Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {66055016 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 \sqrt {2+3 x}}-\frac {66055016 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}-\frac {1986944 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{27783} \]
-66055016/83349*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)-1986944/83349*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)+14/27*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+512/81*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+20420/567*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3* x)^(5/2)+950584/3969*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+66055016/27 783*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.48 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (557240459+3296666850 x+7318104714 x^2+7223771916 x^3+2675228148 x^4\right )}{4 (2+3 x)^{9/2}}+i \sqrt {33} \left (8256877 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-8505245 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{83349} \]
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(557240459 + 3296666850*x + 7318104714* x^2 + 7223771916*x^3 + 2675228148*x^4))/(4*(2 + 3*x)^(9/2)) + I*Sqrt[33]*( 8256877*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 8505245*EllipticF[I* ArcSinh[Sqrt[9 + 15*x]], -2/33])))/83349
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {109, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{11/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{27} \int \frac {(194-157 x) \sqrt {1-2 x}}{(3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{27} \left (\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}-\frac {2}{21} \int -\frac {7 (4451-6086 x)}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \int \frac {4451-6086 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {2}{35} \int \frac {15 (22591-25525 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \int \frac {22591-25525 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {2}{21} \int \frac {1960139-1188230 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \int \frac {1960139-1188230 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (8256877 x+5227331)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {16513754 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {8256877 x+5227331}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {16513754 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1366024}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {8256877}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {16513754 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {1366024}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {8256877}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {16513754 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{27} \left (\frac {1}{3} \left (\frac {6}{7} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {248368}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {8256877}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {16513754 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {237646 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {10210 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{5/2}}\right )+\frac {256 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}\) |
(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (2*((256*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + ((10210*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(5/2)) + (6*((237646*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21* (2 + 3*x)^(3/2)) + ((16513754*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x ]) + (10*((-8256877*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (248368*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/7)/21))/7)/3))/27
3.28.94.3.1 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[(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 = 1.32 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {20420 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15309 \left (\frac {2}{3}+x \right )^{3}}+\frac {950584 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35721 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {660550160}{27783} x^{2}-\frac {66055016}{27783} x +\frac {66055016}{9261}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {83637296 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{583443 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {132110032 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{583443 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{5}}+\frac {484 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(291\) |
| default | \(-\frac {2 \left (2598230844 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2675228148 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6928615584 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-7133941728 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6928615584 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-7133941728 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3079384704 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3170640768 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+513230784 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-528440128 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-80256844440 x^{6}-224738841924 x^{5}-217137403836 x^{4}-55840372398 x^{3}+39255728106 x^{2}+27998280273 x +5015164131\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{83349 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(504\) |
(-(-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 )*(20420/15309*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+950584/35721*(-30*x^ 3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+66055016/83349*(-30*x^2-3*x+9)/((2/3+x)*(- 30*x^2-3*x+9))^(1/2)+83637296/583443*(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))+132110032/583443*(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)))+98/19683*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^5+484/6561*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4)
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 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (135 \, {\left (2675228148 \, x^{4} + 7223771916 \, x^{3} + 7318104714 \, x^{2} + 3296666850 \, x + 557240459\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 561103238 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 1486237860 \, \sqrt {-30} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{3750705 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
2/3750705*(135*(2675228148*x^4 + 7223771916*x^3 + 7318104714*x^2 + 3296666 850*x + 557240459)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 561103238* sqrt(-30)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstras sPInverse(1159/675, 38998/91125, x + 23/90) + 1486237860*sqrt(-30)*(243*x^ 5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*weierstrassZeta(1159/675, 3 8998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(243*x ^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}\,\sqrt {5\,x+3}} \,d x \]