3.2266 \(\int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4 + (121*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(40*Sqrt[10])

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Rubi [A]  time = 0.0163551, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{4} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{11}{40} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*Sqrt[3 + 5*x],x]

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4 + (121*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(40*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 \sqrt{1-2 x} \sqrt{3+5 x} \, dx &=-\frac{1}{4} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{11}{8} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{11}{40} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{121}{80} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{11}{40} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{121 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{40 \sqrt{5}}\\ &=\frac{11}{40} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{121 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{40 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0473167, size = 64, normalized size = 0.89 \[ \frac{10 \sqrt{5 x+3} \left (-40 x^2+18 x+1\right )-121 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{400 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(1 + 18*x - 40*x^2) - 121*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(400*Sqrt[1 - 2*
x])

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Maple [A]  time = 0.005, size = 72, normalized size = 1. \begin{align*}{\frac{1}{10} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{11}{40}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{121\,\sqrt{10}}{800}\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((1-2*x)^(1/2)*(3+5*x)^(1/2),x)

[Out]

1/10*(3+5*x)^(3/2)*(1-2*x)^(1/2)-11/40*(1-2*x)^(1/2)*(3+5*x)^(1/2)+121/800*((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 = 2.33051, size = 55, normalized size = 0.76 \begin{align*} \frac{1}{2} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{121}{800} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1}{40} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-10*x^2 - x + 3)*x - 121/800*sqrt(10)*arcsin(-20/11*x - 1/11) + 1/40*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.83758, size = 194, normalized size = 2.69 \begin{align*} \frac{1}{40} \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{121}{800} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 121/800*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*
sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]  time = 2.57494, size = 184, normalized size = 2.56 \begin{align*} \begin{cases} \frac{5 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{10 x - 5}} - \frac{33 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{10 x - 5}} + \frac{121 i \sqrt{x + \frac{3}{5}}}{40 \sqrt{10 x - 5}} - \frac{121 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{121 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{400} - \frac{5 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{5 - 10 x}} + \frac{33 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4 \sqrt{5 - 10 x}} - \frac{121 \sqrt{x + \frac{3}{5}}}{40 \sqrt{5 - 10 x}} & \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)**(1/2),x)

[Out]

Piecewise((5*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) - 33*I*(x + 3/5)**(3/2)/(4*sqrt(10*x - 5)) + 121*I*sqrt(x + 3/5
)/(40*sqrt(10*x - 5)) - 121*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/400, 10*Abs(x + 3/5)/11 > 1), (121*sq
rt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/400 - 5*(x + 3/5)**(5/2)/sqrt(5 - 10*x) + 33*(x + 3/5)**(3/2)/(4*sqrt(
5 - 10*x)) - 121*sqrt(x + 3/5)/(40*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.13852, size = 61, normalized size = 0.85 \begin{align*} \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/400*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))