Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {988 \sqrt {1-2 x} \sqrt {3+5 x}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac {4418}{945} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {988}{945} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-2/15*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2)-4418/2835*EllipticE(1/7*21 ^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+988/2835*EllipticF(1/7*21^( 1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+74/45*(3+5*x)^(3/2)*(1-2*x)^( 1/2)/(2+3*x)^(3/2)+988/945*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.48 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (6449+20754 x+16731 x^2\right )}{(2+3 x)^{5/2}}+i \sqrt {33} \left (2209 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1715 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2835} \]
(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(6449 + 20754*x + 16731*x^2))/(2 + 3*x) ^(5/2) + I*Sqrt[33]*(2209*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 17 15*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/2835
Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {108, 27, 167, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{15} \int -\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{2 (3 x+2)^{5/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{(3 x+2)^{5/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \int \frac {\sqrt {5 x+3} (245 x+81)}{\sqrt {1-2 x} (3 x+2)^{3/2}}dx+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \left (\frac {2}{21} \int \frac {5 (2209 x+782)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {494 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \left (\frac {5}{21} \int \frac {2209 x+782}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {494 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \left (\frac {5}{21} \left (\frac {2209}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2717}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {494 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \left (\frac {5}{21} \left (-\frac {2717}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2209}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {494 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{9} \left (\frac {5}{21} \left (\frac {494}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2209}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {494 \sqrt {1-2 x} \sqrt {5 x+3}}{21 \sqrt {3 x+2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}\) |
(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(15*(2 + 3*x)^(5/2)) + ((74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + (2*((494*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*Sqrt[2 + 3*x]) + (5*((-2209*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3 /7]*Sqrt[1 - 2*x]], 35/33])/5 + (494*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7] *Sqrt[1 - 2*x]], 35/33])/5))/21))/9)/5
3.28.14.3.1 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[((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]
Leaf count of result is larger than twice the leaf count of optimal. \(253\) vs. \(2(116)=232\).
Time = 1.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.59
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3645 \left (\frac {2}{3}+x \right )^{3}}-\frac {148 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7436}{189} x^{2}-\frac {3718}{945} x +\frac {3718}{315}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {3128 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{19845 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8836 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{19845 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(254\) |
| default | \(-\frac {2 \left (14553 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-19881 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+19404 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-26508 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6468 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-8836 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-501930 x^{4}-672813 x^{3}-105153 x^{2}+167439 x +58041\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2835 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(314\) |
-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)*(14/3645*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-148/1215 *(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+3718/2835*(-30*x^2-3*x+9)/((2/3+x) *(-30*x^2-3*x+9))^(1/2)+3128/19845*(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))+8836/19845*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-2 3*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*Elli pticF((10+15*x)^(1/2),1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {270 \, {\left (16731 \, x^{2} + 20754 \, x + 6449\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 19573 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 198810 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{127575 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/127575*(270*(16731*x^2 + 20754*x + 6449)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqr t(-2*x + 1) - 19573*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassPInv erse(1159/675, 38998/91125, x + 23/90) + 198810*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(11 59/675, 38998/91125, x + 23/90)))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \]