\(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{\sqrt {3+2 x}} \, dx\) [1044]

Optimal result

Integrand size = 29, antiderivative size = 175 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx=-\frac {\sqrt {3+2 x} (107+12429 x) \sqrt {2+5 x+3 x^2}}{5670}+\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {11123 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1620 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {20501 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{2268 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:

-1/5670*(3+2*x)^(1/2)*(107+12429*x)*(3*x^2+5*x+2)^(1/2)+1/63*(52-7*x)*(3+2 
*x)^(1/2)*(3*x^2+5*x+2)^(3/2)-11123/4860*(-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)+20501/6804*(- 
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 = 203, normalized size of antiderivative = 1.16 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx=-\frac {2 \sqrt {3+2 x} \left (-10832-312914 x-1043385 x^2-1306791 x^3-687798 x^4-88290 x^5+34020 x^6\right )+77861 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-16358 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{34020 (3+2 x) \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/34020*(2*Sqrt[3 + 2*x]*(-10832 - 312914*x - 1043385*x^2 - 1306791*x^3 - 
 687798*x^4 - 88290*x^5 + 34020*x^6) + 77861*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] - 16358*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 = 0.55 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1231, 1231, 25, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{126} \int \frac {(1381 x+1204) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}dx\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \int -\frac {77861 x+65539}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{126} \left (-\frac {1}{90} \int \frac {77861 x+65539}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (12429 x+107)\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \left (\frac {102505}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {77861}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \left (\frac {102505 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {77861 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \left (\frac {102505 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {77861 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \left (\frac {102505 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {77861 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{126} \left (\frac {1}{90} \left (\frac {102505 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {77861 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {1}{45} \sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

((52 - 7*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))/63 + (-1/45*(Sqrt[3 + 2 
*x]*(107 + 12429*x)*Sqrt[2 + 5*x + 3*x^2]) + ((-77861*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]) + (102505*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + 
x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]))/90)/126
 

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 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 1231

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

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 = 1.80 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (1020600 x^{6}-2648700 x^{5}+36966 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-77861 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-20633940 x^{4}-39203730 x^{3}-38309040 x^{2}-21066570 x -4996620\right )}{510300 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(151\)
risch	\(-\frac {\left (1890 x^{3}-10890 x^{2}-9711 x -9253\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{5670}-\frac {\left (\frac {65539 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{170100 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {11123 \sqrt {30 x +45}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\operatorname {EllipticF}\left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{24300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(208\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3}+\frac {121 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{63}+\frac {1079 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{630}+\frac {9253 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5670}+\frac {65539 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{170100 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {11123 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{24300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(253\)

Input:

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

Output:

-1/510300*(3*x^2+5*x+2)^(1/2)*(2*x+3)^(1/2)*(1020600*x^6-2648700*x^5+36966 
*(-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))-77861*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(2 
*x+3)^(1/2)*EllipticE(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-20633940*x^4-3920 
3730*x^3-38309040*x^2-21066570*x-4996620)/(6*x^3+19*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.35 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx=-\frac {1}{5670} \, {\left (1890 \, x^{3} - 10890 \, x^{2} - 9711 \, x - 9253\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {299657}{612360} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {11123}{4860} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \] Input:

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

Output:

-1/5670*(1890*x^3 - 10890*x^2 - 9711*x - 9253)*sqrt(3*x^2 + 5*x + 2)*sqrt( 
2*x + 3) + 299657/612360*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 1 
9/18) + 11123/4860*sqrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInv 
erse(19/27, -28/729, x + 19/18))
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \] Input:

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

Output:

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-23*x*sq 
rt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-10*x**2*sqrt(3*x**2 + 5 
*x + 2)/sqrt(2*x + 3), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/sqrt(2* 
x + 3), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx=-\frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{3}+\frac {121 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{63}+\frac {1079 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{630}+\frac {10139 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{7980}+\frac {11123 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{3420}-\frac {657 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{6 x^{3}+19 x^{2}+19 x +6}d x \right )}{280} \] Input:

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

Output:

( - 15960*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**3 + 91960*sqrt(2*x + 3)* 
sqrt(3*x**2 + 5*x + 2)*x**2 + 82004*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x 
 + 60834*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 155722*int((sqrt(2*x + 3)* 
sqrt(3*x**2 + 5*x + 2)*x**2)/(6*x**3 + 19*x**2 + 19*x + 6),x) - 112347*int 
((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(6*x**3 + 19*x**2 + 19*x + 6),x))/ 
47880