∫ 1_________√ −3 + 2x√___

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

1/3*arcsinh(1/13*39^(1/2)*(-3+2*x)^(1/2))*6^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {56, 221} \begin {gather*} \sqrt {\frac {2}{3}} \sinh ^{-1}\left (\sqrt {\frac {3}{13}} \sqrt {2 x-3}\right ) \end {gather*}

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]

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

\begin {align*} \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx &=\sqrt {2} \text {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*}

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Mathematica [A]
time = 0.06, size = 26, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-9+6 x}{4+6 x}}}\right ) \end {gather*}

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.37, size = 43, normalized size = 1.65 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {6} \text {ArcCosh}\left [\frac {\sqrt {26} \sqrt {2+3 x}}{13}\right ]}{3},\text {Abs}\left [\frac {2}{3}+x\right ]>\frac {13}{6}\right \}\right \},-\frac {I \sqrt {6} \text {ArcSin}\left [\frac {\sqrt {78} \sqrt {\frac {2}{3}+x}}{13}\right ]}{3}\right ] \end {gather*}

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

Piecewise[{{Sqrt[6] ArcCosh[Sqrt[26] Sqrt[2 + 3 x] / 13] / 3, Abs[2 / 3 + x] > 13 / 6}}, -I Sqrt[6] ArcSin[Sqr

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).
time = 0.17, size = 57, normalized size = 2.19


method	result	size

default	\(\frac {\sqrt {\left (2 x -3\right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (-\frac {5}{2}+6 x \right ) \sqrt {6}}{6}+\sqrt {6 x^{2}-5 x -6}\right ) \sqrt {6}}{6 \sqrt {2 x -3}\, \sqrt {2+3 x}}\)	\(57\)

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

1/6*((2*x-3)*(2+3*x))^(1/2)/(2*x-3)^(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 = 0.36, size = 28, normalized size = 1.08 \begin {gather*} \frac {1}{6} \, \sqrt {6} \log \left (2 \, \sqrt {6} \sqrt {6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \end {gather*}

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).
time = 0.29, size = 46, normalized size = 1.77 \begin {gather*} \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 {gather*}

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

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 = 0.66, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {\sqrt {6} \operatorname {acosh}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | > \frac {13}{6} \\- \frac {\sqrt {6} i \operatorname {asin}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 0.00, size = 46, normalized size = 1.77 \begin {gather*} -\frac {2 \ln \left (\sqrt {3 \left (2 x-3\right )+13}-\sqrt {3} \sqrt {2 x-3}\right )}{\sqrt {2} \sqrt {3}} \end {gather*}

-1/3*sqrt(3)*sqrt(2)*log(-sqrt(3)*sqrt(2*x - 3) + sqrt(6*x + 4))

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Mupad [B]
time = 0.12, size = 43, normalized size = 1.65 \begin {gather*} \frac {2\,\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}\right )}{3} \end {gather*}

(2*6^(1/2)*atanh((6^(1/2)*(3^(1/2)*1i - (2*x - 3)^(1/2)))/(2*(2^(1/2) - (3*x + 2)^(1/2)))))/3

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3.16.53 \(\int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx\) [1553]