3.1413 \(\int \frac{5-x}{(3+2 x) (2+3 x^2)^{3/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0267461, antiderivative size = 53, 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 = {823, 12, 725, 206} \[ \frac{41 x+26}{70 \sqrt{3 x^2+2}}-\frac{26 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

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 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])

Rule 12

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

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) \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*}

Mathematica [A]  time = 0.0264823, size = 53, normalized size = 1. \[ \frac{123 x+78}{210 \sqrt{3 x^2+2}}-\frac{26 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{35 \sqrt{35}} \]

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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Maple [A]  time = 0.008, size = 77, normalized size = 1.5 \begin{align*} -{\frac{x}{4}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{13}{35}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{117\,x}{140}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{26\,\sqrt{35}}{1225}{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x/(3*x^2+2)^(1/2)+13/35/(3*(x+3/2)^2-9*x-19/4)^(1/2)+117/140*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)-26/1225*35^(1
/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 1.5335, size = 78, normalized size = 1.47 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.54541, size = 219, normalized size = 4.13 \begin{align*} \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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 - \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}}\, dx \end{align*}

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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Giac [A]  time = 1.32556, size = 113, normalized size = 2.13 \begin{align*} \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{align*}

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)