Integrand size = 28, antiderivative size = 253 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {20549264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{9261} \]
683150096/27783*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)+20549264/27783*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))* 33^(1/2)+14/27*(1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2)+652/81*(1-2*x)^(1 /2)/(2+3*x)^(7/2)/(3+5*x)^(1/2)+11660/189*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5 *x)^(1/2)+813208/1323*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+113020952/ 9261*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-3415750480/27783*(1-2*x)^(1 /2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.84 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (17289178827+131099014240 x+397527527442 x^2+602551975428 x^3+456548966244 x^4+138337894440 x^5\right )}{(2+3 x)^{9/2} \sqrt {3+5 x}}-8 i \sqrt {33} \left (42696881 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-43981210 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{27783} \]
(2*((-3*Sqrt[1 - 2*x]*(17289178827 + 131099014240*x + 397527527442*x^2 + 6 02551975428*x^3 + 456548966244*x^4 + 138337894440*x^5))/((2 + 3*x)^(9/2)*S qrt[3 + 5*x]) - (8*I)*Sqrt[33]*(42696881*EllipticE[I*ArcSinh[Sqrt[9 + 15*x ]], -2/33] - 43981210*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/27783
Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{11/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{27} \int \frac {(229-227 x) \sqrt {1-2 x}}{(3 x+2)^{9/2} (5 x+3)^{3/2}}dx+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{27} \left (\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}-\frac {2}{21} \int -\frac {77 (661-996 x)}{2 \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \int \frac {661-996 x}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{35} \int \frac {5 (14473-19875 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \int \frac {14473-19875 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {2}{21} \int \frac {3 (729869-831690 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \int \frac {729869-831690 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (6227584-3852987 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {6227584-3852987 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {2}{11} \int \frac {3 (85393762 x+54061781)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {85393762 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \int \frac {85393762 x+54061781}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {85393762 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {14127619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {85393762}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {85393762 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (\frac {14127619}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {85393762}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {85393762 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{27} \left (\frac {11}{3} \left (\frac {2}{7} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {3}{11} \left (-\frac {2568658}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {85393762}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {85393762 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {7705974 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {55446 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {1590 \sqrt {1-2 x}}{7 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {326 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\right )+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}\) |
(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (2*((326*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (11*((1590*Sqrt[1 - 2*x])/(7*( 2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2*((55446*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/ 2)*Sqrt[3 + 5*x]) + ((7705974*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x ]) + (10*((-85393762*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (3* ((-85393762*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/ 5 - (2568658*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]) /5))/11))/7)/7))/7))/3))/27
3.29.5.3.1 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 = 4.86 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{5}}-\frac {1114 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{4}}-\frac {73970 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5103 \left (\frac {2}{3}+x \right )^{3}}-\frac {5028574 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11907 \left (\frac {2}{3}+x \right )^{2}}-\frac {515062946 \left (-30 x^{2}-3 x +9\right )}{27783 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {864988496 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1366300192 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6050 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(319\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (26871262704 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-27667578888 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+71656700544 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-73780210368 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+71656700544 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-73780210368 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+31847422464 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-32791204608 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5307903744 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5465200768 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-830027366640 x^{6}-2324280114144 x^{5}-2245664953836 x^{4}-577509238368 x^{3}+405988496886 x^{2}+289561969758 x +51867536481\right )}{27783 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\) | \(504\) |
(-(-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 )*(-98/6561*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5-1114/2187*(-30*x^3-23*x ^2+7*x+6)^(1/2)/(2/3+x)^4-73970/5103*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^ 3-5028574/11907*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-515062946/27783*(-3 0*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)-864988496/194481*(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))-1366300192/194481*(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+1 5*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2)))-60 50*(-30*x^2-5*x+10)/((x+3/5)*(-30*x^2-5*x+10))^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (138337894440 \, x^{5} + 456548966244 \, x^{4} + 602551975428 \, x^{3} + 397527527442 \, x^{2} + 131099014240 \, x + 17289178827\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5803007528 \, \sqrt {-30} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 15370877160 \, \sqrt {-30} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1250235 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]
-2/1250235*(135*(138337894440*x^5 + 456548966244*x^4 + 602551975428*x^3 + 397527527442*x^2 + 131099014240*x + 17289178827)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 5803007528*sqrt(-30)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*weierstrassPInverse(1159/675, 38998/911 25, x + 23/90) + 15370877160*sqrt(-30)*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6 840*x^3 + 3360*x^2 + 880*x + 96)*weierstrassZeta(1159/675, 38998/91125, we ierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]