Integrand size = 28, antiderivative size = 218 \[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {6478333 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{38500}-\frac {139163 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3850}-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {112543103 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{8250 \sqrt {35}}+\frac {6478333 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{16500 \sqrt {35}} \] Output:
-6478333/38500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-139163/3850*(1-2* x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-1327/154*(1-2*x)^(1/2)*(2+3*x)^(5/2)* (3+5*x)^(1/2)-166/33*(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+1/3*(2+3*x) ^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-112543103/288750*EllipticE(1/11*55^(1/2 )*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+6478333/577500*EllipticF(1/11*55 ^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 8.00 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {5 \sqrt {2+3 x} \sqrt {3+5 x} \left (35797779-94671446 x+19375686 x^2+6664680 x^3+1336500 x^4\right )+225086206 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-231856835 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{577500 (1-2 x)^{3/2}} \] Input:
Integrate[((2 + 3*x)^(9/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]
Output:
-1/577500*(5*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(35797779 - 94671446*x + 19375686 *x^2 + 6664680*x^3 + 1336500*x^4) + (225086206*I)*Sqrt[33 - 66*x]*(-1 + 2* x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (231856835*I)*Sqrt[33 - 6 6*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*x)^(3/ 2)
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {108, 27, 167, 27, 171, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{9/2} \sqrt {5 x+3}}{(1-2 x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{9/2} \sqrt {5 x+3}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3 x+2)^{7/2} (150 x+91)}{2 (1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{9/2} \sqrt {5 x+3}}{3 (1-2 x)^{3/2}}-\frac {1}{6} \int \frac {(3 x+2)^{7/2} (150 x+91)}{(1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{6} \left (-\frac {1}{11} \int -\frac {3 (3 x+2)^{5/2} (6635 x+4036)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {332 \sqrt {5 x+3} (3 x+2)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \int \frac {(3 x+2)^{5/2} (6635 x+4036)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (-\frac {1}{35} \int -\frac {5 (3 x+2)^{3/2} (278326 x+170069)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1327}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \int \frac {(3 x+2)^{3/2} (278326 x+170069)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (6478333 x+3994175)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {278326}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \int \frac {\sqrt {3 x+2} (6478333 x+3994175)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \left (-\frac {1}{15} \int -\frac {450172412 x+284998831}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6478333}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \left (\frac {1}{30} \int \frac {450172412 x+284998831}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6478333}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {74476919}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {450172412}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {6478333}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {74476919}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {450172412}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {6478333}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{11} \left (\frac {1}{14} \left (\frac {3}{25} \left (\frac {1}{30} \left (-\frac {13541258}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {450172412}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {6478333}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {278326}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {1327}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {332 (3 x+2)^{7/2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}\) |
Input:
Int[((2 + 3*x)^(9/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(5/2),x]
Output:
((2 + 3*x)^(9/2)*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) + ((-332*(2 + 3*x)^(7/ 2)*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*((-1327*Sqrt[1 - 2*x]*(2 + 3*x)^ (5/2)*Sqrt[3 + 5*x])/7 + ((-278326*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + (3*((-6478333*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ( (-450172412*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/ 5 - (13541258*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5)/30))/25)/14))/11)/6
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[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 = 0.68 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(-\frac {\left (446861514 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-900344824 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+200475000 x^{6}-223430757 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+450172412 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1253637000 x^{5}+4252832100 x^{4}-10119455760 x^{3}-11455366730 x^{2}+1121291250 x +2147866740\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}}{1155000 \left (1-2 x \right )^{\frac {3}{2}} \left (15 x^{2}+19 x +6\right )}\) | \(232\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {12123 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{700}-\frac {819477 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14000}+\frac {284998831 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1617000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {112543103 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{404250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {81 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28}+\frac {2401 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{192 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {49735}{11} x^{2}-\frac {188993}{33} x -\frac {19894}{11}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(286\) |
Input:
int((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/1155000*(446861514*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))* x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-900344824*2^(1/2)*EllipticE(1 /7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1 /2)+200475000*x^6-223430757*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^( 1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+450172412*2^(1/2)*(2+3*x) ^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^( 1/2))+1253637000*x^5+4252832100*x^4-10119455760*x^3-11455366730*x^2+112129 1250*x+2147866740)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(15*x^2+19*x+ 6)
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=-\frac {450 \, {\left (1336500 \, x^{4} + 6664680 \, x^{3} + 19375686 \, x^{2} - 94671446 \, x + 35797779\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 7647964657 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 20257758540 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{51975000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Output:
-1/51975000*(450*(1336500*x^4 + 6664680*x^3 + 19375686*x^2 - 94671446*x + 35797779)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 7647964657*sqrt(-30 )*(4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 20257758540*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675, 38998/ 91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4* x + 1)
Timed out. \[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(9/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2), x)
\[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 3)*(3*x + 2)^(9/2)/(-2*x + 1)^(5/2), x)
Timed out. \[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \] Input:
int(((3*x + 2)^(9/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(5/2),x)
Output:
int(((3*x + 2)^(9/2)*(5*x + 3)^(1/2))/(1 - 2*x)^(5/2), x)
\[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx=\frac {-21465000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}-107038800 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-311185260 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-1371535908 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +994049282 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-189236098800 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}+189236098800 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x -47309024700 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )+75716239228 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}-75716239228 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x +18929059807 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )}{7420000 x^{2}-7420000 x +1855000} \] Input:
int((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)
Output:
( - 21465000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 107038800 *sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 311185260*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 1371535908*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 994049282*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2 *x + 1) - 189236098800*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x **2)/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 + 1892360 98800*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x - 47309024700*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28*x**4 - 90*x**3 + 2 7*x**2 + 17*x - 6),x) + 75716239228*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 - 75716239228*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x + 18929059807*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x **2 + 17*x - 6),x))/(1855000*(4*x**2 - 4*x + 1))