\(\int \frac {2-5 x}{x^{7/2} (2+5 x+3 x^2)^{3/2}} \, dx\) [216]

Optimal result

Integrand size = 25, antiderivative size = 222 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2}{5 x^{5/2} \sqrt {2+5 x+3 x^2}}+\frac {11}{3 x^{3/2} \sqrt {2+5 x+3 x^2}}-\frac {487}{15 \sqrt {x} \sqrt {2+5 x+3 x^2}}+\frac {2693 \sqrt {x} (2+3 x)}{30 \sqrt {2+5 x+3 x^2}}-\frac {\sqrt {x} (8755+8079 x)}{30 \sqrt {2+5 x+3 x^2}}-\frac {2693 \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{15 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x}}+\frac {157 \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x+3 x^2}} \] Output:

-2/5/x^(5/2)/(3*x^2+5*x+2)^(1/2)+11/3/x^(3/2)/(3*x^2+5*x+2)^(1/2)-487/15/x 
^(1/2)/(3*x^2+5*x+2)^(1/2)+2693/30*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)-1/3 
0*x^(1/2)*(8755+8079*x)/(3*x^2+5*x+2)^(1/2)-2693/30*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)+157/2*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 = 21.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.68 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-12+110 x+4412 x^2+4710 x^3+2693 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-338 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{30 x^{5/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(3/2)),x]
 

Output:

(-12 + 110*x + 4412*x^2 + 4710*x^3 + (2693*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqr 
t[3 + 2/x]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (338*I)* 
Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[2/ 
3]/Sqrt[x]], 3/2])/(30*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1235, 25, 1237, 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.

\(\displaystyle \int \frac {2-5 x}{x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\int -\frac {225 x+191}{x^{7/2} \sqrt {3 x^2+5 x+2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {225 x+191}{x^{7/2} \sqrt {3 x^2+5 x+2}}dx+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {1}{5} \int \frac {1719 x+1570}{2 x^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{10} \int \frac {1719 x+1570}{x^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \int \frac {2355 x+2693}{x^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (-\int -\frac {3 (2693 x+1570)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {2693 x+1570}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (3 \int \frac {2693 x+1570}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (3 \left (1570 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+2693 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (3 \left (2693 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {785 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (3 \left (\frac {785 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+2693 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2693 \sqrt {3 x^2+5 x+2}}{\sqrt {x}}\right )+\frac {1570 \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )+\frac {2 (45 x+38)}{x^{5/2} \sqrt {3 x^2+5 x+2}}-\frac {191 \sqrt {3 x^2+5 x+2}}{5 x^{5/2}}\)

Input:

Int[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(3/2)),x]
 

Output:

(2*(38 + 45*x))/(x^(5/2)*Sqrt[2 + 5*x + 3*x^2]) - (191*Sqrt[2 + 5*x + 3*x^ 
2])/(5*x^(5/2)) + ((1570*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/2)) + ((-2693*Sqrt 
[2 + 5*x + 3*x^2])/Sqrt[x] + 3*(2693*((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])) + (785*Sqrt[2]*(1 + x)*Sqrt[(2 + 3 
*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/3)/ 
10
 

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 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 = 1.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {3369 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-2693 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+48474 x^{4}+52530 x^{3}+5844 x^{2}-660 x +72}{180 x^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}\)	\(124\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {\sqrt {3 x^{3}+5 x^{2}+2 x}}{5 x^{3}}+\frac {7 \sqrt {3 x^{3}+5 x^{2}+2 x}}{3 x^{2}}-\frac {653 \left (3 x^{2}+5 x +2\right )}{30 \sqrt {\left (3 x^{2}+5 x +2\right ) x}}-\frac {2 x \left (\frac {95}{3}+34 x \right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}+\frac {157 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {2693 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{60 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(250\)

