Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715} \]
10312712/5145*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/ 2)+310208/5145*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)+2/3*(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2)+176/35*(1-2*x)^(1/2)/(2+ 3*x)^(5/2)/(3+5*x)^(1/2)+12276/245*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/ 2)+1706144/1715*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-10312712/1029*(1 -2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (130497191+793777840 x+1809835578 x^2+1833255216 x^3+696108060 x^4\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (1289089 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1327865 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{5145} \]
(2*((-3*Sqrt[1 - 2*x]*(130497191 + 793777840*x + 1809835578*x^2 + 18332552 16*x^3 + 696108060*x^4))/((2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) - (4*I)*Sqrt[33]* (1289089*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1327865*EllipticF[I *ArcSinh[Sqrt[9 + 15*x]], -2/33])))/5145
Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^{9/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{21} \int \frac {77 (2-3 x)}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{3} \int \frac {2-3 x}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{3/2}}dx+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{3} \left (\frac {2}{35} \int \frac {437-600 x}{2 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \int \frac {437-600 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{21} \int \frac {3 (11018-12555 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \int \frac {11018-12555 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {2}{7} \int \frac {5 (188021-116328 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \int \frac {188021-116328 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{11} \int \frac {3 (1289089 x+816107)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2578178 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {6}{11} \int \frac {1289089 x+816107}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2578178 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {6}{11} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1289089}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2578178 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {6}{11} \left (\frac {213268}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2578178 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {22}{3} \left (\frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {6}{11} \left (-\frac {38776}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1289089}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2578178 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {116328 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {1674 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {24 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\right )+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}\) |
(2*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (22*((24*Sqrt[1 - 2* x])/(35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + ((1674*Sqrt[1 - 2*x])/(7*(2 + 3*x )^(3/2)*Sqrt[3 + 5*x]) + (2*((116328*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[ 3 + 5*x]) + (5*((-2578178*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (6*((-1289089*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5 - (38776*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33 ])/5))/11))/7))/7)/35))/3
3.28.45.3.1 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 = 1.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}-\frac {878 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945 \left (\frac {2}{3}+x \right )^{3}}-\frac {67558 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{2}}-\frac {7482962 \left (-30 x^{2}-3 x +9\right )}{5145 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {13057712 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20625424 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {550 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (135214596 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-139221612 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+270429192 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-278443224 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+180286128 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-185628816 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+40063584 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-41250848 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4176648360 x^{5}-8911207116 x^{4}-5359247820 x^{3}+666839694 x^{2}+1598350374 x +391491573\right )}{5145 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\) | \(409\) |
(-(-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 )*(-2/81*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-878/945*(-30*x^3-23*x^2+7* x+6)^(1/2)/(2/3+x)^3-67558/2205*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-748 2962/5145*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)-13057712/36015*( 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))-20625424/36015*(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*Ellipti cE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1 /2)))-550*(-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 = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (696108060 \, x^{4} + 1833255216 \, x^{3} + 1809835578 \, x^{2} + 793777840 \, x + 130497191\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 87601166 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 232036020 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{231525 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
-2/231525*(135*(696108060*x^4 + 1833255216*x^3 + 1809835578*x^2 + 79377784 0*x + 130497191)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 87601166*sqr t(-30)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*weierstrass PInverse(1159/675, 38998/91125, x + 23/90) + 232036020*sqrt(-30)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*weierstrassZeta(1159/675, 3 8998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(405*x ^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]