3.1362 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\sqrt{3 x^2+2} (x+8)}{2 (2 x+3)}+\frac{19 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{\sqrt{35}}+2 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

-((8 + x)*Sqrt[2 + 3*x^2])/(2*(3 + 2*x)) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/Sqrt[35]

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Rubi [A]  time = 0.0430619, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {813, 844, 215, 725, 206} \[ -\frac{\sqrt{3 x^2+2} (x+8)}{2 (2 x+3)}+\frac{19 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{\sqrt{35}}+2 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((8 + x)*Sqrt[2 + 3*x^2])/(2*(3 + 2*x)) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sq
rt[2 + 3*x^2])])/Sqrt[35]

Rule 813

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

Rule 844

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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) \sqrt{2+3 x^2}}{(3+2 x)^2} \, dx &=-\frac{(8+x) \sqrt{2+3 x^2}}{2 (3+2 x)}-\frac{1}{8} \int \frac{8-96 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{(8+x) \sqrt{2+3 x^2}}{2 (3+2 x)}+6 \int \frac{1}{\sqrt{2+3 x^2}} \, dx-19 \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=-\frac{(8+x) \sqrt{2+3 x^2}}{2 (3+2 x)}+2 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+19 \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=-\frac{(8+x) \sqrt{2+3 x^2}}{2 (3+2 x)}+2 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )+\frac{19 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )}{\sqrt{35}}\\ \end{align*}

Mathematica [A]  time = 0.092543, size = 71, normalized size = 0.97 \[ -\frac{\sqrt{3 x^2+2} (x+8)}{4 x+6}+\frac{19 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{\sqrt{35}}+2 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((8 + x)*Sqrt[2 + 3*x^2])/(6 + 4*x)) + 2*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + (19*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt
[2 + 3*x^2])])/Sqrt[35]

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Maple [A]  time = 0.01, size = 98, normalized size = 1.3 \begin{align*} -{\frac{19}{35}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+2\,{\it Arcsinh} \left ( 1/2\,x\sqrt{6} \right ) \sqrt{3}+{\frac{19\,\sqrt{35}}{35}{\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}{70} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{39\,x}{70}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-19/35*(12*(x+3/2)^2-36*x-19)^(1/2)+2*arcsinh(1/2*x*6^(1/2))*3^(1/2)+19/35*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1
/2)/(12*(x+3/2)^2-36*x-19)^(1/2))-13/70/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+39/70*x*(3*(x+3/2)^2-9*x-19/4)^(1
/2)

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Maxima [A]  time = 1.51104, size = 103, normalized size = 1.41 \begin{align*} 2 \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{19}{35} \, \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{1}{4} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{4 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 19/35*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3
)) - 1/4*sqrt(3*x^2 + 2) - 13/4*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 2.26792, size = 294, normalized size = 4.03 \begin{align*} \frac{70 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 19 \, \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 ) - 35 \, \sqrt{3 \, x^{2} + 2}{\left (x + 8\right )}}{70 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70*(70*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 19*sqrt(35)*(2*x + 3)*log((sqrt(35)*s
qrt(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)*(x + 8))/(2*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{5 \sqrt{3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx - \int \frac{x \sqrt{3 x^{2} + 2}}{4 x^{2} + 12 x + 9}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.54464, size = 385, normalized size = 5.27 \begin{align*} \frac{19}{35} \, \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 ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - 2 \, \sqrt{3} \log \left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{13}{8} \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{3 \,{\left (3 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \sqrt{35} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{4 \,{\left ({\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2} - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

19/35*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x +
3)) - 2*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(s
qrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sgn(1/(2*x + 3)) - 13/8*sqrt(-18/(2*x
+ 3) + 35/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 3/4*(3*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*
x + 3))*sgn(1/(2*x + 3)) - sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2
*x + 3))^2 - 3)