Input:

int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/180*(3369*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))*(6*x+4)^(1/2)*(3+3*x)^ 
(1/2)*6^(1/2)*(-x)^(1/2)*x^2-2693*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*( 
6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*x^2+48474*x^4+52530*x^3+5844 
*x^2-660*x+72)/x^(5/2)/(3*x^2+5*x+2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.52 \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {665 \, \sqrt {3} {\left (3 \, x^{5} + 5 \, x^{4} + 2 \, x^{3}\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 24237 \, \sqrt {3} {\left (3 \, x^{5} + 5 \, x^{4} + 2 \, x^{3}\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (8079 \, x^{4} + 8755 \, x^{3} + 974 \, x^{2} - 110 \, x + 12\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{270 \, {\left (3 \, x^{5} + 5 \, x^{4} + 2 \, x^{3}\right )}} \] Input:

integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
 

Output:

1/270*(665*sqrt(3)*(3*x^5 + 5*x^4 + 2*x^3)*weierstrassPInverse(28/27, 80/7 
29, x + 5/9) - 24237*sqrt(3)*(3*x^5 + 5*x^4 + 2*x^3)*weierstrassZeta(28/27 
, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(8079*x^4 + 875 
5*x^3 + 974*x^2 - 110*x + 12)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/(3*x^5 + 5*x^ 
4 + 2*x^3)
 

Sympy [F]

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

integrate((2-5*x)/x**(7/2)/(3*x**2+5*x+2)**(3/2),x)
 

Output:

-Integral(5/(3*x**(9/2)*sqrt(3*x**2 + 5*x + 2) + 5*x**(7/2)*sqrt(3*x**2 + 
5*x + 2) + 2*x**(5/2)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-2/(3*x**(11/ 
2)*sqrt(3*x**2 + 5*x + 2) + 5*x**(9/2)*sqrt(3*x**2 + 5*x + 2) + 2*x**(7/2) 
*sqrt(3*x**2 + 5*x + 2)), x)
 

Maxima [F]

\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \] Input:

integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")
 

Output:

-integrate((5*x - 2)/((3*x^2 + 5*x + 2)^(3/2)*x^(7/2)), x)
 

Giac [F]

\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \] Input:

integrate((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")
 

Output:

integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(3/2)*x^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int -\frac {5\,x-2}{x^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \] Input:

int(-(5*x - 2)/(x^(7/2)*(5*x + 3*x^2 + 2)^(3/2)),x)
                                                                                    
                                                                                    
 

Output:

int(-(5*x - 2)/(x^(7/2)*(5*x + 3*x^2 + 2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {750 \sqrt {3 x^{2}+5 x +2}\, x^{3}+100 \sqrt {3 x^{2}+5 x +2}\, x -24 \sqrt {3 x^{2}+5 x +2}-1080 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{6}+30 \sqrt {x}\, x^{5}+37 \sqrt {x}\, x^{4}+20 \sqrt {x}\, x^{3}+4 \sqrt {x}\, x^{2}}d x \right ) x^{4}-1800 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{6}+30 \sqrt {x}\, x^{5}+37 \sqrt {x}\, x^{4}+20 \sqrt {x}\, x^{3}+4 \sqrt {x}\, x^{2}}d x \right ) x^{3}-720 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{6}+30 \sqrt {x}\, x^{5}+37 \sqrt {x}\, x^{4}+20 \sqrt {x}\, x^{3}+4 \sqrt {x}\, x^{2}}d x \right ) x^{2}+2244 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{5}+30 \sqrt {x}\, x^{4}+37 \sqrt {x}\, x^{3}+20 \sqrt {x}\, x^{2}+4 \sqrt {x}\, x}d x \right ) x^{4}+3740 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{5}+30 \sqrt {x}\, x^{4}+37 \sqrt {x}\, x^{3}+20 \sqrt {x}\, x^{2}+4 \sqrt {x}\, x}d x \right ) x^{3}+1496 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{9 \sqrt {x}\, x^{5}+30 \sqrt {x}\, x^{4}+37 \sqrt {x}\, x^{3}+20 \sqrt {x}\, x^{2}+4 \sqrt {x}\, x}d x \right ) x^{2}+3375 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}\, x^{2}}{9 \sqrt {x}\, x^{4}+30 \sqrt {x}\, x^{3}+37 \sqrt {x}\, x^{2}+20 \sqrt {x}\, x +4 \sqrt {x}}d x \right ) x^{4}+5625 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}\, x^{2}}{9 \sqrt {x}\, x^{4}+30 \sqrt {x}\, x^{3}+37 \sqrt {x}\, x^{2}+20 \sqrt {x}\, x +4 \sqrt {x}}d x \right ) x^{3}+2250 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}\, x^{2}}{9 \sqrt {x}\, x^{4}+30 \sqrt {x}\, x^{3}+37 \sqrt {x}\, x^{2}+20 \sqrt {x}\, x +4 \sqrt {x}}d x \right ) x^{2}}{60 \sqrt {x}\, x^{2} \left (3 x^{2}+5 x +2\right )} \] Input:

