Integrand size = 28, antiderivative size = 187 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {18551}{550} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1289089 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{500 \sqrt {33}}-\frac {9694 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{125 \sqrt {33}} \]
-1289089/16500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-9694/4125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/ 2)+1/3*(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-133/33*(2+3*x)^(5/2)*(3+5 *x)^(1/2)/(1-2*x)^(1/2)-797/110*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)- 18551/550*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.65 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (101763-275587 x+45342 x^2+8910 x^3\right )+1289089 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1327865 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{16500 (1-2 x)^{3/2}} \]
-1/16500*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(101763 - 275587*x + 45342*x^2 + 8910*x^3) + (1289089*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticE[I*ArcSinh[Sqr t[9 + 15*x]], -2/33] - (1327865*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[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 = {108, 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.
\(\displaystyle \int \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{(1-2 x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3 x+2)^{5/2} (120 x+73)}{2 (1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{3 (1-2 x)^{3/2}}-\frac {1}{6} \int \frac {(3 x+2)^{5/2} (120 x+73)}{(1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 167 |
\(\displaystyle \frac {1}{6} \left (-\frac {1}{11} \int -\frac {15 (3 x+2)^{3/2} (797 x+487)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {266 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \int \frac {(3 x+2)^{3/2} (797 x+487)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (37102 x+22875)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {797}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (37102 x+22875)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {1289089 x+816107}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {37102}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {1289089 x+816107}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {37102}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1289089}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {37102}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {37102}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {1}{6} \left (\frac {15}{11} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {38776}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {37102}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {797}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {266 (3 x+2)^{5/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) + ((-266*(2 + 3*x)^(5/ 2)*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (15*((-797*Sqrt[1 - 2*x]*(2 + 3*x)^ (3/2)*Sqrt[3 + 5*x])/25 + (3*((-37102*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ((-1289089*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5 - (38776*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5)/15))/50))/11)/6
3.30.46.3.1 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*((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 = 1.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.27
method | result | size |
default | \(-\frac {\left (2503974 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2578178 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1251987 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+1289089 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+1336500 x^{5}+8494200 x^{4}-32188470 x^{3}-34376560 x^{2}+2799750 x +6105780\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{16500 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(238\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {27 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{20}-\frac {411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}+\frac {816107 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{57750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1289089 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{57750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {89425}{88} x^{2}-\frac {339815}{264} x -\frac {17885}{44}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96 \left (x -\frac {1}{2}\right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(258\) |
-1/16500*(2503974*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)-2578178*5^(1/2)*7^(1/2)*Elli pticE((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)-1251987*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))+1289089*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) )+1336500*x^5+8494200*x^4-32188470*x^3-34376560*x^2+2799750*x+6105780)*(1- 2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*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 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {900 \, {\left (8910 \, x^{3} + 45342 \, x^{2} - 275587 \, x + 101763\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 43800583 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 116018010 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{1485000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
-1/1485000*(900*(8910*x^3 + 45342*x^2 - 275587*x + 101763)*sqrt(5*x + 3)*s qrt(3*x + 2)*sqrt(-2*x + 1) + 43800583*sqrt(-30)*(4*x^2 - 4*x + 1)*weierst rassPInverse(1159/675, 38998/91125, x + 23/90) - 116018010*sqrt(-30)*(4*x^ 2 - 4*x + 1)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(11 59/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1)
Timed out. \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]