Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {3+2 x} (11597+12083 x)}{243 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 \sqrt {3+2 x} (66911+83523 x)}{243 \sqrt {2+5 x+3 x^2}}-\frac {32}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \] Output:
-2/243*(3+2*x)^(1/2)*(11597+12083*x)/(3*x^2+5*x+2)^(3/2)+4/243*(3+2*x)^(1/ 2)*(66911+83523*x)/(3*x^2+5*x+2)^(1/2)-32/81*(3+2*x)^(1/2)*(3*x^2+5*x+2)^( 1/2)-110516/243*(-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)+148780/243*(-3*x^2-5*x-2)^(1/2)*Ellipt icF((1+x)^(1/2)*3^(1/2),1/3*I*6^(1/2))*3^(1/2)/(3*x^2+5*x+2)^(1/2)
Time = 31.64 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \left (3 (3+2 x) \left (-85285-330053 x-411640 x^2-166566 x^3+144 x^4\right )+2 \left (2+5 x+3 x^2\right ) \left (55258 \left (2+5 x+3 x^2\right )+27629 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-5312 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )\right )}{243 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \] Input:
Integrate[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(-2*(3*(3 + 2*x)*(-85285 - 330053*x - 411640*x^2 - 166566*x^3 + 144*x^4) + 2*(2 + 5*x + 3*x^2)*(55258*(2 + 5*x + 3*x^2) + 27629*Sqrt[5]*Sqrt[(1 + x) /(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqr t[5/3]/Sqrt[3 + 2*x]], 3/5] - 5312*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2* x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x ]], 3/5])))/(243*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))
Time = 0.65 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1233, 1233, 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.
| \(\displaystyle \int \frac {(5-x) (2 x+3)^{9/2}}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int \frac {(2 x+3)^{5/2} (411 x+4)}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \left (\frac {2}{3} \int -\frac {3 \sqrt {2 x+3} (7439 x+6081)}{\sqrt {3 x^2+5 x+2}}dx+\frac {2 (2571 x+2164) (2 x+3)^{3/2}}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \int \frac {\sqrt {2 x+3} (7439 x+6081)}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {2}{9} \int \frac {27629 x+22846}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \int \frac {27629 x+22846}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \left (\frac {27629}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {37195}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \left (\frac {27629 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {37195 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \left (\frac {27629 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {37195 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \left (\frac {27629 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {37195 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2}{9} \left (\frac {2 (2 x+3)^{3/2} (2571 x+2164)}{\sqrt {3 x^2+5 x+2}}-2 \left (\frac {1}{9} \left (\frac {27629 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {37195 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {14878}{9} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}\right )\right )-\frac {2 (2 x+3)^{7/2} (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(-2*(3 + 2*x)^(7/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (2*((2*(3 + 2*x)^(3/2)*(2164 + 2571*x))/Sqrt[2 + 5*x + 3*x^2] - 2*((14878*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/9 + ((27629*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[A rcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (3719 5*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sq rt[3]*Sqrt[2 + 5*x + 3*x^2]))/9)))/9
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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !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*(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])
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 = 1.72 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) \left (2 x +3\right )}\, \left (\frac {\left (-\frac {23194}{2187}-\frac {24166 x}{2187}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (6 x +9\right ) \left (-\frac {133822}{729}-\frac {55682 x}{243}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (6 x +9\right )}}-\frac {32 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{81}+\frac {91384 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {110516 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {30 x +45}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}\) | \(250\) |
| default | \(-\frac {2 \left (86094 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-165774 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+143490 \sqrt {15}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}-276290 \sqrt {15}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {2 x +3}+57396 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticF}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )-110516 \sqrt {-30 x -20}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {2 x +3}\, \operatorname {EllipticE}\left (\frac {\sqrt {-30 x -20}}{5}, \frac {\sqrt {10}}{2}\right )+12960 x^{5}-14971500 x^{4}-59534010 x^{3}-85276170 x^{2}-52232805 x -11513475\right ) \sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}}{3645 \left (x +1\right ) \left (2 x^{2}+5 x +3\right ) \left (3 x +2\right )^{2}}\) | \(325\) |
Input:
int((5-x)*(2*x+3)^(9/2)/(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)*((-23194/2 187-24166/2187*x)*(6*x^3+19*x^2+19*x+6)^(1/2)/(x^2+5/3*x+2/3)^2-2*(6*x+9)* (-133822/729-55682/243*x)/((x^2+5/3*x+2/3)*(6*x+9))^(1/2)-32/81*(6*x^3+19* x^2+19*x+6)^(1/2)+91384/1215*(-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)) +110516/1215*(-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))))
Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (113723 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 497322 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 27 \, {\left (144 \, x^{4} - 166566 \, x^{3} - 411640 \, x^{2} - 330053 \, x - 85285\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{2187 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:
integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
2/2187*(113723*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPIn verse(19/27, -28/729, x + 19/18) + 497322*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -2 8/729, x + 19/18)) - 27*(144*x^4 - 166566*x^3 - 411640*x^2 - 330053*x - 85 285)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**(5/2),x)
Output:
Timed out
\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
-integrate((2*x + 3)^(9/2)*(x - 5)/(3*x^2 + 5*x + 2)^(5/2), x)
\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
integrate(-(2*x + 3)^(9/2)*(x - 5)/(3*x^2 + 5*x + 2)^(5/2), x)
Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^{9/2}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:
int(-((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^(5/2),x)
Output:
-int(((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^(5/2), x)
\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {-63936 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}+351648 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}-1715616 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}-3969952 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x -2245674 \sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}+9780750 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{4}+32602500 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{3}+40209750 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{2}+21735000 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x +4347000 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right )-3316275 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{4}-11054250 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{3}-13633575 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x^{2}-7369500 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right ) x -1473900 \left (\int \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}}{54 x^{7}+351 x^{6}+963 x^{5}+1447 x^{4}+1287 x^{3}+678 x^{2}+196 x +24}d x \right )}{161838 x^{4}+539460 x^{3}+665334 x^{2}+359640 x +71928} \] Input:
int((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x)
Output:
( - 63936*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**4 + 351648*sqrt(2*x + 3) *sqrt(3*x**2 + 5*x + 2)*x**3 - 1715616*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2 )*x**2 - 3969952*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x - 2245674*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 9780750*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5* x + 2)*x**2)/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678* x**2 + 196*x + 24),x)*x**4 + 32602500*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x **2 + 196*x + 24),x)*x**3 + 40209750*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x* *2 + 196*x + 24),x)*x**2 + 21735000*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x**2)/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x** 2 + 196*x + 24),x)*x + 4347000*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)*x **2)/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x**2 + 1 96*x + 24),x) - 3316275*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(54*x** 7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x**2 + 196*x + 24),x )*x**4 - 11054250*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(54*x**7 + 35 1*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x**2 + 196*x + 24),x)*x**3 - 13633575*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(54*x**7 + 351*x**6 + 963*x**5 + 1447*x**4 + 1287*x**3 + 678*x**2 + 196*x + 24),x)*x**2 - 736 9500*int((sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2))/(54*x**7 + 351*x**6 + 9...