2860\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{243 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:

integrate((2-5*x)*x^(5/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
 

Output:

-2/243*(4210*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInve 
rse(28/27, 80/729, x + 5/9) - 15588*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*(5196*x^3 + 13045*x^2 + 10688*x + 2860)*sqrt(3*x^2 + 5*x + 2 
)*sqrt(x))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((2-5*x)*x**(5/2)/(3*x**2+5*x+2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {5}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2-5*x)*x^(5/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
 

Output:

-integrate((5*x - 2)*x^(5/2)/(3*x^2 + 5*x + 2)^(5/2), x)
 

Giac [F]

\[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (5 \, x - 2\right )} x^{\frac {5}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2-5*x)*x^(5/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
 

Output:

integrate(-(5*x - 2)*x^(5/2)/(3*x^2 + 5*x + 2)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{5/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:

int(-(x^(5/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

-int((x^(5/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {450 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}+440 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x +264 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}-2376 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{4}-7920 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{3}-9768 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x^{2}-5280 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right ) x -1056 \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{27 \sqrt {x}\, x^{6}+135 \sqrt {x}\, x^{5}+279 \sqrt {x}\, x^{4}+305 \sqrt {x}\, x^{3}+186 \sqrt {x}\, x^{2}+60 \sqrt {x}\, x +8 \sqrt {x}}d x \right )-2430 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{4}-8100 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{3}-9990 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x^{2}-5400 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right ) x -1080 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{27 x^{6}+135 x^{5}+279 x^{4}+305 x^{3}+186 x^{2}+60 x +8}d x \right )}{1215 x^{4}+4050 x^{3}+4995 x^{2}+2700 x +540} \] Input:

int((2-5*x)*x^(5/2)/(3*x^2+5*x+2)^(5/2),x)
 

Output:

(2*(225*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**2 + 220*sqrt(x)*sqrt(3*x**2 + 5* 
x + 2)*x + 132*sqrt(x)*sqrt(3*x**2 + 5*x + 2) - 1188*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**4 - 3960*int(sq 
rt(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 - 4884*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 - 2640*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 - 528*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) - 1215*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**4 - 4050*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 - 4995*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 - 2700*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 - 540*i 
nt((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(27*x**6 + 135*x**5 + 279*x**4 + ...