\(\int \frac {5-x}{(3+2 x)^{3/2} (2+5 x+3 x^2)^{5/2}} \, dx\) [1084]

Optimal result

Integrand size = 29, antiderivative size = 202 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {26}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2 \sqrt {3+2 x} (164+279 x)}{25 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (7489+8799 x)}{125 \sqrt {2+5 x+3 x^2}}-\frac {11732 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {3212 \sqrt {3} \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}} \] Output:

-26/5/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(3/2)-2/25*(3+2*x)^(1/2)*(164+279*x)/(3* 
x^2+5*x+2)^(3/2)+4/125*(3+2*x)^(1/2)*(7489+8799*x)/(3*x^2+5*x+2)^(1/2)-117 
32/125*(-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)+3212/25*(-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 = 31.49 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.06 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (5 \left (8031+29941 x+36414 x^2+14454 x^3\right )-5866 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} \sqrt {3+2 x} \sqrt {\frac {2+3 x}{3+2 x}} \left (6+19 x+19 x^2+6 x^3\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+1048 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} \sqrt {3+2 x} \sqrt {\frac {2+3 x}{3+2 x}} \left (6+19 x+19 x^2+6 x^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{125 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:

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

Output:

(2*(5*(8031 + 29941*x + 36414*x^2 + 14454*x^3) - 5866*Sqrt[5]*Sqrt[(1 + x) 
/(3 + 2*x)]*Sqrt[3 + 2*x]*Sqrt[(2 + 3*x)/(3 + 2*x)]*(6 + 19*x + 19*x^2 + 6 
*x^3)*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 1048*Sqrt[5]*Sqrt[ 
(1 + x)/(3 + 2*x)]*Sqrt[3 + 2*x]*Sqrt[(2 + 3*x)/(3 + 2*x)]*(6 + 19*x + 19* 
x^2 + 6*x^3)*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(125*Sqrt[3 
 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1235, 27, 1235, 1237, 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.

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

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {2}{15} \int \frac {3 (235 x+306)}{(2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{5} \int \frac {235 x+306}{(2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \int \frac {2409 x+2147}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int \frac {3 (2933 x+2392)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \int \frac {2933 x+2392}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2933}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {4015}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2933 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4015 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2933 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {4015 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2933 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {4015 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \left (\frac {5866 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {2933 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {4015 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 (2409 x+2054)}{5 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}\)

Input:

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

Output:

(-2*(37 + 47*x))/(5*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(2054 
 + 2409*x))/(5*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) - (2*((5866*Sqrt[2 + 5 
*x + 3*x^2])/(5*Sqrt[3 + 2*x]) - (3*((2933*Sqrt[-2 - 5*x - 3*x^2]*Elliptic 
E[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (4 
015*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])))/5))/5))/5
 

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 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 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 = 1.92 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.28

method	result	size
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (\frac {\left (-\frac {302}{225}-\frac {134 x}{75}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (6 x +9\right ) \left (-\frac {1002}{25}-\frac {1194 x}{25}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (6 x +9\right )}}-\frac {208 \left (6 x^{2}+10 x +4\right )}{125 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {9568 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{625 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {11732 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{625 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\)	\(259\)
default	\(-\frac {2 \sqrt {3 x^{2}+5 x +2}\, \left (9738 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-17598 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+16230 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-29330 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+6492 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-11732 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-1583820 x^{4}-6363450 x^{3}-9242310 x^{2}-5765175 x -1306245\right )}{1875 \left (x +1\right )^{2} \left (3 x +2\right )^{2} \sqrt {2 x +3}}\)	\(308\)

Input:

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

Output:

((3*x^2+5*x+2)*(2*x+3))^(1/2)/(2*x+3)^(1/2)/(3*x^2+5*x+2)^(1/2)*((-302/225 
-134/75*x)*(6*x^3+19*x^2+19*x+6)^(1/2)/(x^2+5/3*x+2/3)^2-2*(6*x+9)*(-1002/ 
25-1194/25*x)/((x^2+5/3*x+2/3)*(6*x+9))^(1/2)-208/125*(6*x^2+10*x+4)/((x+3 
/2)*(6*x^2+10*x+4))^(1/2)+9568/625*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*(30*x+45 
)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^ 
(1/2))+11732/625*(-30*x-20)^(1/2)*(3+3*x)^(1/2)*(30*x+45)^(1/2)/(6*x^3+19* 
x^2+19*x+6)^(1/2)*(1/3*EllipticE(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))-Ellipt 
icF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (12671 \, \sqrt {6} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 52794 \, \sqrt {6} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 9 \, {\left (105588 \, x^{4} + 424230 \, x^{3} + 616154 \, x^{2} + 384345 \, x + 87083\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{1125 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \] Input:

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

Output:

2/1125*(12671*sqrt(6)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*we 
ierstrassPInverse(19/27, -28/729, x + 19/18) + 52794*sqrt(6)*(18*x^5 + 87* 
x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*weierstrassZeta(19/27, -28/729, weier 
strassPInverse(19/27, -28/729, x + 19/18)) + 9*(105588*x^4 + 424230*x^3 + 
616154*x^2 + 384345*x + 87083)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(18*x^ 
5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)
 

Sympy [F]

\[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(x/(18*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 87*x**4*sqrt(2 
*x + 3)*sqrt(3*x**2 + 5*x + 2) + 164*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x 
+ 2) + 151*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(2*x + 3)* 
sqrt(3*x**2 + 5*x + 2) + 12*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x) - In 
tegral(-5/(18*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 87*x**4*sqrt(2*x 
 + 3)*sqrt(3*x**2 + 5*x + 2) + 164*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 
2) + 151*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(2*x + 3)*sq 
rt(3*x**2 + 5*x + 2) + 12*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x}{108 x^{8}+864 x^{7}+2979 x^{6}+5783 x^{5}+6915 x^{4}+5217 x^{3}+2426 x^{2}+636 x +72}d x \right )+5 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{108 x^{8}+864 x^{7}+2979 x^{6}+5783 x^{5}+6915 x^{4}+5217 x^{3}+2426 x^{2}+636 x +72}d x \right ) \] Input:

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

Output:

 - int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x)/(108*x**8 + 864*x**7 + 297 
9*x**6 + 5783*x**5 + 6915*x**4 + 5217*x**3 + 2426*x**2 + 636*x + 72),x) + 
5*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(108*x**8 + 864*x**7 + 2979*x 
**6 + 5783*x**5 + 6915*x**4 + 5217*x**3 + 2426*x**2 + 636*x + 72),x)