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

Optimal. Leaf size=35 \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} \sqrt{3 x-2}}{\sqrt{5 x+3}}\right ) \]

[Out]

Sqrt[2/3]*ArcTan[(Sqrt[3/2]*Sqrt[-2 + 3*x])/Sqrt[3 + 5*x]]

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Rubi [A]  time = 0.0079861, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {93, 203} \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} \sqrt{3 x-2}}{\sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/3]*ArcTan[(Sqrt[3/2]*Sqrt[-2 + 3*x])/Sqrt[3 + 5*x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{-2+3 x} \sqrt{3+5 x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,\frac{\sqrt{-2+3 x}}{\sqrt{3+5 x}}\right )\\ &=\sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} \sqrt{-2+3 x}}{\sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0128208, size = 30, normalized size = 0.86 \[ \sqrt{\frac{2}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{9 x}{2}-3}}{\sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/3]*ArcTan[Sqrt[-3 + (9*x)/2]/Sqrt[3 + 5*x]]

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Maple [B]  time = 0.018, size = 53, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{6}}{6}\sqrt{-2+3\,x}\sqrt{3+5\,x}\arctan \left ({\frac{ \left ( 12+x \right ) \sqrt{6}}{12}{\frac{1}{\sqrt{15\,{x}^{2}-x-6}}}} \right ){\frac{1}{\sqrt{15\,{x}^{2}-x-6}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-2+3*x)^(1/2)*(3+5*x)^(1/2)/(15*x^2-x-6)^(1/2)*6^(1/2)*arctan(1/12*(12+x)*6^(1/2)/(15*x^2-x-6)^(1/2))

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Maxima [A]  time = 1.76264, size = 27, normalized size = 0.77 \begin{align*} -\frac{1}{6} \, \sqrt{6} \arcsin \left (\frac{x}{19 \,{\left | x \right |}} + \frac{12}{19 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(6)*arcsin(1/19*x/abs(x) + 12/19/abs(x))

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Fricas [A]  time = 1.51318, size = 142, normalized size = 4.06 \begin{align*} -\frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{\sqrt{3} \sqrt{2} \sqrt{5 \, x + 3} \sqrt{3 \, x - 2}{\left (x + 12\right )}}{12 \,{\left (15 \, x^{2} - x - 6\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*sqrt(5*x + 3)*sqrt(3*x - 2)*(x + 12)/(15*x^2 - x - 6))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{3 x - 2} \sqrt{5 x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.43884, size = 57, normalized size = 1.63 \begin{align*} -\frac{1}{15} \, \sqrt{10} \sqrt{5} \sqrt{3} \arctan \left (\frac{1}{60} \, \sqrt{10}{\left ({\left (\sqrt{3} \sqrt{5 \, x + 3} - \sqrt{15 \, x - 10}\right )}^{2} + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*sqrt(10)*sqrt(5)*sqrt(3)*arctan(1/60*sqrt(10)*((sqrt(3)*sqrt(5*x + 3) - sqrt(15*x - 10))^2 + 1))