3.1404 \(\int \frac{5-x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{281 \sqrt{3 x^2+2}}{2450 (2 x+3)}-\frac{13 \sqrt{3 x^2+2}}{70 (2 x+3)^2}-\frac{291 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(70*(3 + 2*x)^2) - (281*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x)) - (291*ArcTanh[(4 - 9*x)/(Sqrt
[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

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Rubi [A]  time = 0.0387895, antiderivative size = 77, normalized size of antiderivative = 1., 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 = {835, 807, 725, 206} \[ -\frac{281 \sqrt{3 x^2+2}}{2450 (2 x+3)}-\frac{13 \sqrt{3 x^2+2}}{70 (2 x+3)^2}-\frac{291 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1225 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(70*(3 + 2*x)^2) - (281*Sqrt[2 + 3*x^2])/(2450*(3 + 2*x)) - (291*ArcTanh[(4 - 9*x)/(Sqrt
[35]*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

Rule 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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]

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 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])

Rubi steps

\begin{align*} \int \frac{5-x}{(3+2 x)^3 \sqrt{2+3 x^2}} \, dx &=-\frac{13 \sqrt{2+3 x^2}}{70 (3+2 x)^2}-\frac{1}{70} \int \frac{-82+39 x}{(3+2 x)^2 \sqrt{2+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+3 x^2}}{70 (3+2 x)^2}-\frac{281 \sqrt{2+3 x^2}}{2450 (3+2 x)}+\frac{291 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx}{1225}\\ &=-\frac{13 \sqrt{2+3 x^2}}{70 (3+2 x)^2}-\frac{281 \sqrt{2+3 x^2}}{2450 (3+2 x)}-\frac{291 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )}{1225}\\ &=-\frac{13 \sqrt{2+3 x^2}}{70 (3+2 x)^2}-\frac{281 \sqrt{2+3 x^2}}{2450 (3+2 x)}-\frac{291 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{1225 \sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.0695316, size = 60, normalized size = 0.78 \[ \frac{-\frac{35 \sqrt{3 x^2+2} (281 x+649)}{(2 x+3)^2}-291 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*(649 + 281*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - 291*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])
/42875

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Maple [A]  time = 0.01, size = 74, normalized size = 1. \begin{align*} -{\frac{281}{4900}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{291\,\sqrt{35}}{42875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }-{\frac{13}{280}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-281/4900/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)-291/42875*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-
36*x-19)^(1/2))-13/280/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [A]  time = 1.49522, size = 103, normalized size = 1.34 \begin{align*} \frac{291}{42875} \, \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}}{70 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{281 \, \sqrt{3 \, x^{2} + 2}}{2450 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

291/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/70*sqrt(3*x^2 + 2)/(4*x
^2 + 12*x + 9) - 281/2450*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 1.98242, size = 243, normalized size = 3.16 \begin{align*} \frac{291 \, \sqrt{35}{\left (4 \, x^{2} + 12 \, x + 9\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 ) - 70 \, \sqrt{3 \, x^{2} + 2}{\left (281 \, x + 649\right )}}{85750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/85750*(291*sqrt(35)*(4*x^2 + 12*x + 9)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2
+ 12*x + 9)) - 70*sqrt(3*x^2 + 2)*(281*x + 649))/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{8 x^{3} \sqrt{3 x^{2} + 2} + 36 x^{2} \sqrt{3 x^{2} + 2} + 54 x \sqrt{3 x^{2} + 2} + 27 \sqrt{3 x^{2} + 2}}\, dx - \int - \frac{5}{8 x^{3} \sqrt{3 x^{2} + 2} + 36 x^{2} \sqrt{3 x^{2} + 2} + 54 x \sqrt{3 x^{2} + 2} + 27 \sqrt{3 x^{2} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(8*x**3*sqrt(3*x**2 + 2) + 36*x**2*sqrt(3*x**2 + 2) + 54*x*sqrt(3*x**2 + 2) + 27*sqrt(3*x**2 + 2))
, x) - Integral(-5/(8*x**3*sqrt(3*x**2 + 2) + 36*x**2*sqrt(3*x**2 + 2) + 54*x*sqrt(3*x**2 + 2) + 27*sqrt(3*x**
2 + 2)), x)

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Giac [B]  time = 1.23148, size = 247, normalized size = 3.21 \begin{align*} \frac{291}{42875} \, \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 ) - \frac{1164 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 6463 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 17904 \, \sqrt{3} x + 2248 \, \sqrt{3} + 17904 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

291/42875*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))) - 1/4900*(1164*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 6463*sqrt(3)*(sqrt(3)*x - sq
rt(3*x^2 + 2))^2 - 17904*sqrt(3)*x + 2248*sqrt(3) + 17904*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 +
3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^2