3.1403 \(\int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx\)

Optimal. Leaf size=55 \[ -\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {41 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \]

[Out]

-41/1225*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-13/35*(3*x^2+2)^(1/2)/(3+2*x)

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Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {807, 725, 206} \[ -\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {41 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (41*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*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 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 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx &=-\frac {13 \sqrt {2+3 x^2}}{35 (3+2 x)}+\frac {41}{35} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+3 x^2}}{35 (3+2 x)}-\frac {41}{35} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=-\frac {13 \sqrt {2+3 x^2}}{35 (3+2 x)}-\frac {41 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{35 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 55, normalized size = 1.00 \[ -\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}-\frac {41 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (41*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])

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fricas [A]  time = 0.58, size = 74, normalized size = 1.35 \[ \frac {41 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 910 \, \sqrt {3 \, x^{2} + 2}}{2450 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2450*(41*sqrt(35)*(2*x + 3)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9
)) - 910*sqrt(3*x^2 + 2))/(2*x + 3)

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giac [B]  time = 0.27, size = 125, normalized size = 2.27 \[ \frac {1}{2450} \, \sqrt {35} {\left (13 \, \sqrt {35} \sqrt {3} + 82 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {41 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{1225 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {13 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}}{70 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2450*sqrt(35)*(13*sqrt(35)*sqrt(3) + 82*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) - 41/1225*sqrt(35)*log(s
qrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3)) - 13/70*sqrt(-18
/(2*x + 3) + 35/(2*x + 3)^2 + 3)/sgn(1/(2*x + 3))

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maple [A]  time = 0.06, size = 53, normalized size = 0.96 \[ -\frac {41 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1225}-\frac {13 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{70 \left (x +\frac {3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/70/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-41/1225*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 = 1.35, size = 53, normalized size = 0.96 \[ \frac {41}{1225} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{35 \, {\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

41/1225*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/35*sqrt(3*x^2 + 2)/(2*x +
3)

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mupad [B]  time = 1.92, size = 53, normalized size = 0.96 \[ \frac {41\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1225}-\frac {41\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1225}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70\,\left (x+\frac {3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(41*35^(1/2)*log(x + 3/2))/1225 - (41*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1225 - (
13*3^(1/2)*(x^2 + 2/3)^(1/2))/(70*(x + 3/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{4 x^{2} \sqrt {3 x^{2} + 2} + 12 x \sqrt {3 x^{2} + 2} + 9 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {3 x^{2} + 2} + 12 x \sqrt {3 x^{2} + 2} + 9 \sqrt {3 x^{2} + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(4*x**2*sqrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2) + 9*sqrt(3*x**2 + 2)), x) - Integral(-5/(4*x**2*s
qrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2) + 9*sqrt(3*x**2 + 2)), x)

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