Integrand size = 28, antiderivative size = 127 \[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {23}{25} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {1}{5} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1597}{250} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {8}{125} \sqrt {33} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-1597/750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-8 /125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1/5*(2 +3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-23/25*(1-2*x)^(1/2)*(2+3*x)^(1/2)* (3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 2.65 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\frac {1}{750} \left (-90 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (11+5 x)+1597 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1645 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]
(-90*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(11 + 5*x) + (1597*I)*Sqrt[ 33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1645*I)*Sqrt[33]*Ellipt icF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/750
Time = 0.22 (sec) , antiderivative size = 137, 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.
| \(\displaystyle \int \frac {(3 x+2)^{3/2} \sqrt {5 x+3}}{\sqrt {1-2 x}} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {1}{5} \int \frac {\sqrt {3 x+2} (138 x+85)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \frac {\sqrt {3 x+2} (138 x+85)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{10} \left (-\frac {1}{15} \int -\frac {3 (1597 x+1011)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {46}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{5} \int \frac {1597 x+1011}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {46}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{5} \left (\frac {264}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1597}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {46}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{5} \left (\frac {264}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1597}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {46}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{5} \left (-\frac {16}{5} \sqrt {33} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1597}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {46}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1}{5} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
-1/5*(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + ((-46*Sqrt[1 - 2*x]*S qrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((-1597*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3 /7]*Sqrt[1 - 2*x]], 35/33])/5 - (16*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sq rt[1 - 2*x]], 35/33])/5)/5)/10
3.29.16.3.1 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 = 1.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (1551 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1597 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+13500 x^{4}+40050 x^{3}+19620 x^{2}-9630 x -5940\right )}{750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {3 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5}-\frac {33 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}+\frac {337 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1597 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(206\) |
| risch | \(\frac {3 \left (11+5 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{25 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1011 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1597 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(246\) |
-1/750*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1551*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))-1597*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ell ipticE((10+15*x)^(1/2),1/35*70^(1/2))+13500*x^4+40050*x^3+19620*x^2-9630*x -5940)/(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 = 0.43 \[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=-\frac {3}{25} \, {\left (5 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {54259}{67500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1597}{750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-3/25*(5*x + 11)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 54259/67500* sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1597/750 *sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159 /675, 38998/91125, x + 23/90))
\[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\sqrt {1 - 2 x}}\, dx \]
\[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}}{\sqrt {1-2\,x}} \,d x \]