3.1557 \(\int \frac{1}{\sqrt{3-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=26 \[ \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{21}} \sqrt{5 x+3}\right ) \]

[Out]

Sqrt[2/5]*ArcSin[Sqrt[2/21]*Sqrt[3 + 5*x]]

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Rubi [A]  time = 0.0089491, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {54, 216} \[ \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{21}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/5]*ArcSin[Sqrt[2/21]*Sqrt[3 + 5*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{1}{\sqrt{3-2 x} \sqrt{3+5 x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{21-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{\sqrt{5}}\\ &=\sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{21}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0090089, size = 27, normalized size = 1.04 \[ -\sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{21}} \sqrt{3-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2/5]*ArcSin[Sqrt[5/21]*Sqrt[3 - 2*x]])

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Maple [B]  time = 0.007, size = 39, normalized size = 1.5 \begin{align*}{\frac{\sqrt{10}}{10}\sqrt{ \left ( 3-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{21}}-{\frac{3}{7}} \right ){\frac{1}{\sqrt{3-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*((3-2*x)*(3+5*x))^(1/2)/(3-2*x)^(1/2)/(3+5*x)^(1/2)*10^(1/2)*arcsin(20/21*x-3/7)

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Maxima [A]  time = 1.42553, size = 15, normalized size = 0.58 \begin{align*} -\frac{1}{10} \, \sqrt{10} \arcsin \left (-\frac{20}{21} \, x + \frac{3}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(10)*arcsin(-20/21*x + 3/7)

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Fricas [B]  time = 1.98971, size = 140, normalized size = 5.38 \begin{align*} -\frac{1}{5} \, \sqrt{5} \sqrt{2} \arctan \left (\frac{\sqrt{5} \sqrt{2} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 3} - 3 \, \sqrt{5} \sqrt{2}}{10 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.0972, size = 58, normalized size = 2.23 \begin{align*} \begin{cases} - \frac{\sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{210} \sqrt{x + \frac{3}{5}}}{21} \right )}}{5} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{21} > 1 \\\frac{\sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{210} \sqrt{x + \frac{3}{5}}}{21} \right )}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(10)*I*acosh(sqrt(210)*sqrt(x + 3/5)/21)/5, 10*Abs(x + 3/5)/21 > 1), (sqrt(10)*asin(sqrt(210)*
sqrt(x + 3/5)/21)/5, True))

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Giac [A]  time = 1.06205, size = 28, normalized size = 1.08 \begin{align*} \frac{1}{5} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{1}{21} \, \sqrt{42} \sqrt{5 \, x + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*sqrt(2)*arcsin(1/21*sqrt(42)*sqrt(5*x + 3))