\(\int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\) [104]

Optimal result

Integrand size = 37, antiderivative size = 71 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {2 \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]

[Out]

2/253*(1/(4+2*(1+4*x)/(2-3*x)))^(1/2)*(4+2*(1+4*x)/(2-3*x))^(1/2)*EllipticF((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*897^(1/2))*253^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/((7+5*x)/(5-2*x))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {176, 429} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {2 \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}} \]

[In]

Int[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]),x]

[Out]

(2*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(Sqrt[253]*Sqrt[-5 + 2*x]*S
qrt[(7 + 5*x)/(5 - 2*x)])

Rule 176

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[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]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\frac {2}{253}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{\sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = \frac {2 \sqrt {7+5 x} F\left (\tan ^{-1}\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}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=-\frac {2 \sqrt {1+4 x} \sqrt {\frac {5-2 x}{7+5 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {23}{11}} \sqrt {2-3 x}}{\sqrt {7+5 x}}\right ),-\frac {39}{23}\right )}{\sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{7+5 x}}} \]

[In]

Integrate[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]),x]

[Out]

(-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]], -39/2
3])/(Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(7 + 5*x)])

Maple [A] (verified)

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	\(\frac {2 \sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{305877 \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}\, \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\)	\(137\)

[In]

int(1/(7+5*x)^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]

[In]

integrate(1/(7+5*x)^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int \frac {1}{\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \sqrt {5 x + 7}}\, dx \]

[In]

integrate(1/(7+5*x)**(1/2)/(2-3*x)**(1/2)/(-5+2*x)**(1/2)/(1+4*x)**(1/2),x)

[Out]

Integral(1/(sqrt(2 - 3*x)*sqrt(2*x - 5)*sqrt(4*x + 1)*sqrt(5*x + 7)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]

[In]

integrate(1/(7+5*x)^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int { \frac {1}{\sqrt {5 \, x + 7} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}} \,d x } \]

[In]

integrate(1/(7+5*x)^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\int \frac {1}{\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\sqrt {5\,x+7}} \,d x \]

[In]

int(1/((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7)^(1/2)),x)

[Out]

int(1/((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7)^(1/2)), x)