\(\int \frac {(2-5 x) x^{9/2}}{(2+5 x+3 x^2)^{5/2}} \, dx\) [219]

Optimal result

Integrand size = 25, antiderivative size = 206 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {x} (1390+1957 x)}{243 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {17512 \sqrt {x} (2+3 x)}{243 \sqrt {2+5 x+3 x^2}}+\frac {14 \sqrt {x} (3335+3846 x)}{243 \sqrt {2+5 x+3 x^2}}-\frac {10}{81} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {17512 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {1+x} \sqrt {2+3 x}}-\frac {7540 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{81 \sqrt {2+5 x+3 x^2}} \] Output:

-2/243*x^(1/2)*(1390+1957*x)/(3*x^2+5*x+2)^(3/2)-17512/243*x^(1/2)*(2+3*x) 
/(3*x^2+5*x+2)^(1/2)+14/243*x^(1/2)*(3335+3846*x)/(3*x^2+5*x+2)^(1/2)-10/8 
1*x^(1/2)*(3*x^2+5*x+2)^(1/2)+17512/243*2^(1/2)*(3*x^2+5*x+2)^(1/2)*Ellipt 
icE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))/(1+x)^(1/2)/(2+3*x)^(1/2)-7540/81*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 = 21.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-2 \left (35024+129880 x+155660 x^2+58590 x^3-1512 x^4+135 x^5\right )-17512 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-5108 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{243 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:

Integrate[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
 

Output:

(-2*(35024 + 129880*x + 155660*x^2 + 58590*x^3 - 1512*x^4 + 135*x^5) - (17 
512*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*A 
rcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (5108*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^( 
3/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(243* 
Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1233, 25, 1233, 27, 1236, 25, 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.

\(\displaystyle \int \frac {(2-5 x) x^{9/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2}{9} \int -\frac {x^{5/2} (150 x+259)}{\left (3 x^2+5 x+2\right )^{3/2}}dx+\frac {2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \int \frac {x^{5/2} (150 x+259)}{\left (3 x^2+5 x+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2}{3} \int -\frac {3 \sqrt {x} (1885 x+1608)}{2 \sqrt {3 x^2+5 x+2}}dx+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2 x^{3/2} (645 x+536)}{\sqrt {3 x^2+5 x+2}}-\int \frac {\sqrt {x} (1885 x+1608)}{\sqrt {3 x^2+5 x+2}}dx\right )\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (-\frac {2}{9} \int -\frac {2189 x+1885}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {2}{9} \int \frac {2189 x+1885}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \int \frac {2189 x+1885}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (1885 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+2189 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (2189 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {1885 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2 x^{7/2} (95 x+74)}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac {2}{9} \left (\frac {4}{9} \left (\frac {1885 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+2189 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {3770}{9} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {2 (645 x+536) x^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )\)

Input:

Int[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
 

Output:

(2*x^(7/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((2*x^(3/2)*(536 
+ 645*x))/Sqrt[2 + 5*x + 3*x^2] - (3770*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/9 + 
 (4*(2189*((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])) + (1885*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt 
[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/9))/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 1236

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

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 = 1.02 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.13

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (\frac {\left (-\frac {2780}{2187}-\frac {3914 x}{2187}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (-\frac {23345}{729}-\frac {8974 x}{243}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}-\frac {10 \sqrt {3 x^{3}+5 x^{2}+2 x}}{81}-\frac {7540 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {8756 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(232\)
default	\(\frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (5472 \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}-13134 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{2}+9120 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -21890 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +3648 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-8756 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-405 x^{5}+240948 x^{4}+612270 x^{3}+504936 x^{2}+135720 x \right )}{729 \sqrt {x}\, \left (3 x +2\right )^{2} \left (x +1\right )^{2}}\)	\(302\)

Input:

int((2-5*x)*x^(9/2)/(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)*((-2780/2187-3914/2187 
*x)*(3*x^3+5*x^2+2*x)^(1/2)/(x^2+5/3*x+2/3)^2-2*x*(-23345/729-8974/243*x)* 
3^(1/2)/(x*(x^2+5/3*x+2/3))^(1/2)-10/81*(3*x^3+5*x^2+2*x)^(1/2)-7540/243*( 
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))-8756/243*(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))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.62 \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (24080 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 78804 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 27 \, {\left (45 \, x^{4} - 26772 \, x^{3} - 68030 \, x^{2} - 56104 \, x - 15080\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{2187 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:

integrate((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
 

Output:

-2/2187*(24080*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPIn 
verse(28/27, 80/729, x + 5/9) - 78804*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 2 
0*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, 
 x + 5/9)) + 27*(45*x^4 - 26772*x^3 - 68030*x^2 - 56104*x - 15080)*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. \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((2-5*x)*x**(9/2)/(3*x**2+5*x+2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
 

Output:

-integrate((5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2), x)
 

Giac [F]

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

integrate((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
 

Output:

integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x^{9/2}\,\left (5\,x-2\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:

int(-(x^(9/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

-int((x^(9/2)*(5*x - 2))/(5*x + 3*x^2 + 2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-1350 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}+15120 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}+92700 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}+126520 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x +75912 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}-683208 \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}-2277360 \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}-2808744 \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}-1518240 \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 -303648 \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 )+952560 \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}+3175200 \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}+3916080 \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}+2116800 \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 +423360 \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 )}{10935 x^{4}+36450 x^{3}+44955 x^{2}+24300 x +4860} \] Input:

int((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x)
 

Output:

(2*( - 675*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**4 + 7560*sqrt(x)*sqrt(3*x**2 
+ 5*x + 2)*x**3 + 46350*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**2 + 63260*sqrt(x 
)*sqrt(3*x**2 + 5*x + 2)*x + 37956*sqrt(x)*sqrt(3*x**2 + 5*x + 2) - 341604 
*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 - 1138680*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 - 1404372*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 - 759120*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 - 151824*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) + 4 
76280*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 + 1587600*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 + 1958040*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 + 105840 
0*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(27*x**6 + 135*x**5 + 279*x**4...