3.2313 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) - (2*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5

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Rubi [A]  time = 0.0109016, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {47, 54, 216} \[ -\frac{2 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/(3 + 5*x)^(3/2),x]

[Out]

(-2*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) - (2*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}-\frac{2}{5} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x}}{5 \sqrt{3+5 x}}-\frac{2}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.038456, size = 70, normalized size = 1.35 \[ \frac{2 \left (5 \sqrt{5 x+3} (2 x-1)+\sqrt{10-20 x} (5 x+3) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{25 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*(-1 + 2*x)*Sqrt[3 + 5*x] + Sqrt[10 - 20*x]*(3 + 5*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]))/(25*Sqrt[1 - 2*x
]*(3 + 5*x))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53214, size = 49, normalized size = 0.94 \begin{align*} -\frac{1}{25} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{5 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/5*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [B]  time = 1.22852, size = 221, normalized size = 4.25 \begin{align*} \frac{\sqrt{5} \sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 10 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(sqrt(5)*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 10*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [C]  time = 1.49344, size = 151, normalized size = 2.9 \begin{align*} \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} - \frac{\sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{25} - \frac{2 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{25} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{25} - \frac{\sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{25} + \frac{2 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{25} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/25 - sqrt(10)*I*log(1/(x + 3/5))/25 - sqrt(10)*I*log(x + 3
/5)/25 - 2*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/25, 11/(10*Abs(x + 3/5)) > 1), (-2*sqrt(10)*I*sqrt(1 - 11
/(10*(x + 3/5)))/25 - sqrt(10)*I*log(1/(x + 3/5))/25 + 2*sqrt(10)*I*log(sqrt(1 - 11/(10*(x + 3/5))) + 1)/25, T
rue))

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Giac [B]  time = 2.30786, size = 112, normalized size = 2.15 \begin{align*} -\frac{1}{50} \, \sqrt{5}{\left (4 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{2} \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/50*sqrt(5)*(4*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(2)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))