Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2}}{25 \sqrt {3+5 x}}+\frac {178}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {496}{625} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {582}{625} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-2/15*(1-2*x)^(3/2)*(2+3*x)^(3/2)/(3+5*x)^(3/2)-582/6875*EllipticF(1/7*21^ (1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+496/1875*EllipticE(1/7*21^(1 /2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-62/25*(2+3*x)^(3/2)*(1-2*x)^(1 /2)/(3+5*x)^(1/2)+178/125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\frac {2 \left (-\frac {55 \sqrt {1-2 x} \sqrt {2+3 x} \left (437+800 x+150 x^2\right )}{(3+5 x)^{3/2}}-2728 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+1855 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{20625} \]
(2*((-55*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(437 + 800*x + 150*x^2))/(3 + 5*x)^(3 /2) - (2728*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (185 5*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/20625
Time = 0.24 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 27, 167, 25, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^{3/2}}{(5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {3 \sqrt {1-2 x} \sqrt {3 x+2} (12 x+1)}{2 (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \frac {\sqrt {1-2 x} \sqrt {3 x+2} (12 x+1)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{5} \left (-\frac {2}{5} \int -\frac {(25-267 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \int \frac {(25-267 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{15} \int \frac {3 (496 x+123)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{10} \int \frac {496 x+123}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {873}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {496}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {873}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {496}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {496}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {582}{5} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )+\frac {89}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}\) |
(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + ((-62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(5*Sqrt[3 + 5*x]) + (2*((89*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((496*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (582*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/10))/5)/5
3.28.48.3.1 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[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 = 1.29 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(\frac {2 \left (795 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1240 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+477 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-744 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4500 x^{4}-24750 x^{3}-15610 x^{2}+5815 x +4370\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{1875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(224\) |
| elliptic | \(-\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (-\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}-\frac {82 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{4375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {992 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{13125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9375 \left (x +\frac {3}{5}\right )^{2}}-\frac {248 \left (-30 x^{2}-5 x +10\right )}{1875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(249\) |
2/1875*(795*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+ 3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)-1240*5^(1/2)*7^(1/2)*EllipticE((10 +15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)+4 77*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((1 0+15*x)^(1/2),1/35*70^(1/2))-744*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))-4500*x^4-24750* x^3-15610*x^2+5815*x+4370)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x) ^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (150 \, x^{2} + 800 \, x + 437\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 169 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 22320 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{84375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
-1/84375*(450*(150*x^2 + 800*x + 437)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2* x + 1) + 169*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInverse(1159/675, 3 8998/91125, x + 23/90) + 22320*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZe ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 2 3/90)))/(25*x^2 + 30*x + 9)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]