Integrand size = 28, antiderivative size = 187 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {317384 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{15125}-\frac {27271 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6050}-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {44109377 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27500 \sqrt {33}}-\frac {663409 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{13750 \sqrt {33}} \]
-44109377/907500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^ (1/2)-663409/453750*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))* 33^(1/2)+7/33*(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-910/363*(2+3*x)^(5 /2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)-27271/6050*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+ 5*x)^(1/2)-317384/15125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (3478434-9445541 x+1528956 x^2+294030 x^3\right )+44109377 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-45436195 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{907500 (1-2 x)^{3/2}} \]
-1/907500*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3478434 - 9445541*x + 1528956*x ^2 + 294030*x^3) + (44109377*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticE[I*Arc Sinh[Sqrt[9 + 15*x]], -2/33] - (45436195*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*Ell ipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*x)^(3/2)
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {109, 27, 167, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{9/2}}{(1-2 x)^{5/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{7/2} \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(3 x+2)^{5/2} (822 x+499)}{2 (1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{7/2} \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(3 x+2)^{5/2} (822 x+499)}{(1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{66} \left (-\frac {1}{11} \int -\frac {3 (3 x+2)^{3/2} (27271 x+16664)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1820 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \int \frac {(3 x+2)^{3/2} (27271 x+16664)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (1269536 x+782725)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {27271}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (1269536 x+782725)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {44109377 x+27925126}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1269536}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {44109377 x+27925126}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1269536}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {7297499}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {44109377}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1269536}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {7297499}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {44109377}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1269536}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {1326818}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {44109377}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1269536}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {27271}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1820 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}\) |
(7*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + ((-1820*(2 + 3*x) ^(5/2)*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*((-27271*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + (3*((-1269536*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*S qrt[3 + 5*x])/15 + ((-44109377*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[ 1 - 2*x]], 35/33])/5 - (1326818*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt [1 - 2*x]], 35/33])/5)/15))/50))/11)/66
3.30.71.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 = 1.37 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {\left (85679682 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-88218754 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-42839841 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+44109377 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+44104500 x^{5}+285209100 x^{4}-1108687710 x^{3}-1181150330 x^{2}+94170000 x +208706040\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{907500 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(238\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{100}-\frac {2511 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}+\frac {13962563 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1588125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {44109377 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3176250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2401 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1056 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {615685}{968} x^{2}-\frac {2339603}{2904} x -\frac {123137}{484}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(258\) |
-1/907500*(85679682*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2 ))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)-88218754*5^(1/2)*7^(1/2)*E llipticE((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3- 5*x)^(1/2)-42839841*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^( 1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+44109377*5^(1/2)*(2+3*x)^(1/ 2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^ (1/2))+44104500*x^5+285209100*x^4-1108687710*x^3-1181150330*x^2+94170000*x +208706040)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+1 9*x+6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {900 \, {\left (294030 \, x^{3} + 1528956 \, x^{2} - 9445541 \, x + 3478434\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1498745669 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 3969843930 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{81675000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/81675000*(900*(294030*x^3 + 1528956*x^2 - 9445541*x + 3478434)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 1498745669*sqrt(-30)*(4*x^2 - 4*x + 1 )*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 3969843930*sqrt( -30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675, 38998/91125, weierstrassP Inverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1)
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]