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

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

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Rubi [A]  time = 0.04, antiderivative size = 77, 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 = {835, 807, 725, 206} \begin {gather*} -\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}} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 0.78 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.60, size = 76, normalized size = 0.99 \begin {gather*} \frac {\sqrt {3 x^2+2} (-281 x-649)}{1225 (2 x+3)^2}+\frac {582 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{1225 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-649 - 281*x)*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)^2) + (582*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 +
3*x^2])/Sqrt[35]])/(1225*Sqrt[35])

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fricas [A]  time = 0.41, size = 89, normalized size = 1.16 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.29, size = 183, normalized size = 2.38 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.06, size = 74, normalized size = 0.96 \begin {gather*} -\frac {291 \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 )}{42875}-\frac {13 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{280 \left (x +\frac {3}{2}\right )^{2}}-\frac {281 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{4900 \left (x +\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/280/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-281/4900/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(1/2)-291/42875*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 = 1.45, size = 76, normalized size = 0.99 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.86, size = 77, normalized size = 1.00 \begin {gather*} \frac {291\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {291\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}-\frac {281\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{280\,\left (x^2+3\,x+\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \left (- \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}}\right )\, dx \end {gather*}

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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