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

Optimal. Leaf size=52 \[ -\frac {13 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}-\frac {\sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {844, 215, 725, 206} \begin {gather*} -\frac {13 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}-\frac {\sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSinh[Sqrt[3/2]*x]/(2*Sqrt[3]) - (13*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])

Rule 206

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \sqrt {2+3 x^2}} \, dx &=-\left (\frac {1}{2} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\right )+\frac {13}{2} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {13}{2} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {\sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {13 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{2 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 52, normalized size = 1.00 \begin {gather*} -\frac {13 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}-\frac {\sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*ArcSinh[Sqrt[3/2]*x]/Sqrt[3] - (13*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35])

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IntegrateAlgebraic [A]  time = 0.31, size = 77, normalized size = 1.48 \begin {gather*} \frac {\log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{2 \sqrt {3}}+\frac {13 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{\sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/Sqrt[35] + Log[-(Sqrt[3]*x) + Sqrt[
2 + 3*x^2]]/(2*Sqrt[3])

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fricas [A]  time = 0.43, size = 76, normalized size = 1.46 \begin {gather*} \frac {1}{12} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {13}{140} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 13/140*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x
- 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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giac [B]  time = 0.25, size = 90, normalized size = 1.73 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {13}{70} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 13/70*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) +
2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2)))

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maple [A]  time = 0.05, size = 44, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{6}-\frac {13 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{70} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-13/70*35^(1/2)*arctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2
))

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maxima [A]  time = 1.60, size = 47, normalized size = 0.90 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {13}{70} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 13/70*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x
+ 3))

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mupad [B]  time = 0.12, size = 49, normalized size = 0.94 \begin {gather*} \frac {\sqrt {35}\,\left (26\,\ln \left (x+\frac {3}{2}\right )-26\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{140}-\frac {\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35^(1/2)*(26*log(x + 3/2) - 26*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)))/140 - (3^(1/2)*asinh((
2^(1/2)*3^(1/2)*x)/2))/6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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