3.1553 \(\int \frac{1}{\sqrt{-3+2 x} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=26 \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]

[Out]

Sqrt[2/3]*ArcSinh[Sqrt[3/13]*Sqrt[-3 + 2*x]]

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Rubi [A]  time = 0.0080107, 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, 215} \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/3]*ArcSinh[Sqrt[3/13]*Sqrt[-3 + 2*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 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-3+2 x} \sqrt{2+3 x}} \, dx &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{13+3 x^2}} \, dx,x,\sqrt{-3+2 x}\right )\\ &=\sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{-3+2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0083718, size = 26, normalized size = 1. \[ \sqrt{\frac{2}{3}} \sinh ^{-1}\left (\sqrt{\frac{3}{13}} \sqrt{2 x-3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/3]*ArcSinh[Sqrt[3/13]*Sqrt[-3 + 2*x]]

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Maple [B]  time = 0.005, size = 57, normalized size = 2.2 \begin{align*}{\frac{\sqrt{6}}{6}\sqrt{ \left ( -3+2\,x \right ) \left ( 2+3\,x \right ) }\ln \left ({\frac{\sqrt{6}}{6} \left ( -{\frac{5}{2}}+6\,x \right ) }+\sqrt{6\,{x}^{2}-5\,x-6} \right ){\frac{1}{\sqrt{-3+2\,x}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((-3+2*x)*(2+3*x))^(1/2)/(-3+2*x)^(1/2)/(2+3*x)^(1/2)*ln(1/6*(-5/2+6*x)*6^(1/2)+(6*x^2-5*x-6)^(1/2))*6^(1/
2)

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Maxima [A]  time = 1.44174, size = 38, normalized size = 1.46 \begin{align*} \frac{1}{6} \, \sqrt{6} \log \left (2 \, \sqrt{6} \sqrt{6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(6)*log(2*sqrt(6)*sqrt(6*x^2 - 5*x - 6) + 12*x - 5)

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Fricas [B]  time = 1.98652, size = 146, normalized size = 5.62 \begin{align*} \frac{1}{12} \, \sqrt{3} \sqrt{2} \log \left (4 \, \sqrt{3} \sqrt{2}{\left (12 \, x - 5\right )} \sqrt{3 \, x + 2} \sqrt{2 \, x - 3} + 288 \, x^{2} - 240 \, x - 119\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*sqrt(2)*log(4*sqrt(3)*sqrt(2)*(12*x - 5)*sqrt(3*x + 2)*sqrt(2*x - 3) + 288*x^2 - 240*x - 119)

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Sympy [A]  time = 1.0987, size = 58, normalized size = 2.23 \begin{align*} \begin{cases} \frac{\sqrt{6} \operatorname{acosh}{\left (\frac{\sqrt{78} \sqrt{x + \frac{2}{3}}}{13} \right )}}{3} & \text{for}\: \frac{6 \left |{x + \frac{2}{3}}\right |}{13} > 1 \\- \frac{\sqrt{6} i \operatorname{asin}{\left (\frac{\sqrt{78} \sqrt{x + \frac{2}{3}}}{13} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(6)*acosh(sqrt(78)*sqrt(x + 2/3)/13)/3, 6*Abs(x + 2/3)/13 > 1), (-sqrt(6)*I*asin(sqrt(78)*sqrt(
x + 2/3)/13)/3, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError