Integrand size = 28, antiderivative size = 191 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {74 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{5/2}}+\frac {3184 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 (2+3 x)^{3/2}}+\frac {220076 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 \sqrt {2+3 x}}-\frac {220076 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36015}-\frac {6584 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{36015} \]
-220076/108045*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-6584/108045*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)-2/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+74/735*(1-2*x)^(1/2)*( 3+5*x)^(1/2)/(2+3*x)^(5/2)+3184/5145*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^( 3/2)+220076/36015*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 4.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (926791+4100535 x+6042348 x^2+2971026 x^3\right )}{2 (2+3 x)^{7/2}}+i \sqrt {33} \left (55019 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-56665 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{108045} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(926791 + 4100535*x + 6042348*x^2 + 297 1026*x^3))/(2*(2 + 3*x)^(7/2)) + I*Sqrt[33]*(55019*EllipticE[I*ArcSinh[Sqr t[9 + 15*x]], -2/33] - 56665*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])) )/108045
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 27, 169, 27, 169, 27, 169, 27, 176, 123, 129}
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.
\(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^{9/2}} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {2}{21} \int -\frac {20 x+1}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{21} \int \frac {20 x+1}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{21} \left (\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}-\frac {2}{35} \int -\frac {3 (142-185 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \int \frac {142-185 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {13033-7960 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {13033-7960 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (55019 x+34822)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {110038 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {55019 x+34822}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {110038 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {9053}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {55019}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {110038 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {9053}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {55019}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {110038 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {1646}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {55019}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {110038 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1592 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {74 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\) |
(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + ((74*Sqrt[1 - 2*x] *Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (6*((1592*Sqrt[1 - 2*x]*Sqrt[3 + 5* x])/(21*(2 + 3*x)^(3/2)) + ((110038*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (10*((-55019*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5 - (1646*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5))/7)/21))/35)/21
3.27.52.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.24 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.40
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701 \left (\frac {2}{3}+x \right )^{4}}+\frac {74 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19845 \left (\frac {2}{3}+x \right )^{3}}+\frac {3184 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46305 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {440152}{7203} x^{2}-\frac {220076}{36015} x +\frac {220076}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {278576 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{756315 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {440152 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{756315 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
default | \(-\frac {2 \left (2885058 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2971026 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5770116 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-5942052 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3846744 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3961368 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+854832 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-880304 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-89130780 x^{5}-190183518 x^{4}-114403860 x^{3}+14275797 x^{2}+34124442 x +8341119\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{108045 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(409\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(-2/1701*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+74/19845*(-30*x^3-23*x^2 +7*x+6)^(1/2)/(2/3+x)^3+3184/46305*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+ 220076/108045*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+278576/75631 5*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^( 1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+440152/756315*(10+15*x)^(1/2 )*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*Ellip ticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^ (1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\frac {2 \, {\left (135 \, {\left (2971026 \, x^{3} + 6042348 \, x^{2} + 4100535 \, x + 926791\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1868543 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4951710 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{4862025 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
2/4862025*(135*(2971026*x^3 + 6042348*x^2 + 4100535*x + 926791)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1868543*sqrt(-30)*(81*x^4 + 216*x^3 + 2 16*x^2 + 96*x + 16)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 4951710*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassZe ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 2 3/90)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]