Integrand size = 28, antiderivative size = 189 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {36 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{5/2}}-\frac {26 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{3/2}}+\frac {5636 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 \sqrt {2+3 x}}-\frac {5636 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {4364 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{12005 \sqrt {33}} \]
-5636/36015*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) -4364/396165*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2 )+2/7*(3+5*x)^(1/2)/(2+3*x)^(5/2)/(1-2*x)^(1/2)-36/245*(1-2*x)^(1/2)*(3+5* x)^(1/2)/(2+3*x)^(5/2)-26/1715*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+5 636/12005*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\frac {2 \left (-\frac {33 \sqrt {3+5 x} \left (-11923-13127 x+41724 x^2+50724 x^3\right )}{\sqrt {1-2 x} (2+3 x)^{5/2}}+2 i \sqrt {33} \left (15499 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16590 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{396165} \]
(2*((-33*Sqrt[3 + 5*x]*(-11923 - 13127*x + 41724*x^2 + 50724*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (2*I)*Sqrt[33]*(15499*EllipticE[I*ArcSinh[Sqrt[ 9 + 15*x]], -2/33] - 16590*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/ 396165
Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {110, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}-\frac {2}{7} \int -\frac {75 x+44}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {75 x+44}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{35} \int \frac {540 x+347}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \int \frac {540 x+347}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{21} \int \frac {3 (65 x+513)}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \int \frac {65 x+513}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {2}{7} \int \frac {5 (2818 x+1909)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \int \frac {2818 x+1909}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {1091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2818}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {1091}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2818}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {2182 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {2818}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {26 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {36 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{5/2}}\) |
(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + ((-36*Sqrt[1 - 2*x]* Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + ((-26*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/( 7*(2 + 3*x)^(3/2)) + (2*((2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3* x]) + (5*((-2818*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 33])/5 - (2182*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[ 33])))/7))/7)/35)/7
3.29.98.3.1 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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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 = 1.36 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.43
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{3}}+\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {11512}{2401} x^{2}-\frac {5756}{12005} x +\frac {17268}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {7636 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{252105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {11272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{252105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8 \left (-30 x^{2}-38 x -12\right )}{2401 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(271\) |
| default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (25596 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-25362 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+34128 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-33816 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+11376 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-11272 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-760860 x^{4}-1082376 x^{3}-178611 x^{2}+296988 x +107307\right )}{36015 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\) | \(314\) |
(-(-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 )*(-2/2205*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+34/15435*(-30*x^3-23*x^2 +7*x+6)^(1/2)/(2/3+x)^2+5756/36015*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9 ))^(1/2)+7636/252105*(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))+11272/252 105*(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*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15 *x)^(1/2),1/35*70^(1/2)))-8/2401*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x- 12))^(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 = 0.68 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\frac {2 \, {\left (135 \, {\left (50724 \, x^{3} + 41724 \, x^{2} - 13127 \, x - 11923\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 53498 \, \sqrt {-30} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 126810 \, \sqrt {-30} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1620675 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
2/1620675*(135*(50724*x^3 + 41724*x^2 - 13127*x - 11923)*sqrt(5*x + 3)*sqr t(3*x + 2)*sqrt(-2*x + 1) - 53498*sqrt(-30)*(54*x^4 + 81*x^3 + 18*x^2 - 20 *x - 8)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 126810*sqr t(-30)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*weierstrassZeta(1159/675, 389 98/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{7/2}} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}} \,d x \]