3.2523 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}-\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0099029, antiderivative size = 47, 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} \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}-\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 5*x]/(1 - 2*x)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0204431, size = 46, normalized size = 0.98 \[ \frac{\sqrt{5 x+3}}{\sqrt{1-2 x}}+\sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 4.05169, size = 49, normalized size = 1.04 \begin{align*} -\frac{1}{4} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{2 \, x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [B]  time = 1.72706, size = 219, normalized size = 4.66 \begin{align*} \frac{\sqrt{5} \sqrt{2}{\left (2 \, x - 1\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 ) - 4 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(5)*sqrt(2)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 4*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]  time = 1.58118, size = 95, normalized size = 2.02 \begin{align*} \begin{cases} - \frac{5 i \sqrt{x + \frac{3}{5}}}{\sqrt{10 x - 5}} + \frac{\sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2} + \frac{5 \sqrt{x + \frac{3}{5}}}{\sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-5*I*sqrt(x + 3/5)/sqrt(10*x - 5) + sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/2, 10*Abs(x + 3/5)
/11 > 1), (-sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/2 + 5*sqrt(x + 3/5)/sqrt(5 - 10*x), True))

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Giac [A]  time = 2.5618, size = 61, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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