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

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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} \begin {gather*} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {3 x-2}}{\sqrt {5 x+3}}\right ) \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.86 \begin {gather*} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {9 x}{2}-3}}{\sqrt {5 x+3}}\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.09, size = 36, normalized size = 1.03 \begin {gather*} -\sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt {5 x+3}}{\sqrt {3 x-2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.38, size = 46, normalized size = 1.31 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.28, size = 42, normalized size = 1.20 \begin {gather*} -\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 {gather*}

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))

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maple [B]  time = 0.02, size = 53, normalized size = 1.51 \begin {gather*} -\frac {\sqrt {3 x -2}\, \sqrt {5 x +3}\, \sqrt {6}\, \arctan \left (\frac {\left (x +12\right ) \sqrt {6}}{12 \sqrt {15 x^{2}-x -6}}\right )}{6 \sqrt {15 x^{2}-x -6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-2+3*x)^(1/2)*(5*x+3)^(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 = 1.38, size = 20, normalized size = 0.57 \begin {gather*} -\frac {1}{6} \, \sqrt {6} \arcsin \left (\frac {x}{19 \, {\left | x \right |}} + \frac {12}{19 \, {\left | x \right |}}\right ) \end {gather*}

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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mupad [B]  time = 4.96, size = 118, normalized size = 3.37 \begin {gather*} -\frac {\sqrt {6}\,\left (\ln \left (\frac {{\left (-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}^2}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3}{5}-\frac {\sqrt {6}\,\left (-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{30\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )-\ln \left (\frac {-\sqrt {3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {3}-\sqrt {5\,x+3}}\right )\right )\,1{}\mathrm {i}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6^(1/2)*(log((2^(1/2)*1i - (3*x - 2)^(1/2))^2/(3^(1/2) - (5*x + 3)^(1/2))^2 - (6^(1/2)*(2^(1/2)*1i - (3*x -
2)^(1/2))*1i)/(30*(3^(1/2) - (5*x + 3)^(1/2))) + 3/5) - log((2^(1/2)*1i - (3*x - 2)^(1/2))/(3^(1/2) - (5*x + 3
)^(1/2))))*1i)/6

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {3 x - 2} \sqrt {5 x + 3}}\, dx \end {gather*}

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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