Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {16}{75} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {458}{375} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {178}{375} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
458/1125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-17 8/1125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5* (1-2*x)^(3/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)-16/75*(1-2*x)^(1/2)*(2+3*x)^(1/2 )*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 5.62 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} (39+10 x)}{\sqrt {3+5 x}}-229 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+140 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{1125} \]
(2*((-15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(39 + 10*x))/Sqrt[3 + 5*x] - (229*I)* Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (140*I)*Sqrt[33]*El lipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/1125
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} \sqrt {3 x+2}}{(5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{5} \int -\frac {3 \sqrt {1-2 x} (8 x+3)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} \sqrt {3 x+2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{5} \int \frac {\sqrt {1-2 x} (8 x+3)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {3}{5} \left (\frac {2}{45} \int \frac {458 x+79}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {16}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{5} \left (\frac {1}{45} \int \frac {458 x+79}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {16}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {3}{5} \left (\frac {1}{45} \left (\frac {458}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {979}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {16}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {3}{5} \left (\frac {1}{45} \left (-\frac {979}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {458}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {16}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {3}{5} \left (\frac {1}{45} \left (\frac {178}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {458}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {16}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\) |
(-2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (3*((16*Sqrt[1 - 2* x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + ((-458*Sqrt[11/3]*EllipticE[ArcSin[Sq rt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (178*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[ 3/7]*Sqrt[1 - 2*x]], 35/33])/5)/45))/5
3.28.40.3.1 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 = 1.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (132 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-229 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-900 x^{3}-3660 x^{2}-285 x +1170\right )}{1125 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(140\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75}-\frac {158 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {916 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
2/1125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(132*5^(1/2)*(2+3*x)^(1/2 )*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^( 1/2))-229*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellip ticE((10+15*x)^(1/2),1/35*70^(1/2))-900*x^3-3660*x^2-285*x+1170)/(30*x^3+2 3*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (675 \, {\left (10 \, x + 39\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 856 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 10305 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{50625 \, {\left (5 \, x + 3\right )}} \]
-2/50625*(675*(10*x + 39)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 856 *sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 10305*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125, weiers trassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)
\[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \]