\(\int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx\) [429]

Optimal result

Integrand size = 23, antiderivative size = 51 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right ),-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]

[Out]

2/3*EllipticF(1/2*6^(1/2)*x^(1/2),1/3*(-6*e/d)^(1/2))*(1+e*x/d)^(1/2)*3^(1/2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {728, 12, 118, 116} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right ),-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]

[In]

Int[1/(Sqrt[d + e*x]*Sqrt[2*x - 3*x^2]),x]

[Out]

(2*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[Sqrt[3/2]*Sqrt[x]], (-2*e)/(3*d)])/(Sqrt[3]*Sqrt[d + e*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (GtQ[-b/d, 0] || LtQ[-b/f, 0])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {2} \sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx \\ & = \frac {\int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx}{\sqrt {2}} \\ & = \frac {\sqrt {1+\frac {e x}{d}} \int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {2} \sqrt {d+e x}} \\ & = \frac {2 \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=-\frac {\sqrt {6-\frac {4}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),-\frac {3 d}{2 e}\right )}{\sqrt {-x (-2+3 x)} \sqrt {d+e x}} \]

[In]

Integrate[1/(Sqrt[d + e*x]*Sqrt[2*x - 3*x^2]),x]

[Out]

-((Sqrt[6 - 4/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[ArcSin[Sqrt[2/3]/Sqrt[x]], (-3*d)/(2*e)])/(Sqrt[-(x*(-2 +
3*x))]*Sqrt[d + e*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(41)=82\).

Time = 2.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.25

method	result	size
default	\(-\frac {2 F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, d \sqrt {e x +d}\, \sqrt {-x \left (-2+3 x \right )}}{e x \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\)	\(115\)
elliptic	\(\frac {2 \sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{\sqrt {-x \left (-2+3 x \right )}\, \sqrt {e x +d}\, e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\)	\(136\)

[In]

int(1/(e*x+d)^(1/2)/(-3*x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1/2))*(-e*x/d)^(1/2)*(-(-2+3*x)*e/(3*d+2*e))^(1/2)*((e*x
+d)/d)^(1/2)*d*(e*x+d)^(1/2)*(-x*(-2+3*x))^(1/2)/e/x/(3*e*x^2+3*d*x-2*e*x-2*d)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=-\frac {2 \, \sqrt {3} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right )}{3 \, e} \]

[In]

integrate(1/(e*x+d)^(1/2)/(-3*x^2+2*x)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(3)*sqrt(-e)*weierstrassPInverse(4/27*(9*d^2 + 6*d*e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2*e - 18*d*e
^2 - 8*e^3)/e^3, 1/9*(9*e*x + 3*d - 2*e)/e)/e

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int \frac {1}{\sqrt {- x \left (3 x - 2\right )} \sqrt {d + e x}}\, dx \]

[In]

integrate(1/(e*x+d)**(1/2)/(-3*x**2+2*x)**(1/2),x)

[Out]

Integral(1/(sqrt(-x*(3*x - 2))*sqrt(d + e*x)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int { \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]

[In]

integrate(1/(e*x+d)^(1/2)/(-3*x^2+2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 + 2*x)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int { \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]

[In]

integrate(1/(e*x+d)^(1/2)/(-3*x^2+2*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(e*x + d)*sqrt(-3*x^2 + 2*x)), x)

Mupad [F(-1)]

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

[In]

int(1/((2*x - 3*x^2)^(1/2)*(d + e*x)^(1/2)),x)

[Out]

int(1/((2*x - 3*x^2)^(1/2)*(d + e*x)^(1/2)), x)