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

Optimal. Leaf size=51 \[ \frac {2 \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]

[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)

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {714, 12, 117, 115} \[ \frac {2 \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[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 115

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

Rule 117

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 714

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c*x)/b]), 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*} \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx &=\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*}

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Mathematica [A]  time = 0.11, size = 76, normalized size = 1.49 \[ -\frac {\sqrt {6-\frac {4}{x}} x^{3/2} \sqrt {\frac {d}{e x}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|-\frac {3 d}{2 e}\right )}{\sqrt {-x (3 x-2)} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[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]))

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fricas [F]  time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}}{3 \, e x^{3} + {\left (3 \, d - 2 \, e\right )} x^{2} - 2 \, d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.08, size = 115, normalized size = 2.25 \[ -\frac {2 \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (3 x -2\right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {e x +d}\, \sqrt {-\left (3 x -2\right ) x}\, d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )}{\left (3 e \,x^{2}+3 d x -2 e x -2 d \right ) e x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {2\,x-3\,x^2}\,\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x \left (3 x - 2\right )} \sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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