\(\int \frac {2-5 x}{x^{5/2} (2+5 x+3 x^2)^{5/2}} \, dx\) [226]

Optimal result

Integrand size = 25, antiderivative size = 220 \[ \int \frac {2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{3 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {15}{\sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {\sqrt {x} (86+75 x)}{\left (2+5 x+3 x^2\right )^{3/2}}+\frac {625 \sqrt {x} (2+3 x)}{2 \sqrt {2+5 x+3 x^2}}-\frac {5 \sqrt {x} (307+375 x)}{2 \sqrt {2+5 x+3 x^2}}-\frac {625 \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {1+x} \sqrt {2+3 x}}+\frac {795 \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x+3 x^2}} \] Output:

-2/3/x^(3/2)/(3*x^2+5*x+2)^(3/2)+15/x^(1/2)/(3*x^2+5*x+2)^(3/2)+x^(1/2)*(8 
6+75*x)/(3*x^2+5*x+2)^(3/2)+625/2*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)-5/2* 
x^(1/2)*(307+375*x)/(3*x^2+5*x+2)^(1/2)-625/2*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)+795 
/2*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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.77 \[ \int \frac {2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-4+7590 x+28806 x^2+35550 x^3+14310 x^4+1875 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{5/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+510 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{5/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{6 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:

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

Output:

(-4 + 7590*x + 28806*x^2 + 35550*x^3 + 14310*x^4 + (1875*I)*Sqrt[2 + 2/x]* 
Sqrt[3 + 2/x]*x^(5/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt 
[x]], 3/2] + (510*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(5/2)*(2 + 5*x + 3*x^2) 
*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(6*x^(3/2)*(2 + 5*x + 3*x^2 
)^(3/2))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1235, 27, 1235, 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.

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}-\frac {1}{3} \int -\frac {9 (35 x+13)}{x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int \frac {35 x+13}{x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle 3 \left (-\int \frac {5 (135 x+106)}{2 x^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (-\frac {5}{2} \int \frac {135 x+106}{x^{5/2} \sqrt {3 x^2+5 x+2}}dx-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle 3 \left (-\frac {5}{2} \left (-\frac {1}{3} \int \frac {159 x+125}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\int -\frac {3 (125 x+106)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-\frac {3}{2} \int \frac {125 x+106}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1240

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-3 \int \frac {125 x+106}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1503

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-3 \left (106 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+125 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1413

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-3 \left (125 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {53 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1456

\(\displaystyle 3 \left (-\frac {5}{2} \left (\frac {1}{3} \left (\frac {125 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}-3 \left (\frac {53 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+125 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )-\frac {106 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {225 x+181}{x^{3/2} \sqrt {3 x^2+5 x+2}}\right )+\frac {2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}\)

Input:

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

Output:

(2*(38 + 45*x))/(3*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2)) + 3*(-((181 + 225*x)/( 
x^(3/2)*Sqrt[2 + 5*x + 3*x^2])) - (5*((-106*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3 
/2)) + ((125*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x] - 3*(125*((Sqrt[x]*(2 + 3*x))/ 
(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ellip 
ticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (53*Sqrt[2]*(1 + 
 x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x 
 + 3*x^2]))/3))/2)
 

Defintions of rubi rules used

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 1235

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] 
)
 

rule 1237

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])
 

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 = 261, normalized size of antiderivative = 1.19

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {\sqrt {3 x^{3}+5 x^{2}+2 x}}{6 x^{2}}+\frac {\frac {55}{4} x^{2}+\frac {275}{12} x +\frac {55}{6}}{\sqrt {\left (3 x^{2}+5 x +2\right ) x}}+\frac {\left (\frac {136}{27}+\frac {50 x}{9}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (\frac {9479}{72}+\frac {3805 x}{24}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}+\frac {265 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{2 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {625 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{4 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(261\)
default	\(-\frac {855 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-1875 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{3}+1425 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-3125 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+570 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -1250 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +33750 x^{5}+83880 x^{4}+67650 x^{3}+17388 x^{2}-180 x +8}{12 x^{\frac {3}{2}} \left (x +1\right ) \left (3 x +2\right ) \sqrt {3 x^{2}+5 x +2}}\)	\(309\)

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)*(-1/6/x^2*(3*x^3+5*x^2 
+2*x)^(1/2)+55/12*(3*x^2+5*x+2)/((3*x^2+5*x+2)*x)^(1/2)+(136/27+50/9*x)*(3 
*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(9479/72+3805/24*x)*3^(1/2)/(x 
*(x^2+5/3*x+2/3))^(1/2)+265/2*(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))+625/4*(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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.68 \[ \int \frac {2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {1645 \, \sqrt {3} {\left (9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 5625 \, \sqrt {3} {\left (9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 3 \, {\left (16875 \, x^{5} + 41940 \, x^{4} + 33825 \, x^{3} + 8694 \, x^{2} - 90 \, x + 4\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{18 \, {\left (9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}\right )}} \] Input:

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

Output:

1/18*(1645*sqrt(3)*(9*x^6 + 30*x^5 + 37*x^4 + 20*x^3 + 4*x^2)*weierstrassP 
Inverse(28/27, 80/729, x + 5/9) - 5625*sqrt(3)*(9*x^6 + 30*x^5 + 37*x^4 + 
20*x^3 + 4*x^2)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 
80/729, x + 5/9)) - 3*(16875*x^5 + 41940*x^4 + 33825*x^3 + 8694*x^2 - 90*x 
 + 4)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/(9*x^6 + 30*x^5 + 37*x^4 + 20*x^3 + 4 
*x^2)
 

Sympy [F]

\[ \int \frac {2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {5}{9 x^{\frac {11}{2}} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2} + 20 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2} + 4 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {2}{9 x^{\frac {13}{2}} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{\frac {11}{2}} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 20 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2} + 4 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(5/(9*x**(11/2)*sqrt(3*x**2 + 5*x + 2) + 30*x**(9/2)*sqrt(3*x**2 
+ 5*x + 2) + 37*x**(7/2)*sqrt(3*x**2 + 5*x + 2) + 20*x**(5/2)*sqrt(3*x**2 
+ 5*x + 2) + 4*x**(3/2)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-2/(9*x**(1 
3/2)*sqrt(3*x**2 + 5*x + 2) + 30*x**(11/2)*sqrt(3*x**2 + 5*x + 2) + 37*x** 
(9/2)*sqrt(3*x**2 + 5*x + 2) + 20*x**(7/2)*sqrt(3*x**2 + 5*x + 2) + 4*x**( 
5/2)*sqrt(3*x**2 + 5*x + 2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(210*sqrt(3*x**2 + 5*x + 2)*x**3 + 60*sqrt(3*x**2 + 5*x + 2)*x - 8*sqrt(3* 
x**2 + 5*x + 2) - 1080*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 - 3600*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*s 
qrt(x)*x**4 + 186*sqrt(x)*x**3 + 60*sqrt(x)*x**2 + 8*sqrt(x)*x),x)*x**4 - 
4440*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**3 - 2400*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(27*sq 
rt(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**2 - 480*sqrt(x)*int(sq 
rt(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 + 4428*sqrt(x)*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**5 + 14760*sqrt(x)*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 + 18 
6*sqrt(x)*x**2 + 60*sqrt(x)*x + 8*sqrt(x)),x)*x**4 + 18204*sqrt(x)*int(sqr 
t(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...