3.1399 \(\int \frac{(5-x) (3+2 x)^2}{\sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{1}{9} \sqrt{3 x^2+2} (2 x+3)^2+\frac{2}{27} (36 x+251) \sqrt{3 x^2+2}+\frac{127 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

-((3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 + (2*(251 + 36*x)*Sqrt[2 + 3*x^2])/27 + (127*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3]
)

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Rubi [A]  time = 0.0247848, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {833, 780, 215} \[ -\frac{1}{9} \sqrt{3 x^2+2} (2 x+3)^2+\frac{2}{27} (36 x+251) \sqrt{3 x^2+2}+\frac{127 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 + (2*(251 + 36*x)*Sqrt[2 + 3*x^2])/27 + (127*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3]
)

Rule 833

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

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

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]

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)^2}{\sqrt{2+3 x^2}} \, dx &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+3 x^2}+\frac{1}{9} \int \frac{(3+2 x) (143+72 x)}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+3 x^2}+\frac{2}{27} (251+36 x) \sqrt{2+3 x^2}+\frac{127}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+3 x^2}+\frac{2}{27} (251+36 x) \sqrt{2+3 x^2}+\frac{127 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0395794, size = 45, normalized size = 0.73 \[ \frac{1}{27} \left (381 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (12 x^2-36 x-475\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-475 - 36*x + 12*x^2)) + 381*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/27

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Maple [A]  time = 0.005, size = 51, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{475}{27}\sqrt{3\,{x}^{2}+2}}+{\frac{4\,x}{3}\sqrt{3\,{x}^{2}+2}}+{\frac{127\,\sqrt{3}}{9}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*x^2*(3*x^2+2)^(1/2)+475/27*(3*x^2+2)^(1/2)+4/3*x*(3*x^2+2)^(1/2)+127/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 1.57186, size = 68, normalized size = 1.1 \begin{align*} -\frac{4}{9} \, \sqrt{3 \, x^{2} + 2} x^{2} + \frac{4}{3} \, \sqrt{3 \, x^{2} + 2} x + \frac{127}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{475}{27} \, \sqrt{3 \, x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*sqrt(3*x^2 + 2)*x^2 + 4/3*sqrt(3*x^2 + 2)*x + 127/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 475/27*sqrt(3*x^2 +
2)

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Fricas [A]  time = 1.7678, size = 143, normalized size = 2.31 \begin{align*} -\frac{1}{27} \,{\left (12 \, x^{2} - 36 \, x - 475\right )} \sqrt{3 \, x^{2} + 2} + \frac{127}{18} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(12*x^2 - 36*x - 475)*sqrt(3*x^2 + 2) + 127/18*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 0.640268, size = 63, normalized size = 1.02 \begin{align*} - \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{9} + \frac{4 x \sqrt{3 x^{2} + 2}}{3} + \frac{475 \sqrt{3 x^{2} + 2}}{27} + \frac{127 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x**2*sqrt(3*x**2 + 2)/9 + 4*x*sqrt(3*x**2 + 2)/3 + 475*sqrt(3*x**2 + 2)/27 + 127*sqrt(3)*asinh(sqrt(6)*x/2)
/9

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Giac [A]  time = 1.18178, size = 57, normalized size = 0.92 \begin{align*} -\frac{1}{27} \,{\left (12 \,{\left (x - 3\right )} x - 475\right )} \sqrt{3 \, x^{2} + 2} - \frac{127}{9} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(12*(x - 3)*x - 475)*sqrt(3*x^2 + 2) - 127/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))