3.2462 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=50 \[ \frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}}-\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3} \]

[Out]

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

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Rubi [A]  time = 0.0095897, antiderivative size = 50, 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 = {50, 54, 216} \[ \frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}}-\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 5*x]/Sqrt[1 - 2*x],x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{2} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{4} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1}{2} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2 \sqrt{5}}\\ &=-\frac{1}{2} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0140032, size = 50, normalized size = 1. \[ -\frac{1}{2} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{11 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 56, normalized size = 1.1 \begin{align*} -{\frac{1}{2}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{11\,\sqrt{10}}{40}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(1-2*x)^(1/2)*(3+5*x)^(1/2)+11/40*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/
11*x+1/11)

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Maxima [A]  time = 2.38812, size = 39, normalized size = 0.78 \begin{align*} \frac{11}{40} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{2} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/40*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.79455, size = 177, normalized size = 3.54 \begin{align*} -\frac{11}{40} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac{1}{2} \, \sqrt{5 \, x + 3} \sqrt{-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)^(1/2),x, algorithm="fricas")

[Out]

-11/40*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 1/2*sqrt(5*x
+ 3)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.84812, size = 141, normalized size = 2.82 \begin{align*} \begin{cases} - \frac{5 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{10 x - 5}} + \frac{11 i \sqrt{x + \frac{3}{5}}}{2 \sqrt{10 x - 5}} - \frac{11 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{20} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{11 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{20} + \frac{5 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{\sqrt{5 - 10 x}} - \frac{11 \sqrt{x + \frac{3}{5}}}{2 \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)**(1/2),x)

[Out]

Piecewise((-5*I*(x + 3/5)**(3/2)/sqrt(10*x - 5) + 11*I*sqrt(x + 3/5)/(2*sqrt(10*x - 5)) - 11*sqrt(10)*I*acosh(
sqrt(110)*sqrt(x + 3/5)/11)/20, 10*Abs(x + 3/5)/11 > 1), (11*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/20 + 5*
(x + 3/5)**(3/2)/sqrt(5 - 10*x) - 11*sqrt(x + 3/5)/(2*sqrt(5 - 10*x)), True))

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Giac [A]  time = 1.99486, size = 54, normalized size = 1.08 \begin{align*} \frac{1}{20} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 2*sqrt(5*x + 3)*sqrt(-10*x + 5))