3.2863 \(\int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{\sqrt{33}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

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Rubi [A]  time = 0.0065586, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {119} \[ -\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

Rule 119

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=-\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}\\ \end{align*}

Mathematica [C]  time = 0.111223, size = 74, normalized size = 2.55 \[ \frac{i \sqrt{3 x+2} \sqrt{\frac{4 x-2}{5 x+3}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{15 x+9}}\right ),-\frac{33}{2}\right )}{\sqrt{1-2 x} \sqrt{\frac{3 x+2}{5 x+3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[2 + 3*x]*Sqrt[(-2 + 4*x)/(3 + 5*x)]*EllipticF[I*ArcSinh[1/Sqrt[9 + 15*x]], -33/2])/(Sqrt[1 - 2*x]*Sqrt
[(2 + 3*x)/(3 + 5*x)])

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Maple [C]  time = 0.013, size = 21, normalized size = 0.7 \begin{align*}{\it EllipticF} \left ({\frac{1}{11}\sqrt{66+110\,x}},{\frac{i}{2}}\sqrt{66} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(30*x^3 + 23*x^2 - 7*x - 6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{1 - 2 x} \sqrt{3 x + 2} \sqrt{5 x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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