Integrand size = 37, antiderivative size = 71 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {2 \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]
2/253*(1/(4+2*(1+4*x)/(2-3*x)))^(1/2)*(4+2*(1+4*x)/(2-3*x))^(1/2)*Elliptic F((1+4*x)^(1/2)*2^(1/2)/(2-3*x)^(1/2)/(4+2*(1+4*x)/(2-3*x))^(1/2),1/23*I*8 97^(1/2))*253^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/((7+5*x)/(5-2*x))^(1/2)
Time = 3.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=-\frac {2 \sqrt {1+4 x} \sqrt {\frac {5-2 x}{7+5 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {23}{11}} \sqrt {2-3 x}}{\sqrt {7+5 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{7+5 x}}} \]
(-2*Sqrt[1 + 4*x]*Sqrt[(5 - 2*x)/(7 + 5*x)]*EllipticF[ArcSin[(Sqrt[23/11]* Sqrt[2 - 3*x])/Sqrt[7 + 5*x]], -39/23])/(Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(7 + 5*x)])
Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(71)=142\).
Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.32, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {188, 27, 320}
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.
\(\displaystyle \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \sqrt {5 x+7}} \, dx\) |
\(\Big \downarrow \) 188 |
\(\displaystyle \frac {\sqrt {\frac {2}{253}} \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {\sqrt {46}}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{\sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \int \frac {1}{\sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23}}d\frac {\sqrt {4 x+1}}{\sqrt {2-3 x}}}{\sqrt {11} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {2 \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {5 x+7} \sqrt {\frac {31 (4 x+1)}{2-3 x}+23} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{2-3 x}} \sqrt {\frac {4 x+1}{2-3 x}+2} \sqrt {\frac {\frac {31 (4 x+1)}{2-3 x}+23}{\frac {4 x+1}{2-3 x}+2}}}\) |
(2*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[7 + 5*x]*Sqrt[23 + (31*(1 + 4*x))/(2 - 3 *x)]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(Sq rt[253]*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(2 - 3*x)]*Sqrt[2 + (1 + 4*x)/(2 - 3 *x)]*Sqrt[(23 + (31*(1 + 4*x))/(2 - 3*x))/(2 + (1 + 4*x)/(2 - 3*x))])
3.2.4.3.1 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[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( -(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))])) Subst[Int[1/(Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Time = 1.63 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.87
method | result | size |
default | \(-\frac {2 F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right ) \sqrt {\frac {1+4 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {3}\, \sqrt {13}\, \left (-2+3 x \right ) \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \sqrt {2-3 x}\, \sqrt {7+5 x}}{9867 \left (40 x^{3}-34 x^{2}-151 x -35\right )}\) | \(133\) |
elliptic | \(\frac {2 \sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{305877 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}\, \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\) | \(137\) |
-2/9867*EllipticF(1/23*(-253*(7+5*x)/(-2+3*x))^(1/2),1/39*I*897^(1/2))*((1 +4*x)/(-2+3*x))^(1/2)*23^(1/2)*((-5+2*x)/(-2+3*x))^(1/2)*3^(1/2)*13^(1/2)* (-2+3*x)*(-253*(7+5*x)/(-2+3*x))^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(2-3*x )^(1/2)*(7+5*x)^(1/2)/(40*x^3-34*x^2-151*x-35)
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
integral(-sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(120*x^ 4 - 182*x^3 - 385*x^2 + 197*x + 70), x)
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \sqrt {5 x + 7}}\, dx \]
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\sqrt {5\,x+7}} \,d x \]