3.2678 \(\int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}} \]

[Out]

(2*Sqrt[7/5]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(3*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0125451, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {114, 113} \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[7/5]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(3*Sqrt[3 + 5*x])

Rule 114

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

Rule 113

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx &=\frac{\left (\sqrt{7} \sqrt{-3-5 x}\right ) \int \frac{\sqrt{\frac{3}{7}-\frac{6 x}{7}}}{\sqrt{-9-15 x} \sqrt{2+3 x}} \, dx}{\sqrt{3+5 x}}\\ &=\frac{2 \sqrt{\frac{7}{5}} \sqrt{-3-5 x} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{2+3 x}\right )|\frac{2}{35}\right )}{3 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [B]  time = 0.228512, size = 121, normalized size = 2.47 \[ \frac{2 \sqrt{1-2 x} \left (5 \left (6 x^2+x-2\right ) \sqrt{5 x+3}+\sqrt{33} \sqrt{\frac{2 x-1}{5 x+3}} \sqrt{\frac{3 x+2}{5 x+3}} (5 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{11}{2}}}{\sqrt{5 x+3}}\right )|-\frac{2}{33}\right )\right )}{15 \sqrt{3 x+2} \left (10 x^2+x-3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - 2*x]*(5*Sqrt[3 + 5*x]*(-2 + x + 6*x^2) + Sqrt[33]*Sqrt[(-1 + 2*x)/(3 + 5*x)]*Sqrt[(2 + 3*x)/(3 + 5
*x)]*(3 + 5*x)^2*EllipticE[ArcSin[Sqrt[11/2]/Sqrt[3 + 5*x]], -2/33]))/(15*Sqrt[2 + 3*x]*(-3 + x + 10*x^2))

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Maple [C]  time = 0.011, size = 43, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{15} \left ( 35\,{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -2\,{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(35*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-2*EllipticE(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{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-2*x + 1)/(sqrt(5*x + 3)*sqrt(3*x + 2)), 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}}{15 \, x^{2} + 19 \, x + 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)/(15*x^2 + 19*x + 6), x)

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

[Out]

Integral(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{\sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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