3.2862 \(\int \frac{\sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=31 \[ -\sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

[Out]

-(Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35])

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Rubi [A]  time = 0.0072197, antiderivative size = 31, 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 = {113} \[ -\sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35])

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{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right )\\ \end{align*}

Mathematica [C]  time = 0.150874, size = 129, normalized size = 4.16 \[ \frac{\sqrt{3 x+2} \sqrt{\frac{2 x-1}{5 x+3}} \left (5 \sqrt{\frac{2 x-1}{5 x+3}} \sqrt{\frac{3 x+2}{5 x+3}} \sqrt{5 x+3}+i \sqrt{2} E\left (i \sinh ^{-1}\left (\frac{1}{\sqrt{15 x+9}}\right )|-\frac{33}{2}\right )\right )}{5 \sqrt{1-2 x} \sqrt{\frac{3 x+2}{5 x+3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.012, size = 22, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{5}{\it EllipticE} \left ({\frac{1}{11}\sqrt{66+110\,x}},{\frac{i}{2}}\sqrt{66} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(3*x + 2)/(sqrt(5*x + 3)*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}}{10 \, x^{2} + x - 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(1-2*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)/(10*x^2 + x - 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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