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

Optimal. Leaf size=53 \[ \frac {41 x+26}{70 \sqrt {3 x^2+2}}-\frac {26 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \]

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Rubi [A]  time = 0.03, antiderivative size = 53, 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 = {823, 12, 725, 206} \begin {gather*} \frac {41 x+26}{70 \sqrt {3 x^2+2}}-\frac {26 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) - (26*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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])

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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx &=\frac {26+41 x}{70 \sqrt {2+3 x^2}}-\frac {1}{210} \int -\frac {156}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {26+41 x}{70 \sqrt {2+3 x^2}}+\frac {26}{35} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {26+41 x}{70 \sqrt {2+3 x^2}}-\frac {26}{35} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {26+41 x}{70 \sqrt {2+3 x^2}}-\frac {26 \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 = 53, normalized size = 1.00 \begin {gather*} \frac {123 x+78}{210 \sqrt {3 x^2+2}}-\frac {26 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(78 + 123*x)/(210*Sqrt[2 + 3*x^2]) - (26*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])

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IntegrateAlgebraic [A]  time = 0.42, size = 69, normalized size = 1.30 \begin {gather*} \frac {41 x+26}{70 \sqrt {3 x^2+2}}+\frac {52 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{35 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*Sqrt[2 + 3*x^2]) + (52*ArcTanh[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/
(35*Sqrt[35])

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fricas [A]  time = 0.43, size = 83, normalized size = 1.57 \begin {gather*} \frac {26 \, \sqrt {35} {\left (3 \, x^{2} + 2\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 ) + 35 \, \sqrt {3 \, x^{2} + 2} {\left (41 \, x + 26\right )}}{2450 \, {\left (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)^(3/2),x, algorithm="fricas")

[Out]

1/2450*(26*sqrt(35)*(3*x^2 + 2)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x +
9)) + 35*sqrt(3*x^2 + 2)*(41*x + 26))/(3*x^2 + 2)

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giac [A]  time = 0.21, size = 84, normalized size = 1.58 \begin {gather*} \frac {26}{1225} \, \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 {41 \, x + 26}{70 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

26/1225*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))) + 1/70*(41*x + 26)/sqrt(3*x^2 + 2)

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maple [A]  time = 0.05, size = 77, normalized size = 1.45 \begin {gather*} -\frac {x}{4 \sqrt {3 x^{2}+2}}+\frac {117 x}{140 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {26 \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}{35 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(3*x^2+2)^(1/2)*x+13/35/(-9*x+3*(x+3/2)^2-19/4)^(1/2)+117/140*x/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-26/1225*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 = 1.24, size = 58, normalized size = 1.09 \begin {gather*} \frac {26}{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 {41 \, x}{70 \, \sqrt {3 \, x^{2} + 2}} + \frac {13}{35 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

26/1225*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 41/70*x/sqrt(3*x^2 + 2) + 13
/35/sqrt(3*x^2 + 2)

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mupad [B]  time = 1.81, size = 106, normalized size = 2.00 \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 )}{1225}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-234+\sqrt {6}\,123{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7560\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (234+\sqrt {6}\,123{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7560\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)*(3*x^2 + 2)^(3/2)),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)))/1225 - (3^(1/2)*6^(1/2
)*(6^(1/2)*123i - 234)*(x^2 + 2/3)^(1/2)*1i)/(7560*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*123i + 23
4)*(x^2 + 2/3)^(1/2)*1i)/(7560*(x - (6^(1/2)*1i)/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{6 x^{3} \sqrt {3 x^{2} + 2} + 9 x^{2} \sqrt {3 x^{2} + 2} + 4 x \sqrt {3 x^{2} + 2} + 6 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{6 x^{3} \sqrt {3 x^{2} + 2} + 9 x^{2} \sqrt {3 x^{2} + 2} + 4 x \sqrt {3 x^{2} + 2} + 6 \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)**(3/2),x)

[Out]

-Integral(x/(6*x**3*sqrt(3*x**2 + 2) + 9*x**2*sqrt(3*x**2 + 2) + 4*x*sqrt(3*x**2 + 2) + 6*sqrt(3*x**2 + 2)), x
) - Integral(-5/(6*x**3*sqrt(3*x**2 + 2) + 9*x**2*sqrt(3*x**2 + 2) + 4*x*sqrt(3*x**2 + 2) + 6*sqrt(3*x**2 + 2)
), x)

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