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

Optimal result

Integrand size = 19, antiderivative size = 26 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arcsinh}\left (\sqrt {\frac {3}{13}} \sqrt {-3+2 x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 221} \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arcsinh}\left (\sqrt {\frac {3}{13}} \sqrt {2 x-3}\right ) \]

[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 56

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 221

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {1}{\sqrt {\frac {-9+6 x}{4+6 x}}}\right ) \]

[In]

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

[Out]

Sqrt[2/3]*ArcTanh[1/Sqrt[(-9 + 6*x)/(4 + 6*x)]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).

Time = 0.66 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19

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

[In]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {1}{12} \, \sqrt {3} \sqrt {2} \log \left (4 \, \sqrt {3} \sqrt {2} {\left (12 \, x - 5\right )} \sqrt {3 \, x + 2} \sqrt {2 \, x - 3} + 288 \, x^{2} - 240 \, x - 119\right ) \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\begin {cases} \frac {\sqrt {6} \operatorname {acosh}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | > \frac {13}{6} \\- \frac {\sqrt {6} i \operatorname {asin}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {otherwise} \end {cases} \]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {1}{6} \, \sqrt {6} \log \left (2 \, \sqrt {6} \sqrt {6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=-\frac {1}{3} \, \sqrt {3} \sqrt {2} \log \left ({\left | -\sqrt {2} \sqrt {3 \, x + 2} + \sqrt {6 \, x - 9} \right |}\right ) \]

[In]

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

[Out]

-1/3*sqrt(3)*sqrt(2)*log(abs(-sqrt(2)*sqrt(3*x + 2) + sqrt(6*x - 9)))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {2\,\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}\right )}{3} \]

[In]

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

[Out]

(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