int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(3/2),x)
 

Output:

(750*sqrt(3*x**2 + 5*x + 2)*x**3 + 100*sqrt(3*x**2 + 5*x + 2)*x - 24*sqrt( 
3*x**2 + 5*x + 2) - 1080*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(9*sqrt(x)*x** 
6 + 30*sqrt(x)*x**5 + 37*sqrt(x)*x**4 + 20*sqrt(x)*x**3 + 4*sqrt(x)*x**2), 
x)*x**4 - 1800*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(9*sqrt(x)*x**6 + 30*sqr 
t(x)*x**5 + 37*sqrt(x)*x**4 + 20*sqrt(x)*x**3 + 4*sqrt(x)*x**2),x)*x**3 - 
720*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(9*sqrt(x)*x**6 + 30*sqrt(x)*x**5 + 
 37*sqrt(x)*x**4 + 20*sqrt(x)*x**3 + 4*sqrt(x)*x**2),x)*x**2 + 2244*sqrt(x 
)*int(sqrt(3*x**2 + 5*x + 2)/(9*sqrt(x)*x**5 + 30*sqrt(x)*x**4 + 37*sqrt(x 
)*x**3 + 20*sqrt(x)*x**2 + 4*sqrt(x)*x),x)*x**4 + 3740*sqrt(x)*int(sqrt(3* 
x**2 + 5*x + 2)/(9*sqrt(x)*x**5 + 30*sqrt(x)*x**4 + 37*sqrt(x)*x**3 + 20*s 
qrt(x)*x**2 + 4*sqrt(x)*x),x)*x**3 + 1496*sqrt(x)*int(sqrt(3*x**2 + 5*x + 
2)/(9*sqrt(x)*x**5 + 30*sqrt(x)*x**4 + 37*sqrt(x)*x**3 + 20*sqrt(x)*x**2 + 
 4*sqrt(x)*x),x)*x**2 + 3375*sqrt(x)*int((sqrt(3*x**2 + 5*x + 2)*x**2)/(9* 
sqrt(x)*x**4 + 30*sqrt(x)*x**3 + 37*sqrt(x)*x**2 + 20*sqrt(x)*x + 4*sqrt(x 
)),x)*x**4 + 5625*sqrt(x)*int((sqrt(3*x**2 + 5*x + 2)*x**2)/(9*sqrt(x)*x** 
4 + 30*sqrt(x)*x**3 + 37*sqrt(x)*x**2 + 20*sqrt(x)*x + 4*sqrt(x)),x)*x**3 
+ 2250*sqrt(x)*int((sqrt(3*x**2 + 5*x + 2)*x**2)/(9*sqrt(x)*x**4 + 30*sqrt 
(x)*x**3 + 37*sqrt(x)*x**2 + 20*sqrt(x)*x + 4*sqrt(x)),x)*x**2)/(60*sqrt(x 
)*x**2*(3*x**2 + 5*x + 2))