Integrand size = 28, antiderivative size = 249 \[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {4373 \sqrt {1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac {294 (2+3 x)^{9/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {150812 \sqrt {1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt {3+5 x}}-\frac {371279941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{45753125}-\frac {31887029 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{18301250}-\frac {51601293223 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{83187500 \sqrt {33}}-\frac {776112041 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{41593750 \sqrt {33}} \]
7/33*(2+3*x)^(11/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)-51601293223/2745187500*Ell ipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-776112041/1372 593750*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-294/ 121*(2+3*x)^(9/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)+4373/19965*(2+3*x)^(7/2)*(1- 2*x)^(1/2)/(3+5*x)^(3/2)+150812/1098075*(2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x )^(1/2)-31887029/18301250*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-371279 941/45753125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.90 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {-\frac {10 \sqrt {2+3 x} \left (36533948644+21979664649 x-215557803774 x^2-222254370925 x^3+53010668700 x^4+8004966750 x^5\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+i \sqrt {33} \left (51601293223 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-53153517305 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{2745187500} \]
((-10*Sqrt[2 + 3*x]*(36533948644 + 21979664649*x - 215557803774*x^2 - 2222 54370925*x^3 + 53010668700*x^4 + 8004966750*x^5))/((1 - 2*x)^(3/2)*(3 + 5* x)^(3/2)) + I*Sqrt[33]*(51601293223*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], - 2/33] - 53153517305*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/27451875 00
Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {109, 27, 167, 25, 167, 27, 167, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{13/2}}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^{9/2} (274 x+157)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^{9/2} (274 x+157)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^{7/2} (26571 x+15068)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \int \frac {(3 x+2)^{7/2} (26571 x+15068)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{165} \int \frac {(3 x+2)^{5/2} (2761719 x+1626869)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \int \frac {(3 x+2)^{5/2} (2761719 x+1626869)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {2}{55} \int \frac {3 (3 x+2)^{3/2} (31887029 x+19498546)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \int \frac {(3 x+2)^{3/2} (31887029 x+19498546)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (1485119764 x+915676775)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {31887029}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (1485119764 x+915676775)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {51601293223 x+32668222424}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1485119764}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {51601293223 x+32668222424}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1485119764}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {8537232451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {51601293223}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1485119764}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {8537232451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {51601293223}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1485119764}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {1552224082}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {51601293223}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1485119764}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {31887029}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {301624 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {8746 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\right )-\frac {588 (3 x+2)^{9/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\) |
(7*(2 + 3*x)^(11/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-588*(2 + 3* x)^(9/2))/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((8746*Sqrt[1 - 2*x]*(2 + 3 *x)^(7/2))/(165*(3 + 5*x)^(3/2)) + ((301624*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)) /(55*Sqrt[3 + 5*x]) + (3*((-31887029*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + (3*((-1485119764*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/1 5 + ((-51601293223*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5 - (1552224082*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5)/15))/50))/55)/165)/11)/22
3.30.90.3.1 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 = 267, normalized size of antiderivative = 1.07
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {1102959343}{1815000000}+\frac {1838265689 x}{1815000000}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {129571567801}{43923000000}-\frac {18225070049 x}{4392300000}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {729 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2500}-\frac {23409 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12500}+\frac {16334111212 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{4804078125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {51601293223 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{9608156250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| default | \(-\frac {\sqrt {1-2 x}\, \left (501161734590 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-516012932230 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+50116173459 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-51601293223 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-150348520377 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+154803879669 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+240149002500 x^{6}+1750419396000 x^{5}-5607417753750 x^{4}-10911821531720 x^{3}-3651766136010 x^{2}+1535611752300 x +730678972880\right )}{2745187500 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\) | \(321\) |
(-(-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 )*((1102959343/1815000000+1838265689/1815000000*x)*(-30*x^3-23*x^2+7*x+6)^ (1/2)/(-3/10+x^2+1/10*x)^2-2*(-20-30*x)*(-129571567801/43923000000-1822507 0049/4392300000*x)/((-3/10+x^2+1/10*x)*(-20-30*x))^(1/2)-729/2500*x*(-30*x ^3-23*x^2+7*x+6)^(1/2)-23409/12500*(-30*x^3-23*x^2+7*x+6)^(1/2)+1633411121 2/4804078125*(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))+51601293223/96081 56250*(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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.55 \[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {900 \, {\left (8004966750 \, x^{5} + 53010668700 \, x^{4} - 222254370925 \, x^{3} - 215557803774 \, x^{2} + 21979664649 \, x + 36533948644\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1753310274031 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 4644116390070 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{247066875000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
-1/247066875000*(900*(8004966750*x^5 + 53010668700*x^4 - 222254370925*x^3 - 215557803774*x^2 + 21979664649*x + 36533948644)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 1753310274031*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 464411639 0070*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassZeta(1159/ 675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/ (100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Timed out. \[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {13}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {13}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{13/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]