Integrand size = 28, antiderivative size = 191 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {175111 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1181250} \]
-2911577/1771875*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^ (1/2)-175111/3543750*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2)) *33^(1/2)-23/1575*(2+3*x)^(3/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)+2/45*(2+3*x)^( 5/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1244/13125*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2 +3*x)^(1/2)-175111/236250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-136987+410490 x+861750 x^2+472500 x^3\right )+5823154 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5998265 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3543750} \]
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-136987 + 410490*x + 861750 *x^2 + 472500*x^3) + (5823154*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15* x]], -2/33] - (5998265*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2 /33])/3543750
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {112, 27, 171, 27, 171, 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.
| \(\displaystyle \int \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{45} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}-\frac {2}{45} \int -\frac {(3 x+2)^{3/2} \sqrt {5 x+3} (23 x+27)}{2 \sqrt {1-2 x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \int \frac {(3 x+2)^{3/2} \sqrt {5 x+3} (23 x+27)}{\sqrt {1-2 x}}dx+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (-\frac {1}{35} \int -\frac {3 \sqrt {3 x+2} \sqrt {5 x+3} (2488 x+1605)}{2 \sqrt {1-2 x}}dx-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \int \frac {\sqrt {3 x+2} \sqrt {5 x+3} (2488 x+1605)}{\sqrt {1-2 x}}dx-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (-\frac {1}{25} \int -\frac {\sqrt {5 x+3} (175111 x+113838)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \int \frac {\sqrt {5 x+3} (175111 x+113838)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (-\frac {1}{9} \int -\frac {11646308 x+7373029}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {175111}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \int \frac {11646308 x+7373029}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {175111}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {11646308}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {175111}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {175111}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{45} \left (\frac {3}{70} \left (\frac {1}{25} \left (\frac {1}{18} \left (-\frac {350222}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {175111}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2488}{25} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {23}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}\right )+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}\) |
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/45 + ((-23*Sqrt[1 - 2*x] *(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/35 + (3*((-2488*Sqrt[1 - 2*x]*Sqrt[2 + 3 *x]*(3 + 5*x)^(3/2))/25 + ((-175111*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5 *x])/9 + ((-11646308*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (350222*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18)/25))/70)/45
3.27.45.3.1 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 = 1.39 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5655507 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5823154 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-212625000 x^{6}-550800000 x^{5}-432411750 x^{4}+53033850 x^{3}+167919465 x^{2}+22560465 x -12328830\right )}{3543750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(155\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {4561 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}-\frac {136987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236250}+\frac {7373029 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{24806250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5823154 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12403125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {383 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105}+2 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(250\) |
| risch | \(-\frac {\left (472500 x^{3}+861750 x^{2}+410490 x -136987\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{236250 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {7373029 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{25987500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2911577 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
-1/3543750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5655507*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))-5823154*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))-212625000*x^6-550800000*x^ 5-432411750*x^4+53033850*x^3+167919465*x^2+22560465*x-12328830)/(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 = 64, normalized size of antiderivative = 0.34 \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\frac {1}{236250} \, {\left (472500 \, x^{3} + 861750 \, x^{2} + 410490 \, x - 136987\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {197853763}{318937500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {2911577}{1771875} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
1/236250*(472500*x^3 + 861750*x^2 + 410490*x - 136987)*sqrt(5*x + 3)*sqrt( 3*x + 2)*sqrt(-2*x + 1) - 197853763/318937500*sqrt(-30)*weierstrassPInvers e(1159/675, 38998/91125, x + 23/90) + 2911577/1771875*sqrt(-30)*weierstras sZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {5}{2}} \sqrt {5 x + 3}\, dx \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int { \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
\[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int { \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1} \,d x } \]
Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3} \,d x \]