\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{9/2}} \, dx\) [1058]

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 = 207 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx=\frac {1}{140} (200-4269 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {(11453+8007 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}+\frac {(134+111 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {4091 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {2505 \sqrt {3} \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{56 \sqrt {2+5 x+3 x^2}} \] Output:

1/140*(200-4269*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)+1/210*(11453+8007*x)* 
(3*x^2+5*x+2)^(3/2)/(3+2*x)^(3/2)+1/35*(134+111*x)*(3*x^2+5*x+2)^(5/2)/(3+ 
2*x)^(7/2)-4091/120*(-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)+2505/56*(-3*x^2-5*x-2)^(1/2)*Ellip 
ticF((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 = 31.47 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx=-\frac {1223436+5250234 x+8516152 x^2+6437058 x^3+2140148 x^4+158172 x^5-29736 x^6+4536 x^7+28637 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{9/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-6092 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{9/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{840 (3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

-1/840*(1223436 + 5250234*x + 8516152*x^2 + 6437058*x^3 + 2140148*x^4 + 15 
8172*x^5 - 29736*x^6 + 4536*x^7 + 28637*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 
 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 
+ 2*x]], 3/5] - 6092*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[ 
(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 
+ 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1230, 25, 1229, 27, 1230, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {1}{14} \int -\frac {(223 x+187) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{7/2}}dx-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{14} \int \frac {(223 x+187) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{7/2}}dx-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {1}{14} \left (-\frac {1}{50} \int -\frac {5 (5469 x+4646) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{3/2}}dx-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \int \frac {(5469 x+4646) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{3/2}}dx-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {(1823 x+6292) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}-\frac {1}{6} \int \frac {3 (28637 x+24168)}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {(1823 x+6292) \sqrt {3 x^2+5 x+2}}{\sqrt {2 x+3}}-\frac {1}{2} \int \frac {28637 x+24168}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {1}{2} \left (\frac {37575}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {28637}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (1823 x+6292)}{\sqrt {2 x+3}}\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {1}{2} \left (\frac {12525 \sqrt {3} \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {28637 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1823 x+6292)}{\sqrt {2 x+3}}\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {1}{2} \left (\frac {37575 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {28637 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1823 x+6292)}{\sqrt {2 x+3}}\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {1}{2} \left (\frac {12525 \sqrt {3} \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {28637 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1823 x+6292)}{\sqrt {2 x+3}}\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \left (\frac {1}{2} \left (\frac {12525 \sqrt {3} \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {28637 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (1823 x+6292)}{\sqrt {2 x+3}}\right )-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^{5/2}}\right )-\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}\)

Input:

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

Output:

-1/35*((43 + 7*x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(7/2) + (-1/15*((3354 
 + 2531*x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^(5/2) + (((6292 + 1823*x)*Sq 
rt[2 + 5*x + 3*x^2])/Sqrt[3 + 2*x] + ((-28637*Sqrt[-2 - 5*x - 3*x^2]*Ellip 
ticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + 
 (12525*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x 
]], -2/3])/Sqrt[2 + 5*x + 3*x^2])/2)/10)/14
 

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 1229
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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1230
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (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 = 1.77 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.49

method result size
elliptic \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (-\frac {325 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{3584 \left (x +\frac {3}{2}\right )^{4}}+\frac {1595 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1792 \left (x +\frac {3}{2}\right )^{3}}-\frac {1403 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{336 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {4949}{40} x^{2}+\frac {4949}{24} x +\frac {4949}{60}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {9 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{80}+\frac {8 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {1007 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{175 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {4091 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{600 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) \(309\)
default \(-\frac {107256 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}-229096 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {3+3 x}\, \sqrt {2 x +3}\, \sqrt {-30 x -20}+482652 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-1030932 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+68040 x^{7}+723978 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-1546398 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-446040 x^{6}+361989 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-773199 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-18246060 x^{5}-95046060 x^{4}-211005510 x^{3}-235661250 x^{2}-130010220 x -28040400}{12600 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {7}{2}}}\) \(399\)

Input:

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^(9/2),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)*(-325/3584 
*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4+1595/1792*(6*x^3+19*x^2+19*x+6)^(1/ 
2)/(x+3/2)^3-1403/336*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2+4949/240*(6*x^ 
2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-9/80*x*(6*x^3+19*x^2+19*x+6)^(1/2 
)+8/5*(6*x^3+19*x^2+19*x+6)^(1/2)+1007/175*(-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))+4091/600*(-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)) 
-EllipticF(1/5*(-30*x-20)^(1/2),1/2*10^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.66 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx=\frac {109079 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 515466 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (756 \, x^{5} - 6216 \, x^{4} - 192878 \, x^{3} - 730460 \, x^{2} - 998487 \, x - 467340\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{15120 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \] Input:

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

Output:

1/15120*(109079*sqrt(6)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*weierstra 
ssPInverse(19/27, -28/729, x + 19/18) + 515466*sqrt(6)*(16*x^4 + 96*x^3 + 
216*x^2 + 216*x + 81)*weierstrassZeta(19/27, -28/729, weierstrassPInverse( 
19/27, -28/729, x + 19/18)) - 36*(756*x^5 - 6216*x^4 - 192878*x^3 - 730460 
*x^2 - 998487*x - 467340)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(16*x^4 + 9 
6*x^3 + 216*x^2 + 216*x + 81)
 

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\, dx \] Input:

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

Output:

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt 
(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3 
)), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96 
*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sq 
rt(2*x + 3)), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt 
(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2* 
x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2) 
/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 
 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-15*x**4*sqrt(3*x* 
*2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sq 
rt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(9*x** 
5*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 
216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 81648*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**5 + 671328*sqrt(2*x + 3) 
*sqrt(3*x**2 + 5*x + 2)*x**4 - 3911544*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2 
)*x**3 - 48683376*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2 - 118863948*sq 
rt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 99202142*sqrt(2*x + 3)*sqrt(3*x**2 
+ 5*x + 2) - 1317803040*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(9 
6*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2835*x 
 + 486),x)*x**4 - 7906818240*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x** 
2)/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x**2 + 2 
835*x + 486),x)*x**3 - 17790341040*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 
2)*x**2)/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939*x* 
*2 + 2835*x + 486),x)*x**2 - 17790341040*int((sqrt(2*x + 3)*sqrt(3*x**2 + 
5*x + 2)*x**2)/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6 
939*x**2 + 2835*x + 486),x)*x - 6671377890*int((sqrt(2*x + 3)*sqrt(3*x**2 
+ 5*x + 2)*x**2)/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 
 6939*x**2 + 2835*x + 486),x) + 1122946640*int((sqrt(2*x + 3)*sqrt(3*x**2 
+ 5*x + 2))/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939 
*x**2 + 2835*x + 486),x)*x**4 + 6737679840*int((sqrt(2*x + 3)*sqrt(3*x**2 
+ 5*x + 2))/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + 6939 
*x**2 + 2835*x + 486),x)*x**3 + 15159779640*int((sqrt(2*x + 3)*sqrt(3*x**2 
 + 5*x + 2))/(96*x**7 + 880*x**6 + 3424*x**5 + 7320*x**4 + 9270*x**3 + ...