Integrand size = 29, antiderivative size = 224 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {6 (37+47 x)}{5 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}}-\frac {4124 \sqrt {2+5 x+3 x^2}}{125 (3+2 x)^{5/2}}-\frac {61468 \sqrt {2+5 x+3 x^2}}{1875 (3+2 x)^{3/2}}-\frac {426748 \sqrt {2+5 x+3 x^2}}{9375 \sqrt {3+2 x}}+\frac {213374 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{3125 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {30734 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{625 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]
-6/5*(37+47*x)/(3+2*x)^(5/2)/(3*x^2+5*x+2)^(1/2)+213374/9375*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)-30734/1875*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)-4124/125*(3*x^2+5*x+2)^(1/2)/(3+2*x) ^(5/2)-61468/1875*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)-426748/9375*(3*x^2+5*x +2)^(1/2)/(3+2*x)^(1/2)
Time = 31.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.81 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 \left (1439445+3957355 x+3383680 x^2+922020 x^3-106687 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+60586 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{9375 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \]
(-2*(1439445 + 3957355*x + 3383680*x^2 + 922020*x^3 - 106687*Sqrt[5]*Sqrt[ (1 + x)/(3 + 2*x)]*(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[Arc Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 60586*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)] *(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt [3 + 2*x]], 3/5]))/(9375*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])
Time = 0.43 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1235, 1237, 27, 1237, 27, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \int \frac {705 x+542}{(2 x+3)^{7/2} \sqrt {3 x^2+5 x+2}}dx-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}-\frac {2}{25} \int -\frac {9279 x+6235}{2 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \int \frac {9279 x+6235}{(2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}dx+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac {2}{15} \int -\frac {46101 x+15808}{2 (2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \int \frac {46101 x+15808}{(2 x+3)^{3/2} \sqrt {3 x^2+5 x+2}}dx+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {2}{5} \int \frac {3 (106687 x+121613)}{2 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \int \frac {106687 x+121613}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {106687}{2} \int \frac {\sqrt {2 x+3}}{\sqrt {3 x^2+5 x+2}}dx-\frac {76835}{2} \int \frac {1}{\sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}dx\right )\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {106687 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {3} \sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {76835 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {3}}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {106687 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {76835 \sqrt {-3 x^2-5 x-2} \int \frac {1}{\sqrt {1-3 (x+1)} \sqrt {6 (x+1)+3}}d\left (\sqrt {3} \sqrt {x+1}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {106687 \sqrt {-3 x^2-5 x-2} \int \frac {\sqrt {6 (x+1)+3}}{\sqrt {1-3 (x+1)}}d\left (\sqrt {3} \sqrt {x+1}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {76835 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2}{5} \left (\frac {1}{25} \left (\frac {1}{15} \left (\frac {213374 \sqrt {3 x^2+5 x+2}}{5 \sqrt {2 x+3}}-\frac {3}{5} \left (\frac {106687 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {76835 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{\sqrt {3} \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {30734 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}\right )+\frac {2062 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}\right )-\frac {6 (47 x+37)}{5 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}}\) |
(-6*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2]) - (2*((2062*Sqr t[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^(5/2)) + ((30734*Sqrt[2 + 5*x + 3*x^2])/ (15*(3 + 2*x)^(3/2)) + ((213374*Sqrt[2 + 5*x + 3*x^2])/(5*Sqrt[3 + 2*x]) - (3*((106687*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]) - (76835*Sqrt[-2 - 5*x - 3*x^2]*El lipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2] )))/5)/15)/25))/5
3.27.20.3.1 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)*(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] )
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])
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 = 0.38 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {13 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{125 \left (x +\frac {3}{2}\right )^{3}}-\frac {1822 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1875 \left (x +\frac {3}{2}\right )^{2}}-\frac {87104 \left (6 x^{2}+10 x +4\right )}{9375 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {2 \left (9+6 x \right ) \left (\frac {2959}{625}+\frac {4209 x}{625}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {243226 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{46875 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {213374 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{46875 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(274\) |
| default | \(-\frac {2 \left (179112 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+426748 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+537336 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+1280244 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+403002 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+960183 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+38407320 x^{4}+193064460 x^{3}+354813150 x^{2}+280202415 x +79202655\right )}{140625 \left (3+2 x \right )^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}\) | \(296\) |
((3+2*x)*(3*x^2+5*x+2))^(1/2)/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-13/125*( 6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3-1822/1875*(6*x^3+19*x^2+19*x+6)^(1/2) /(x+3/2)^2-87104/9375*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-2*(9+6 *x)*(2959/625+4209/625*x)/((x^2+5/3*x+2/3)*(9+6*x))^(1/2)-243226/46875*(-2 0-30*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*El lipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-213374/46875*(-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*10^(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 = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65 \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {161981 \, \sqrt {6} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 1920366 \, \sqrt {6} {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 18 \, {\left (2560488 \, x^{4} + 12870964 \, x^{3} + 23654210 \, x^{2} + 18680161 \, x + 5280177\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{84375 \, {\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} \]
1/84375*(161981*sqrt(6)*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54 )*weierstrassPInverse(19/27, -28/729, x + 19/18) - 1920366*sqrt(6)*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)*weierstrassZeta(19/27, -28/72 9, weierstrassPInverse(19/27, -28/729, x + 19/18)) - 18*(2560488*x^4 + 128 70964*x^3 + 23654210*x^2 + 18680161*x + 5280177)*sqrt(3*x^2 + 5*x + 2)*sqr t(2*x + 3))/(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \frac {x}{24 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 148 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 358 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 423 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 243 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 54 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{24 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 148 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 358 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 423 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 243 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 54 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(24*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 148*x**4*sqrt( 2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 358*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 423*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 243*x*sqrt(2*x + 3 )*sqrt(3*x**2 + 5*x + 2) + 54*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(24*x**5*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 148*x**4*sqrt( 2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 358*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 423*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 243*x*sqrt(2*x + 3 )*sqrt(3*x**2 + 5*x + 2) + 54*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=\int { -\frac {x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^{7/2}\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,d x \]