Integrand size = 26, antiderivative size = 39 \[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=E\left (\left .\arcsin \left (\frac {\sqrt {2-x}}{\sqrt {3}}\right )\right |-6\right )-7 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2-x}}{\sqrt {3}}\right ),-6\right ) \] Output:
EllipticE(1/3*(2-x)^(1/2)*3^(1/2),I*6^(1/2))-7*EllipticF(1/3*(2-x)^(1/2)*3 ^(1/2),I*6^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(39)=78\).
Time = 1.90 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=-\frac {(1+x) \sqrt {\frac {-2+x}{-5+2 x}} \left (2 \sqrt {5-2 x} \sqrt {\frac {-2+x}{-5+2 x}}+\frac {\sqrt {7} E\left (\arcsin \left (\frac {\sqrt {7}}{\sqrt {5-2 x}}\right )|\frac {1}{7}\right )}{\sqrt {\frac {1+x}{-5+2 x}}}\right )}{\sqrt {2+x-x^2}} \] Input:
Integrate[Sqrt[1 + x]/(Sqrt[5 - 2*x]*Sqrt[2 - x]),x]
Output:
-(((1 + x)*Sqrt[(-2 + x)/(-5 + 2*x)]*(2*Sqrt[5 - 2*x]*Sqrt[(-2 + x)/(-5 + 2*x)] + (Sqrt[7]*EllipticE[ArcSin[Sqrt[7]/Sqrt[5 - 2*x]], 1/7])/Sqrt[(1 + x)/(-5 + 2*x)]))/Sqrt[2 + x - x^2])
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {x+1}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {\sqrt {7} \sqrt {x-2} \int \frac {\sqrt {x+1}}{\sqrt {7} \sqrt {5-2 x} \sqrt {x-2}}dx}{\sqrt {2-x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {x-2} \int \frac {\sqrt {x+1}}{\sqrt {5-2 x} \sqrt {x-2}}dx}{\sqrt {2-x}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {\sqrt {6} \sqrt {x-2} E\left (\arcsin \left (\sqrt {2} \sqrt {x-2}\right )|-\frac {1}{6}\right )}{\sqrt {2-x}}\) |
Input:
Int[Sqrt[1 + x]/(Sqrt[5 - 2*x]*Sqrt[2 - x]),x]
Output:
(Sqrt[6]*Sqrt[-2 + x]*EllipticE[ArcSin[Sqrt[2]*Sqrt[-2 + x]], -1/6])/Sqrt[ 2 - x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 0.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\left (6 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \sqrt {7}\right )+\operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \sqrt {7}\right )\right ) \sqrt {-x +2}}{\sqrt {x -2}}\) | \(43\) |
elliptic | \(\frac {\sqrt {\left (-5+2 x \right ) \left (x -2\right ) \left (1+x \right )}\, \left (-\frac {\sqrt {35-14 x}\, \sqrt {2 x -4}\, \sqrt {14+14 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \sqrt {7}\right )}{7 \sqrt {2 x^{3}-7 x^{2}+x +10}}-\frac {\sqrt {35-14 x}\, \sqrt {2 x -4}\, \sqrt {14+14 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {35-14 x}}{7}, \sqrt {7}\right )}{2}+2 \operatorname {EllipticF}\left (\frac {\sqrt {35-14 x}}{7}, \sqrt {7}\right )\right )}{7 \sqrt {2 x^{3}-7 x^{2}+x +10}}\right )}{\sqrt {1+x}\, \sqrt {5-2 x}\, \sqrt {-x +2}}\) | \(156\) |
Input:
int((1+x)^(1/2)/(5-2*x)^(1/2)/(-x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
(6*EllipticF(1/7*(35-14*x)^(1/2),7^(1/2))+EllipticE(1/7*(35-14*x)^(1/2),7^ (1/2)))/(x-2)^(1/2)*(-x+2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\frac {13}{6} \, \sqrt {2} {\rm weierstrassPInverse}\left (\frac {43}{3}, -\frac {260}{27}, x - \frac {7}{6}\right ) - \sqrt {2} {\rm weierstrassZeta}\left (\frac {43}{3}, -\frac {260}{27}, {\rm weierstrassPInverse}\left (\frac {43}{3}, -\frac {260}{27}, x - \frac {7}{6}\right )\right ) \] Input:
integrate((1+x)^(1/2)/(5-2*x)^(1/2)/(2-x)^(1/2),x, algorithm="fricas")
Output:
13/6*sqrt(2)*weierstrassPInverse(43/3, -260/27, x - 7/6) - sqrt(2)*weierst rassZeta(43/3, -260/27, weierstrassPInverse(43/3, -260/27, x - 7/6))
\[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\int \frac {\sqrt {x + 1}}{\sqrt {2 - x} \sqrt {5 - 2 x}}\, dx \] Input:
integrate((1+x)**(1/2)/(5-2*x)**(1/2)/(2-x)**(1/2),x)
Output:
Integral(sqrt(x + 1)/(sqrt(2 - x)*sqrt(5 - 2*x)), x)
\[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\int { \frac {\sqrt {x + 1}}{\sqrt {-x + 2} \sqrt {-2 \, x + 5}} \,d x } \] Input:
integrate((1+x)^(1/2)/(5-2*x)^(1/2)/(2-x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 1)/(sqrt(-x + 2)*sqrt(-2*x + 5)), x)
\[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\int { \frac {\sqrt {x + 1}}{\sqrt {-x + 2} \sqrt {-2 \, x + 5}} \,d x } \] Input:
integrate((1+x)^(1/2)/(5-2*x)^(1/2)/(2-x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 1)/(sqrt(-x + 2)*sqrt(-2*x + 5)), x)
Timed out. \[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\int \frac {\sqrt {x+1}}{\sqrt {5-2\,x}\,\sqrt {2-x}} \,d x \] Input:
int((x + 1)^(1/2)/((5 - 2*x)^(1/2)*(2 - x)^(1/2)),x)
Output:
int((x + 1)^(1/2)/((5 - 2*x)^(1/2)*(2 - x)^(1/2)), x)
\[ \int \frac {\sqrt {1+x}}{\sqrt {5-2 x} \sqrt {2-x}} \, dx=\int \frac {\sqrt {x +1}\, \sqrt {-x +2}\, \sqrt {-2 x +5}}{2 x^{2}-9 x +10}d x \] Input:
int((1+x)^(1/2)/(5-2*x)^(1/2)/(2-x)^(1/2),x)
Output:
int((sqrt(x + 1)*sqrt( - x + 2)*sqrt( - 2*x + 5))/(2*x**2 - 9*x + 10),x)