\(\int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx\) [1063]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [A] (verification not implemented)
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 29, antiderivative size = 192 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {1010}{189} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {10}{7} (3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}-\frac {2}{21} (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}+\frac {865 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{27 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {2525 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{189 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

1010/189*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)+10/7*(3+2*x)^(3/2)*(3*x^2+5*x+2 
)^(1/2)-2/21*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(1/2)+865/81*(-3*x^2-5*x-2)^(1/2) 
*EllipticE((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)- 
2525/567*(-3*x^2-5*x-2)^(1/2)*EllipticF((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2)) 
*3^(1/2)/(3*x^2+5*x+2)^(1/2)
 

Mathematica [A] (verified)

Time = 31.41 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {8 \sqrt {3+2 x} \left (-6758-20111 x-18501 x^2-5526 x^3-216 x^4+162 x^5\right )-6055 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+4540 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{567 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/567*(8*Sqrt[3 + 2*x]*(-6758 - 20111*x - 18501*x^2 - 5526*x^3 - 216*x^4 
+ 162*x^5) - 6055*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3* 
x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 4540*Sqrt[ 
5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF 
[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1236, 27, 1236, 27, 1236, 27, 1269, 1172, 27, 321, 327}

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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{21} \int \frac {25 (2 x+3)^{3/2} (9 x+14)}{2 \sqrt {3 x^2+5 x+2}}dx-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{21} \int \frac {(2 x+3)^{3/2} (9 x+14)}{\sqrt {3 x^2+5 x+2}}dx-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {25}{21} \left (\frac {2}{15} \int \frac {3 \sqrt {2 x+3} (101 x+129)}{2 \sqrt {3 x^2+5 x+2}}dx+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \int \frac {\sqrt {2 x+3} (101 x+129)}{\sqrt {3 x^2+5 x+2}}dx+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {2}{9} \int \frac {1211 x+1564}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \int \frac {1211 x+1564}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1211}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {505}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1211 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {505 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1211 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {505 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1211 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {505 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {25}{21} \left (\frac {1}{5} \left (\frac {1}{9} \left (\frac {1211 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {505 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {202}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )+\frac {6}{5} \sqrt {3 x^2+5 x+2} (2 x+3)^{3/2}\right )-\frac {2}{21} (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}\)

Input:

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

Output:

(-2*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])/21 + (25*((6*(3 + 2*x)^(3/2)*Sq 
rt[2 + 5*x + 3*x^2])/5 + ((202*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/9 + (( 
1211*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/ 
(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (505*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[Ar 
cSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/9)/5))/ 
21
 

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 321
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

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 1269
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 
Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.76

method result size
default \(-\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (3888 x^{5}+1059 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+1211 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-5184 x^{4}-132624 x^{3}-335034 x^{2}-301014 x -89532\right )}{1701 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\) \(146\)
risch \(-\frac {2 \left (36 x^{2}-162 x -829\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{189}-\frac {\left (-\frac {1564 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{567 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {173 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{81 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) \(203\)
elliptic \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {8 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{21}+\frac {12 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{7}+\frac {1658 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{189}-\frac {1564 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{567 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {173 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{81 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) \(231\)

Input:

int((5-x)*(2*x+3)^(5/2)/(3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/1701*(2*x+3)^(1/2)*(3*x^2+5*x+2)^(1/2)*(3888*x^5+1059*(-30*x-20)^(1/2)* 
(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*EllipticF(1/5*(-30*x-20)^(1/2),1/2*10 
^(1/2))+1211*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2*x+3)^(1/2)*Ellipti 
cE(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-5184*x^4-132624*x^3-335034*x^2-30101 
4*x-89532)/(6*x^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.30 \[ \int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {2}{189} \, {\left (36 \, x^{2} - 162 \, x - 829\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {25715}{10206} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - \frac {865}{81} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

integrate((5-x)*(3+2*x)^(5/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
 

Output:

-2/189*(36*x^2 - 162*x - 829)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 25715/ 
10206*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 19/18) - 865/81*sqrt 
(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x 
+ 19/18))
 

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx=- \int \left (- \frac {45 \sqrt {2 x + 3}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {51 x \sqrt {2 x + 3}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2} \sqrt {2 x + 3}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3} \sqrt {2 x + 3}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \] Input:

integrate((5-x)*(3+2*x)**(5/2)/(3*x**2+5*x+2)**(1/2),x)
 

Output:

-Integral(-45*sqrt(2*x + 3)/sqrt(3*x**2 + 5*x + 2), x) - Integral(-51*x*sq 
rt(2*x + 3)/sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2*sqrt(2*x + 3)/sq 
rt(3*x**2 + 5*x + 2), x) - Integral(4*x**3*sqrt(2*x + 3)/sqrt(3*x**2 + 5*x 
 + 2), x)
 

Maxima [F]

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

integrate((5-x)*(3+2*x)^(5/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((5-x)*(3+2*x)^(5/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(-((2*x + 3)^(5/2)*(x - 5))/(5*x + 3*x^2 + 2)^(1/2),x)
 

Output:

-int(((2*x + 3)^(5/2)*(x - 5))/(5*x + 3*x^2 + 2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(5-x) (3+2 x)^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {8 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{21}+\frac {12 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{7}+\frac {1391 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{133}-\frac {865 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{57}+\frac {355 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{14} \] Input:

int((5-x)*(3+2*x)^(5/2)/(3*x^2+5*x+2)^(1/2),x)
 

Output:

( - 304*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 + 1368*sqrt(2*x + 3)*sqr 
t(3*x**2 + 5*x + 2)*x + 8346*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) - 12110* 
int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(6*x**3 + 19*x**2 + 19*x + 
 6),x) + 20235*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(6*x**3 + 19*x** 
2 + 19*x + 6),x))/798