Integrand size = 29, antiderivative size = 175 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=\frac {(8561+6179 x) \sqrt {2+5 x+3 x^2}}{1750 (3+2 x)^{3/2}}+\frac {(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}-\frac {721 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{500 \sqrt {2+5 x+3 x^2}}+\frac {1327 \sqrt {3} \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{700 \sqrt {2+5 x+3 x^2}} \]
1/175*(358+347*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(7/2)-721/500*EllipticE(3^(1 /2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^ (1/2)+1327/700*EllipticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2) ^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+1/1750*(8561+6179*x)*(3*x^2+5*x+2)^(1/2 )/(3+2*x)^(3/2)
Time = 31.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=-\frac {208240+878020 x+1386750 x^2+1009230 x^3+323760 x^4+31500 x^5+5047 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{9/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-1066 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{9/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{3500 (3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}} \]
-1/3500*(208240 + 878020*x + 1386750*x^2 + 1009230*x^3 + 323760*x^4 + 3150 0*x^5 + 5047*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] - 1066*Sqrt[5 ]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ellipt icF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2])
Time = 0.34 (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 = {1229, 1229, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \int \frac {(261 x+250) \sqrt {3 x^2+5 x+2}}{(2 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (-\frac {1}{30} \int -\frac {3 (5047 x+4253)}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (6179 x+8561)}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \int \frac {5047 x+4253}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {5047}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {6635}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {5047 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {6635 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {5047 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {6635 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {5047 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {6635 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}-\frac {3}{350} \left (\frac {1}{10} \left (\frac {5047 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {6635 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {(6179 x+8561) \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )\) |
((358 + 347*x)*(2 + 5*x + 3*x^2)^(3/2))/(175*(3 + 2*x)^(7/2)) - (3*(-1/15* ((8561 + 6179*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(3/2) + ((5047*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]) - (6635*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]))/10))/350
3.26.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
Time = 0.36 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.54
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{896 \left (x +\frac {3}{2}\right )^{4}}+\frac {867 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2240 \left (x +\frac {3}{2}\right )^{3}}-\frac {4317 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5600 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {7527}{1000} x^{2}+\frac {2509}{200} x +\frac {2509}{500}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {4253 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{17500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {721 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{2500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(270\) |
| default | \(-\frac {19056 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-40376 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+85752 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-181692 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+128628 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-272538 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+64314 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-136269 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-3161340 x^{5}-17552280 x^{4}-39066330 x^{3}-43245180 x^{2}-23622330 x -5052540}{52500 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {7}{2}}}\) | \(389\) |
((3+2*x)*(3*x^2+5*x+2))^(1/2)/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-65/896*( 6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4+867/2240*(6*x^3+19*x^2+19*x+6)^(1/2)/ (x+3/2)^3-4317/5600*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2+2509/2000*(6*x^2 +10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)+4253/17500*(-20-30*x)^(1/2)*(3+3*x )^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*EllipticF(1/5*(-20-30* x)^(1/2),1/2*10^(1/2))+721/2500*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^( 1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*EllipticE(1/5*(-20-30*x)^(1/2),1/2*1 0^(1/2))-EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=\frac {19339 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 90846 \, \sqrt {6} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (35126 \, x^{3} + 136482 \, x^{2} + 183183 \, x + 84209\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{63000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
1/63000*(19339*sqrt(6)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*weierstras sPInverse(19/27, -28/729, x + 19/18) + 90846*sqrt(6)*(16*x^4 + 96*x^3 + 21 6*x^2 + 216*x + 81)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19 /27, -28/729, x + 19/18)) + 36*(35126*x^3 + 136482*x^2 + 183183*x + 84209) *sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\, dx \]
-Integral(-10*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(-23*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(-10*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(3*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)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{9/2}} \,d x \]