Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {214}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {4157 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}+\frac {412 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3375} \]
-4157/10125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) +412/10125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+ 2/15*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+214/675*(1-2*x)^(1/2)*(2+3* x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 2.69 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\frac {-60 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (-76+45 x)+4157 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-3745 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{10125} \]
(-60*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-76 + 45*x) + (4157*I)*Sqr t[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (3745*I)*Sqrt[33]*Elli pticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/10125
Time = 0.23 (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 = {112, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{\sqrt {3 x+2}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{15} (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2}{15} \int -\frac {\sqrt {1-2 x} (107 x+62)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \int \frac {\sqrt {1-2 x} (107 x+62)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{15} \left (\frac {2}{45} \int \frac {4157 x+2041}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{45} \int \frac {4157 x+2041}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{45} \left (\frac {4157}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2266}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{45} \left (-\frac {2266}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4157}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{45} \left (\frac {412}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4157}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {214}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}\) |
(2*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ((214*Sqrt[1 - 2*x]*S qrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + ((-4157*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[ 3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (412*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7 ]*Sqrt[1 - 2*x]], 35/33])/5)/45)/15
3.28.2.3.1 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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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 = 145, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (3531 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4157 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+81000 x^{4}-74700 x^{3}-123780 x^{2}+15720 x +27360\right )}{10125 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15}+\frac {304 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{675}+\frac {4082 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{70875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8314 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{70875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(206\) |
| risch | \(\frac {4 \left (-76+45 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{675 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {2041 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{37125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4157 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{37125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(246\) |
-1/10125*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(3531*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*7 0^(1/2))-4157*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*E llipticE((10+15*x)^(1/2),1/35*70^(1/2))+81000*x^4-74700*x^3-123780*x^2+157 20*x+27360)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.42 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=-\frac {4}{675} \, {\left (45 \, x - 76\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {88079}{911250} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {4157}{10125} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-4/675*(45*x - 76)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 88079/9112 50*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 4157/ 10125*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse (1159/675, 38998/91125, x + 23/90))
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\sqrt {3 x + 2}}\, dx \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{\sqrt {3\,x+2}} \,d x \]