3.9.73 \(\int \frac {1}{\sqrt {2-3 x} \sqrt {x} \sqrt {d+e x}} \, dx\) [873]

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

[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 = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {118, 116} \begin {gather*} \frac {2 \sqrt {\frac {e x}{d}+1} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2-3 x} \sqrt {x} \sqrt {d+e x}} \, dx &=\frac {\left (\sqrt {1-\frac {3 x}{2}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {2-3 x} \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 = 1.68, size = 72, normalized size = 1.41 \begin {gather*} -\frac {\sqrt {x} \sqrt {\frac {d+e x}{e (-2+3 x)}} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {1-\frac {3 x}{2}}}\right )|1+\frac {3 d}{2 e}\right )}{\sqrt {\frac {x}{-4+6 x}} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(41)=82\).
time = 0.10, size = 112, normalized size = 2.20

method	result	size
default	\(-\frac {2 \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, d \sqrt {2-3 x}\, \sqrt {e x +d}}{\sqrt {x}\, e \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\)	\(112\)
elliptic	\(\frac {2 \sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {2}{3}}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {x}\, \sqrt {e x +d}\, e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2-3*x)^(1/2)/x^(1/2)/(e*x+d)^(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*(2-3*x)^(1/2)/x^(1/2)*(e*x+d)^(1/2)/e/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x} \sqrt {2 - 3 x} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {2-3\,x}\,\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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