3.1361 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{3+2 x} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{4} \sqrt{3 x^2+2} (13-x)-\frac{13}{8} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{121 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{8 \sqrt{3}} \]

[Out]

((13 - x)*Sqrt[2 + 3*x^2])/4 - (121*ArcSinh[Sqrt[3/2]*x])/(8*Sqrt[3]) - (13*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/8

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Rubi [A]  time = 0.0434245, antiderivative size = 72, 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 = {815, 844, 215, 725, 206} \[ \frac{1}{4} \sqrt{3 x^2+2} (13-x)-\frac{13}{8} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{121 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{8 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((13 - x)*Sqrt[2 + 3*x^2])/4 - (121*ArcSinh[Sqrt[3/2]*x])/(8*Sqrt[3]) - (13*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/8

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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} \, dx &=\frac{1}{4} (13-x) \sqrt{2+3 x^2}+\frac{1}{24} \int \frac{276-726 x}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{1}{4} (13-x) \sqrt{2+3 x^2}-\frac{121}{8} \int \frac{1}{\sqrt{2+3 x^2}} \, dx+\frac{455}{8} \int \frac{1}{(3+2 x) \sqrt{2+3 x^2}} \, dx\\ &=\frac{1}{4} (13-x) \sqrt{2+3 x^2}-\frac{121 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{8 \sqrt{3}}-\frac{455}{8} \operatorname{Subst}\left (\int \frac{1}{35-x^2} \, dx,x,\frac{4-9 x}{\sqrt{2+3 x^2}}\right )\\ &=\frac{1}{4} (13-x) \sqrt{2+3 x^2}-\frac{121 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{8 \sqrt{3}}-\frac{13}{8} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{2+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0416274, size = 68, normalized size = 0.94 \[ \frac{1}{24} \left (-6 \sqrt{3 x^2+2} (x-13)-39 \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-121 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-13 + x)*Sqrt[2 + 3*x^2] - 121*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 39*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqr
t[2 + 3*x^2])])/24

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Maple [A]  time = 0.007, size = 72, normalized size = 1. \begin{align*} -{\frac{x}{4}\sqrt{3\,{x}^{2}+2}}-{\frac{121\,\sqrt{3}}{24}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{8}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{13\,\sqrt{35}}{8}{\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*x^2+2)^(1/2)/(3+2*x),x)

[Out]

-1/4*x*(3*x^2+2)^(1/2)-121/24*arcsinh(1/2*x*6^(1/2))*3^(1/2)+13/8*(12*(x+3/2)^2-36*x-19)^(1/2)-13/8*35^(1/2)*a
rctanh(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 = 1.51656, size = 95, normalized size = 1.32 \begin{align*} -\frac{1}{4} \, \sqrt{3 \, x^{2} + 2} x - \frac{121}{24} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{13}{8} \, \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}{4} \, \sqrt{3 \, x^{2} + 2} \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),x, algorithm="maxima")

[Out]

-1/4*sqrt(3*x^2 + 2)*x - 121/24*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 13/8*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x +
3) - 2/3*sqrt(6)/abs(2*x + 3)) + 13/4*sqrt(3*x^2 + 2)

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Fricas [A]  time = 2.72362, size = 258, normalized size = 3.58 \begin{align*} -\frac{1}{4} \, \sqrt{3 \, x^{2} + 2}{\left (x - 13\right )} + \frac{121}{48} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac{13}{16} \, \sqrt{35} \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 ) \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),x, algorithm="fricas")

[Out]

-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/48*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 13/16*sqrt(35)*log
(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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

[Out]

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

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Giac [A]  time = 1.20313, size = 140, normalized size = 1.94 \begin{align*} -\frac{1}{4} \, \sqrt{3 \, x^{2} + 2}{\left (x - 13\right )} + \frac{121}{24} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{13}{8} \, \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 ) \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),x, algorithm="giac")

[Out]

-1/4*sqrt(3*x^2 + 2)*(x - 13) + 121/24*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 13/8*sqrt(35)*log(-abs(-2*s
qrt(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)))