Integrand size = 28, antiderivative size = 222 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{7/2}}-\frac {2818 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{5/2}}-\frac {5438 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 (2+3 x)^{3/2}}+\frac {189368 \sqrt {1-2 x} \sqrt {3+5 x}}{588245 \sqrt {2+3 x}}-\frac {189368 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{588245}-\frac {23012 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{588245} \]
-189368/1764735*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)-23012/1764735*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)+11/7*(3+5*x)^(1/2)/(2+3*x)^(7/2)/(1-2*x)^(1/2)-229/343*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-2818/12005*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)^(5/2)-5438/84035*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+189368/5882 45*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.71 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\frac {4 \left (-\frac {3 \sqrt {3+5 x} \left (-809083-2279324 x+1004571 x^2+7326810 x^3+5112936 x^4\right )}{2 \sqrt {1-2 x} (2+3 x)^{7/2}}+i \sqrt {33} \left (47342 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-53095 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1764735} \]
(4*((-3*Sqrt[3 + 5*x]*(-809083 - 2279324*x + 1004571*x^2 + 7326810*x^3 + 5 112936*x^4))/(2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + I*Sqrt[33]*(47342*Ellipti cE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 53095*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/1764735
Time = 0.30 (sec) , antiderivative size = 247, 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 = {109, 27, 169, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{3/2} (3 x+2)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {1}{7} \int -\frac {980 x+577}{2 \sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {980 x+577}{\sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \int \frac {5725 x+3347}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {2}{35} \int \frac {3 (14090 x+8487)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \int \frac {14090 x+8487}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {13595 x+24844}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (94684 x+69467)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {94684 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {94684 x+69467}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {94684 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {63283}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {94684}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {94684 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {63283}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {94684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {94684 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{14} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {11506}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {94684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {94684 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {5438 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {2818 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\) |
(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + ((-458*Sqrt[1 - 2*x ]*Sqrt[3 + 5*x])/(49*(2 + 3*x)^(7/2)) + (2*((-2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (3*((-5438*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*( 2 + 3*x)^(3/2)) + (2*((94684*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x] ) + (5*((-94684*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5 - (11506*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33 ])/5))/7))/21))/35))/49)/14
3.30.7.3.1 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 + 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 = 1.37 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27783 \left (\frac {2}{3}+x \right )^{4}}-\frac {508 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{324135 \left (\frac {2}{3}+x \right )^{3}}-\frac {818 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{756315 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {397216}{117649} x^{2}-\frac {198608}{588245} x +\frac {595824}{588245}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {277868 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12353145 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {378736 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12353145 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {88 \left (-30 x^{2}-38 x -12\right )}{16807 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
| default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (2703294 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2556468 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5406588 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-5112936 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3604392 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3408624 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+800976 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-757472 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-76694040 x^{5}-155918574 x^{4}-81009855 x^{3}+25148721 x^{2}+32650161 x +7281747\right )}{1764735 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\) | \(409\) |
(-(-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 )*(2/27783*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-508/324135*(-30*x^3-23*x ^2+7*x+6)^(1/2)/(2/3+x)^3-818/756315*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^ 2+198608/1764735*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+277868/12 353145*(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))+378736/12353145*(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)))-88/16807*(-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 = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\frac {2 \, {\left (135 \, {\left (5112936 \, x^{4} + 7326810 \, x^{3} + 1004571 \, x^{2} - 2279324 \, x - 809083\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 2037149 \, \sqrt {-30} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4260780 \, \sqrt {-30} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{79413075 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
2/79413075*(135*(5112936*x^4 + 7326810*x^3 + 1004571*x^2 - 2279324*x - 809 083)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 2037149*sqrt(-30)*(162*x ^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 4260780*sqrt(-30)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*weierstrassZeta(1159/675, 38998/91125, weierstrassPI nverse(1159/675, 38998/91125, x + 23/90)))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \]