Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {3+5 x}}{673876665 \sqrt {2+3 x}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {3316711588 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}-\frac {103970992 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{61261515 \sqrt {33}} \]
-2/33*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2)-3316711588/2021629995*Ell ipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-103970992/2021 629995*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1329 2/43659*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)+362/891*(3+5*x)^(5/2)*(1 -2*x)^(1/2)/(2+3*x)^(9/2)-1366496/4584195*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)^(5/2)+45748292/96268095*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+3316 711588/673876665*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (55875107717+415681177941 x+1234133449713 x^2+1829570010885 x^3+1356237833922 x^4+402980457942 x^5\right )}{2 (2+3 x)^{11/2}}+i \sqrt {33} \left (829177897 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-855170645 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2021629995} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(55875107717 + 415681177941*x + 1234133 449713*x^2 + 1829570010885*x^3 + 1356237833922*x^4 + 402980457942*x^5))/(2 *(2 + 3*x)^(11/2)) + I*Sqrt[33]*(829177897*EllipticE[I*ArcSinh[Sqrt[9 + 15 *x]], -2/33] - 855170645*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/20 21629995
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 = {108, 27, 167, 25, 167, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{33} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^{11/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^{11/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{33} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {2}{27} \int -\frac {(1993-1995 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \int \frac {(1993-1995 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {2}{147} \int \frac {3 (71718-63235 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{7/2}}dx-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \int \frac {(71718-63235 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{7/2}}dx-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {2}{105} \int \frac {3267421-817405 x}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \int \frac {3267421-817405 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \int \frac {200145479-114370730 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \int \frac {200145479-114370730 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (829177897 x+526098761)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1658355794 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {829177897 x+526098761}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1658355794 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {142960114}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {829177897}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {1658355794 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {142960114}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {829177897}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1658355794 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{33} \left (\frac {2}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {25992748}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {829177897}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {1658355794 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {22874146 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {683248 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {6646 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\) |
(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + ((362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) + (2*((-6646*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(49*(2 + 3*x)^(7/2)) + ((-683248*Sqrt[1 - 2*x]*Sqrt[3 + 5*x ])/(105*(2 + 3*x)^(5/2)) + ((22874146*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + ((1658355794*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x] ) + (10*((-829177897*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (25992748*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5))/7)/21)/105)/49))/27)/33
3.28.27.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 = 1.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {1658 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1948617 \left (\frac {2}{3}+x \right )^{5}}-\frac {309494 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31827411 \left (\frac {2}{3}+x \right )^{4}}+\frac {5986462 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{371319795 \left (\frac {2}{3}+x \right )^{3}}+\frac {45748292 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{866412855 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {6633423176}{134775333} x^{2}-\frac {3316711588}{673876665} x +\frac {3316711588}{224625555}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {4208790088 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{14151409965 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6633423176 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{14151409965 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{649539 \left (\frac {2}{3}+x \right )^{6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(326\) |
| default | \(\frac {2 \left (402980457942 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-391863622986 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1343268193140 \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}-1306212076620 \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}+1791024257520 \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}-1741616102160 \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}+1194016171680 \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}-1161077401440 \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}+398005390560 \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}-387025800480 \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}+12089413738260 x^{7}+53067385408 \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 )-51603440064 \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 )+41896076391486 x^{6}+55328989706838 x^{5}+30306573018747 x^{4}-293294410596 x^{3}-8183904282084 x^{2}-3573505278318 x -502875969453\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2021629995 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5* x)*(2+3*x))^(1/2)*(1658/1948617*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5-309 494/31827411*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+5986462/371319795*(-30 *x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+45748292/866412855*(-30*x^3-23*x^2+7*x+ 6)^(1/2)/(2/3+x)^2+3316711588/2021629995*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2 -3*x+9))^(1/2)+4208790088/14151409965*(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))+6633423176/14151409965*(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)))-14/649539*(-30*x^3-23 *x^2+7*x+6)^(1/2)/(2/3+x)^6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (402980457942 \, x^{5} + 1356237833922 \, x^{4} + 1829570010885 \, x^{3} + 1234133449713 \, x^{2} + 415681177941 \, x + 55875107717\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 28277796859 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 74626010730 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{90973349775 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
2/90973349775*(135*(402980457942*x^5 + 1356237833922*x^4 + 1829570010885*x ^3 + 1234133449713*x^2 + 415681177941*x + 55875107717)*sqrt(5*x + 3)*sqrt( 3*x + 2)*sqrt(-2*x + 1) - 28277796859*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860 *x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 389 98/91125, x + 23/90) + 74626010730*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^ 4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassZeta(1159/675, 38998/9112 5, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(729*x^6 + 2916 *x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]