Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt {3+5 x}}-\frac {703672 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{32484375}+\frac {2020841 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6496875}+\frac {346636 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{259875}-\frac {2972 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{7425}-\frac {48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {264260033 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{29531250 \sqrt {33}}-\frac {7261561 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14765625 \sqrt {33}} \]
-264260033/974531250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2)) *33^(1/2)-7261561/487265625*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155 ^(1/2))*33^(1/2)-2/5*(1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(1/2)-48/275*(1-2 *x)^(3/2)*(2+3*x)^(7/2)*(3+5*x)^(1/2)+2020841/6496875*(2+3*x)^(3/2)*(1-2*x )^(1/2)*(3+5*x)^(1/2)+346636/259875*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1 /2)-2972/7425*(2+3*x)^(7/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-703672/32484375*(1 -2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 9.01 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (26378214+71568535 x-32807925 x^2-141221250 x^3+56227500 x^4+127575000 x^5\right )}{\sqrt {3+5 x}}+264260033 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-278783155 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{974531250} \]
((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(26378214 + 71568535*x - 32807925*x^2 - 1 41221250*x^3 + 56227500*x^4 + 127575000*x^5))/Sqrt[3 + 5*x] + (264260033*I )*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (278783155*I)*Sqr t[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/974531250
Time = 0.31 (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, 171, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^{7/2}}{(5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{5} \int \frac {(1-72 x) (1-2 x)^{3/2} (3 x+2)^{5/2}}{2 \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {(1-72 x) (1-2 x)^{3/2} (3 x+2)^{5/2}}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{165} \int \frac {3 (943-7430 x) \sqrt {1-2 x} (3 x+2)^{5/2}}{2 \sqrt {5 x+3}}dx-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \int \frac {(943-7430 x) \sqrt {1-2 x} (3 x+2)^{5/2}}{\sqrt {5 x+3}}dx-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {2}{135} \int \frac {5 (116107-346636 x) (3 x+2)^{5/2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \int \frac {(116107-346636 x) (3 x+2)^{5/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {346636}{35} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}-\frac {1}{35} \int -\frac {(674816-2020841 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \int \frac {(674816-2020841 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {2020841}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (1407344 x+5653525)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (1407344 x+5653525)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {264260033 x+174531454}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1407344}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {264260033 x+174531454}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1407344}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {79877171}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {264260033}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1407344}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {79877171}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {264260033}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1407344}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{55} \left (\frac {1}{27} \left (\frac {1}{35} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {14523122}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {264260033}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1407344}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2020841}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {346636}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {2972}{27} \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {48}{55} (1-2 x)^{3/2} (3 x+2)^{7/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt {5 x+3}}\) |
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) + ((-48*(1 - 2*x)^( 3/2)*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/55 + ((-2972*Sqrt[1 - 2*x]*(2 + 3*x)^( 7/2)*Sqrt[3 + 5*x])/27 + ((346636*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5 *x])/35 + ((2020841*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + (3*( (-1407344*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ((-264260033*Sqr t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (14523122*S qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/15))/50)/3 5)/27)/55)/5
3.28.96.3.1 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[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 = 1.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-22963500000 x^{7}+262852689 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-264260033 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-13948200000 x^{6}+31387500000 x^{5}+13515714000 x^{4}-20371373550 x^{3}-8863610070 x^{2}+3502765680 x +1582692840\right )}{974531250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(160\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {198269 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{721875}+\frac {8960444 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{32484375}+\frac {174531454 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{3410859375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {264260033 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3410859375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {114694 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{144375}+\frac {216 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{275}-\frac {172 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1375}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{78125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(300\) |
-1/974531250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-22963500000*x^7+2 62852689*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellipt icF((10+15*x)^(1/2),1/35*70^(1/2))-264260033*5^(1/2)*(2+3*x)^(1/2)*7^(1/2) *(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))-139 48200000*x^6+31387500000*x^5+13515714000*x^4-20371373550*x^3-8863610070*x^ 2+3502765680*x+1582692840)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.37 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\frac {2700 \, {\left (127575000 \, x^{5} + 56227500 \, x^{4} - 141221250 \, x^{3} - 32807925 \, x^{2} + 71568535 \, x + 26378214\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 9629850101 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 23783402970 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{87707812500 \, {\left (5 \, x + 3\right )}} \]
1/87707812500*(2700*(127575000*x^5 + 56227500*x^4 - 141221250*x^3 - 328079 25*x^2 + 71568535*x + 26378214)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 9629850101*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/9112 5, x + 23/90) + 23783402970*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \]