\(\int \frac {\sqrt {7+5 x}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx\) [103]

Optimal result

Integrand size = 37, antiderivative size = 100 \[ \int \frac {\sqrt {7+5 x}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {23 \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right ),\frac {39}{62}\right )}{2 \sqrt {682} \sqrt {2-3 x} \sqrt {-5+2 x}} \]

[Out]

23/1364*(7+5*x)*EllipticPi(1/11*341^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2),55/124,1/62*2418^(1/2))*682^(1/2)*((2-3*
x)/(7+5*x))^(1/2)*((5-2*x)/(7+5*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 100, 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} \[ \int \frac {\sqrt {7+5 x}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=\frac {23 \sqrt {\frac {2-3 x}{5 x+7}} \sqrt {\frac {5-2 x}{5 x+7}} (5 x+7) \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {\sqrt {\frac {31}{11}} \sqrt {4 x+1}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{2 \sqrt {682} \sqrt {2-3 x} \sqrt {2 x-5}} \]

[In]

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

[Out]

(23*Sqrt[(2 - 3*x)/(7 + 5*x)]*Sqrt[(5 - 2*x)/(7 + 5*x)]*(7 + 5*x)*EllipticPi[55/124, ArcSin[(Sqrt[31/11]*Sqrt[
1 + 4*x])/Sqrt[7 + 5*x]], 39/62])/(2*Sqrt[682]*Sqrt[2 - 3*x]*Sqrt[-5 + 2*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 (23 \sqrt {2} \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {-\frac {-5+2 x}{7+5 x}} (7+5 x)\right ) \text {Subst}\left (\int \frac {1}{\left (4-5 x^2\right ) \sqrt {1-\frac {31 x^2}{11}} \sqrt {1-\frac {39 x^2}{22}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {7+5 x}}\right )}{11 \sqrt {2-3 x} \sqrt {-5+2 x}} \\ & = \frac {23 \sqrt {\frac {2-3 x}{7+5 x}} \sqrt {\frac {5-2 x}{7+5 x}} (7+5 x) \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{2 \sqrt {682} \sqrt {2-3 x} \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.77 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {7+5 x}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx=-\frac {62 \sqrt {1+4 x} \sqrt {\frac {5-2 x}{7+5 x}} \operatorname {EllipticPi}\left (-\frac {55}{69},\arcsin \left (\frac {\sqrt {\frac {23}{11}} \sqrt {2-3 x}}{\sqrt {7+5 x}}\right ),-\frac {39}{23}\right )}{3 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {1+4 x}{7+5 x}}} \]

[In]

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

[Out]

(-62*Sqrt[1 + 4*x]*Sqrt[(5 - 2*x)/(7 + 5*x)]*EllipticPi[-55/69, ArcSin[(Sqrt[23/11]*Sqrt[2 - 3*x])/Sqrt[7 + 5*
x]], -39/23])/(3*Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(1 + 4*x)/(7 + 5*x)])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.61 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {62 \left (F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-\Pi \left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )\right ) \sqrt {\frac {1+4 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {3}\, \sqrt {13}\, \left (-2+3 x \right ) \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \sqrt {2-3 x}\, \sqrt {7+5 x}}{29601 \left (40 x^{3}-34 x^{2}-151 x -35\right )}\)	\(162\)
elliptic	\(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {14 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{305877 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {10 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{305877 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(250\)

[In]

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

[Out]

-62/29601*(EllipticF(1/23*(-253*(7+5*x)/(-2+3*x))^(1/2),1/39*I*897^(1/2))-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)*(2-3*x)^(1/2)*(7+5*x)^(1/2)/(40*x^3-34*
x^2-151*x-35)

Fricas [F]

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

[In]

integrate((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)/(24*x^3 - 70*x^2 + 21*x + 10), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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