Integrand size = 28, antiderivative size = 81 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\arcsin \left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \]
2/77*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)*(-3-5*x)^(1/2 )/(3+5*x)^(1/2)+4/77*(2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\frac {2}{77} \left (\frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}}-i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )\right ) \]
(2*((2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - I*Sqrt[33]*EllipticE[I *ArcSinh[Sqrt[9 + 15*x]], -2/33]))/77
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {115, 27, 124, 27, 123}
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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}} \, dx\) |
\(\Big \downarrow \) 115 |
\(\displaystyle \frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {15 \sqrt {1-2 x}}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {15}{77} \int \frac {\sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {15 \sqrt {-5 x-3} \int \frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {-5 x-3} \sqrt {3 x+2}}dx}{11 \sqrt {7} \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {15 \sqrt {-5 x-3} \int \frac {\sqrt {1-2 x}}{\sqrt {-5 x-3} \sqrt {3 x+2}}dx}{77 \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\arcsin \left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}}+\frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\) |
(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) + (2*Sqrt[5/7]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(11*Sqrt[3 + 5*x])
3.30.22.3.1 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[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Time = 1.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (\sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+30 x^{2}+38 x +12\right )}{77 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(92\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-38 x -12\right )}{77 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{539 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{539 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(195\) |
-2/77*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(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)) +30*x^2+38*x+12)/(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 = 68, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=-\frac {2 \, {\left (34 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 45 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) + 90 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}\right )}}{3465 \, {\left (2 \, x - 1\right )}} \]
-2/3465*(34*sqrt(-30)*(2*x - 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 45*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/675, 38998/91125 , weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) + 90*sqrt(5*x + 3 )*sqrt(3*x + 2)*sqrt(-2*x + 1))/(2*x - 1)
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,\sqrt {5\,x+3}} \,d x \]