Integrand size = 28, antiderivative size = 218 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {1336 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {806 \sqrt {1-2 x} \sqrt {3+5 x}}{207515 (2+3 x)^{5/2}}+\frac {349904 \sqrt {1-2 x} \sqrt {3+5 x}}{1452605 (2+3 x)^{3/2}}+\frac {26062156 \sqrt {1-2 x} \sqrt {3+5 x}}{10168235 \sqrt {2+3 x}}-\frac {26062156 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{924385 \sqrt {33}}-\frac {837304 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{924385 \sqrt {33}} \]
-26062156/30504705*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)-837304/30504705*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/ 2))*33^(1/2)+4/231*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2)+1336/17787*(3 +5*x)^(1/2)/(2+3*x)^(5/2)/(1-2*x)^(1/2)-806/207515*(1-2*x)^(1/2)*(3+5*x)^( 1/2)/(2+3*x)^(5/2)+349904/1452605*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2 )+26062156/10168235*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (165071409-176797172 x-914077314 x^2+513206712 x^3+1407356424 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^{5/2}}+2 i \sqrt {33} \left (6515539 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-6724865 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{30504705} \]
(2*((Sqrt[3 + 5*x]*(165071409 - 176797172*x - 914077314*x^2 + 513206712*x^ 3 + 1407356424*x^4))/((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + (2*I)*Sqrt[33]*(6 515539*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 6724865*EllipticF[I*A rcSinh[Sqrt[9 + 15*x]], -2/33])))/30504705
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {115, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{7/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}-\frac {2}{231} \int -\frac {210 x+229}{2 (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \int \frac {210 x+229}{(1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}-\frac {2}{77} \int -\frac {3 (16700 x+10999)}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \int \frac {16700 x+10999}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {2}{35} \int \frac {3 (2015 x+30502)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \int \frac {2015 x+30502}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {1588673-874760 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {1588673-874760 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (6515539 x+4139582)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {13031078 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {6515539 x+4139582}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {13031078 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1151293}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {6515539}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {13031078 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1151293}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6515539}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {13031078 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{231} \left (\frac {3}{77} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {209326}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {6515539}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {13031078 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {174952 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {806 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {1336 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{5/2}}\) |
(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)) + ((1336*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) + (3*((-806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + (6*((174952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 1*(2 + 3*x)^(3/2)) + ((13031078*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3 *x]) + (10*((-6515539*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5 - (209326*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] , 35/33])/5))/7)/21))/35))/77)/231
3.30.79.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 = 1.40 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.35
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}+\frac {48 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {431352}{16807} x^{2}-\frac {215676}{84035} x +\frac {647028}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {33116656 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{213532935 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {52124312 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{213532935 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{79233 \left (x -\frac {1}{2}\right )^{2}}-\frac {6928 \left (-30 x^{2}-38 x -12\right )}{6100941 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (228261132 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-234559404 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+190217610 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-195466170 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-50724696 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+52124312 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-50724696 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+52124312 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-7036782120 x^{5}-6788102832 x^{4}+3030766434 x^{3}+3626217802 x^{2}-294965529 x -495214227\right )}{30504705 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(406\) |
(-(-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 )*(2/1715*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+48/1715*(-30*x^3-23*x^2+7 *x+6)^(1/2)/(2/3+x)^2+71892/84035*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9) )^(1/2)+33116656/213532935*(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))+521 24312/213532935*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-2 3*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*Elli pticF((10+15*x)^(1/2),1/35*70^(1/2)))+8/79233*(-30*x^3-23*x^2+7*x+6)^(1/2) /(x-1/2)^2-6928/6100941*(-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 = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (45 \, {\left (1407356424 \, x^{4} + 513206712 \, x^{3} - 914077314 \, x^{2} - 176797172 \, x + 165071409\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 222704983 \, \sqrt {-30} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 586398510 \, \sqrt {-30} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1372711725 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
2/1372711725*(45*(1407356424*x^4 + 513206712*x^3 - 914077314*x^2 - 1767971 72*x + 165071409)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 222704983*s qrt(-30)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*weierstrassPInver se(1159/675, 38998/91125, x + 23/90) + 586398510*sqrt(-30)*(108*x^5 + 108* x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*weierstrassZeta(1159/675, 38998/91125, we ierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \]