Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {10 (1-2 x)^{3/2} \sqrt {3+5 x}}{99 (2+3 x)^{9/2}}+\frac {1900 \sqrt {1-2 x} \sqrt {3+5 x}}{2079 (2+3 x)^{7/2}}+\frac {76492 \sqrt {1-2 x} \sqrt {3+5 x}}{14553 (2+3 x)^{5/2}}+\frac {3560432 \sqrt {1-2 x} \sqrt {3+5 x}}{101871 (2+3 x)^{3/2}}+\frac {247408648 \sqrt {1-2 x} \sqrt {3+5 x}}{713097 \sqrt {2+3 x}}-\frac {247408648 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{64827 \sqrt {33}}-\frac {7442032 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{64827 \sqrt {33}} \]
-247408648/2139291*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3 3^(1/2)-7442032/2139291*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/ 2))*33^(1/2)-2/33*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2)+10/99*(1-2*x) ^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+1900/2079*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)^(7/2)+76492/14553*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+3560432 /101871*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+247408648/713097*(1-2*x) ^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.91 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (4174268813+30956769477 x+91862628912 x^2+136342955970 x^3+101209884912 x^4+30060150732 x^5\right )}{4 (2+3 x)^{11/2}}+i \sqrt {33} \left (30926081 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-31856335 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2139291} \]
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4174268813 + 30956769477*x + 918626289 12*x^2 + 136342955970*x^3 + 101209884912*x^4 + 30060150732*x^5))/(4*(2 + 3 *x)^(11/2)) + I*Sqrt[33]*(30926081*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2 /33] - 31856335*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/2139291
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, 167, 27, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} \sqrt {5 x+3}}{(3 x+2)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{33} \int -\frac {5 (1-2 x)^{3/2} (12 x+5)}{2 (3 x+2)^{11/2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{5/2} \sqrt {5 x+3}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \int \frac {(1-2 x)^{3/2} (12 x+5)}{(3 x+2)^{11/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{27} \int \frac {9 (23-13 x) \sqrt {1-2 x}}{(3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \int \frac {(23-13 x) \sqrt {1-2 x}}{(3 x+2)^{9/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}-\frac {2}{21} \int -\frac {3329-4568 x}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \int \frac {3329-4568 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {2}{35} \int \frac {3 (84608-95615 x)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \int \frac {84608-95615 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {7341667-4450540 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {7341667-4450540 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (30926081 x+19578928)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61852162 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {30926081 x+19578928}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61852162 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {5116397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {30926081}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {61852162 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {5116397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {30926081}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {61852162 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{33} \left (-\frac {2}{3} \left (\frac {1}{21} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {930254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {30926081}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {61852162 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {890108 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38246 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {190 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{7/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{9/2}}\right )-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}\) |
(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) - (5*((-2*(1 - 2* x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(9/2)) - (2*((190*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/(21*(2 + 3*x)^(7/2)) + ((38246*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35* (2 + 3*x)^(5/2)) + (6*((890108*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^ (3/2)) + ((61852162*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (10*( (-30926081*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (930254*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 ))/7)/21))/35)/21))/3))/33
3.28.63.3.1 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 = 1.36 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {568 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{56133 \left (\frac {2}{3}+x \right )^{4}}+\frac {76492 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{392931 \left (\frac {2}{3}+x \right )^{3}}+\frac {3560432 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{916839 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2474086480}{713097} x^{2}-\frac {247408648}{713097} x +\frac {247408648}{237699}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {313262848 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {494817296 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{14975037 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8019 \left (\frac {2}{3}+x \right )^{5}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(315\) |
| default | \(-\frac {2 \left (29194965756 \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}-30060150732 \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}+97316552520 \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}-100200502440 \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}+129755403360 \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}-133600669920 \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}+86503602240 \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}-89067113280 \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}+28834534080 \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}-29689037760 \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}-901804521960 x^{7}+3844604544 \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 )-3958538368 \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 )-3126476999556 x^{6}-4123376977248 x^{5}-2254018771062 x^{4}+22795632684 x^{3}+608665287387 x^{2}+266088118854 x +37568419317\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2139291 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
(-(-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 )*(568/56133*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+76492/392931*(-30*x^3- 23*x^2+7*x+6)^(1/2)/(2/3+x)^3+3560432/916839*(-30*x^3-23*x^2+7*x+6)^(1/2)/ (2/3+x)^2+247408648/2139291*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2 )+313262848/14975037*(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))+494817296 /14975037*(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)))-98/216513*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/ 3+x)^6+14/8019*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5)
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 \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (30060150732 \, x^{5} + 101209884912 \, x^{4} + 136342955970 \, x^{3} + 91862628912 \, x^{2} + 30956769477 \, x + 4174268813\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 2101607314 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 5566694580 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{96268095 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
2/96268095*(135*(30060150732*x^5 + 101209884912*x^4 + 136342955970*x^3 + 9 1862628912*x^2 + 30956769477*x + 4174268813)*sqrt(5*x + 3)*sqrt(3*x + 2)*s qrt(-2*x + 1) - 2101607314*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320 *x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 5566694580*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassZeta(1159/675, 38998/91125, weierstra ssPInverse(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. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]