Integrand size = 28, antiderivative size = 31 \[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=-\frac {2}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {3} \sqrt {-3-5 x}\right )|-\frac {2}{33}\right ) \] Output:
-2/15*33^(1/2)*EllipticE(3^(1/2)*(-3-5*x)^(1/2),1/33*I*66^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(31)=62\).
Time = 1.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=-\frac {2 \left (\frac {3 \left (-3+x+10 x^2\right )}{\sqrt {2+3 x}}+\sqrt {35} \sqrt {\frac {-1+2 x}{2+3 x}} (2+3 x) \sqrt {\frac {3+5 x}{2+3 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {7}{2}}}{\sqrt {2+3 x}}\right )|\frac {2}{35}\right )\right )}{15 \sqrt {-3-5 x} \sqrt {1-2 x}} \] Input:
Integrate[Sqrt[1 - 2*x]/(Sqrt[-3 - 5*x]*Sqrt[2 + 3*x]),x]
Output:
(-2*((3*(-3 + x + 10*x^2))/Sqrt[2 + 3*x] + Sqrt[35]*Sqrt[(-1 + 2*x)/(2 + 3 *x)]*(2 + 3*x)*Sqrt[(3 + 5*x)/(2 + 3*x)]*EllipticE[ArcSin[Sqrt[7/2]/Sqrt[2 + 3*x]], 2/35]))/(15*Sqrt[-3 - 5*x]*Sqrt[1 - 2*x])
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {1-2 x}}{\sqrt {-5 x-3} \sqrt {3 x+2}} \, dx\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {2}{3} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )\) |
Input:
Int[Sqrt[1 - 2*x]/(Sqrt[-3 - 5*x]*Sqrt[2 + 3*x]),x]
Output:
(2*Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/3
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])
Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {\left (33 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )\right ) \sqrt {2}}{15}\) | \(41\) |
elliptic | \(\frac {\sqrt {\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {30 x^{3}+23 x^{2}-7 x -6}}-\frac {2 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21 \sqrt {30 x^{3}+23 x^{2}-7 x -6}}\right )}{\sqrt {1-2 x}\, \sqrt {-3-5 x}\, \sqrt {2+3 x}}\) | \(172\) |
Input:
int((1-2*x)^(1/2)/(-3-5*x)^(1/2)/(2+3*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/15*(33*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+2*EllipticE(1/7*(28+4 2*x)^(1/2),1/2*70^(1/2)))*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=\frac {68}{675} \, \sqrt {30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {2}{15} \, \sqrt {30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(1/2)/(-3-5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="fricas" )
Output:
68/675*sqrt(30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 2/ 15*sqrt(30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(115 9/675, 38998/91125, x + 23/90))
\[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\sqrt {- 5 x - 3} \sqrt {3 x + 2}}\, dx \] Input:
integrate((1-2*x)**(1/2)/(-3-5*x)**(1/2)/(2+3*x)**(1/2),x)
Output:
Integral(sqrt(1 - 2*x)/(sqrt(-5*x - 3)*sqrt(3*x + 2)), x)
\[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2} \sqrt {-5 \, x - 3}} \,d x } \] Input:
integrate((1-2*x)^(1/2)/(-3-5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima" )
Output:
integrate(sqrt(-2*x + 1)/(sqrt(3*x + 2)*sqrt(-5*x - 3)), x)
\[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2} \sqrt {-5 \, x - 3}} \,d x } \] Input:
integrate((1-2*x)^(1/2)/(-3-5*x)^(1/2)/(2+3*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-2*x + 1)/(sqrt(3*x + 2)*sqrt(-5*x - 3)), x)
Timed out. \[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {1-2\,x}}{\sqrt {3\,x+2}\,\sqrt {-5\,x-3}} \,d x \] Input:
int((1 - 2*x)^(1/2)/((3*x + 2)^(1/2)*(- 5*x - 3)^(1/2)),x)
Output:
int((1 - 2*x)^(1/2)/((3*x + 2)^(1/2)*(- 5*x - 3)^(1/2)), x)
\[ \int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {-5 x -3}}{15 x^{2}+19 x +6}d x \right ) \] Input:
int((1-2*x)^(1/2)/(-3-5*x)^(1/2)/(2+3*x)^(1/2),x)
Output:
- int((sqrt(3*x + 2)*sqrt( - 2*x + 1)*sqrt( - 5*x - 3))/(15*x**2 + 19*x + 6),x)