\(\int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx\) [96]

Optimal result

Integrand size = 37, antiderivative size = 101 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \]

[Out]

62/2145*(2-3*x)*EllipticPi(1/23*253^(1/2)*(7+5*x)^(1/2)/(2-3*x)^(1/2),-69/55,1/39*I*897^(1/2))*((5-2*x)/(2-3*x
))^(1/2)*((-1-4*x)/(2-3*x))^(1/2)*429^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 101, 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 = {171, 551} \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {4 x+1}{2-3 x}} \operatorname {EllipticPi}\left (-\frac {69}{55},\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {5 x+7}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {2 x-5} \sqrt {4 x+1}} \]

[In]

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

[Out]

(62*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sqrt[11/23]*Sq
rt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])/(5*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

Rule 171

Int[Sqrt[(a_.) + (b_.)*(x_)]/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S
ymbol] :> Dist[2*(a + b*x)*Sqrt[(b*g - a*h)*((c + d*x)/((d*g - c*h)*(a + b*x)))]*(Sqrt[(b*g - a*h)*((e + f*x)/
((f*g - e*h)*(a + b*x)))]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Subst[Int[1/((h - b*x^2)*Sqrt[1 + (b*c - a*d)*(x^2/(
d*g - c*h))]*Sqrt[1 + (b*e - a*f)*(x^2/(f*g - e*h))]), x], x, Sqrt[g + h*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b,
c, d, e, f, g, h}, x]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (62 (2-3 x) \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {11 x^2}{23}} \sqrt {1+\frac {11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac {\sqrt {7+5 x}}{\sqrt {2-3 x}}\right )}{\sqrt {897} \sqrt {-5+2 x} \sqrt {1+4 x}} \\ & = \frac {62 (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \Pi \left (-\frac {69}{55};\sin ^{-1}\left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right )|-\frac {23}{39}\right )}{5 \sqrt {429} \sqrt {-5+2 x} \sqrt {1+4 x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 5.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx=\frac {\sqrt {\frac {1+4 x}{7+5 x}} (7+5 x)^{3/2} \left (-62 \sqrt {\frac {5-2 x}{7+5 x}} \sqrt {\frac {-2+3 x}{7+5 x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {155-62 x}{77+55 x}}\right ),\frac {23}{62}\right )+117 \sqrt {\frac {-10+19 x-6 x^2}{(7+5 x)^2}} \operatorname {EllipticPi}\left (-\frac {55}{62},\arcsin \left (\sqrt {\frac {155-62 x}{77+55 x}}\right ),\frac {23}{62}\right )\right )}{5 \sqrt {682} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \]

[In]

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

[Out]

(Sqrt[(1 + 4*x)/(7 + 5*x)]*(7 + 5*x)^(3/2)*(-62*Sqrt[(5 - 2*x)/(7 + 5*x)]*Sqrt[(-2 + 3*x)/(7 + 5*x)]*EllipticF
[ArcSin[Sqrt[(155 - 62*x)/(77 + 55*x)]], 23/62] + 117*Sqrt[(-10 + 19*x - 6*x^2)/(7 + 5*x)^2]*EllipticPi[-55/62
, ArcSin[Sqrt[(155 - 62*x)/(77 + 55*x)]], 23/62]))/(5*Sqrt[682]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {62 \Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right ) \sqrt {\frac {1+4 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {3}\, \sqrt {13}\, \left (-2+3 x \right ) \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \sqrt {7+5 x}\, \sqrt {2-3 x}}{49335 \left (40 x^{3}-34 x^{2}-151 x -35\right )}\)	\(134\)
elliptic	\(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {4 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{305877 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {2 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{101959 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(250\)

[In]

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

[Out]

-62/49335*EllipticPi(1/23*(-253*(7+5*x)/(-2+3*x))^(1/2),-69/55,1/39*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)*(7+5*x)^(1/2)*(2-3*x)^(1/2)/(40*x^3-34*x^2-151*x-35)

Fricas [F]

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

[In]

integrate((2-3*x)^(1/2)/(7+5*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)/(40*x^3 - 34*x^2 - 151*x - 35), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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