3.1358 \(\int (5-x) (3+2 x)^2 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac{131}{6} x \sqrt{3 x^2+2}+\frac{131 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

[Out]

(131*x*Sqrt[2 + 3*x^2])/6 - ((3 + 2*x)^2*(2 + 3*x^2)^(3/2))/15 + (2*(431 + 99*x)*(2 + 3*x^2)^(3/2))/135 + (131
*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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Rubi [A]  time = 0.0302865, antiderivative size = 78, 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 = {833, 780, 195, 215} \[ -\frac{1}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{135} (99 x+431) \left (3 x^2+2\right )^{3/2}+\frac{131}{6} x \sqrt{3 x^2+2}+\frac{131 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(131*x*Sqrt[2 + 3*x^2])/6 - ((3 + 2*x)^2*(2 + 3*x^2)^(3/2))/15 + (2*(431 + 99*x)*(2 + 3*x^2)^(3/2))/135 + (131
*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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 (5-x) (3+2 x)^2 \sqrt{2+3 x^2} \, dx &=-\frac{1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{1}{15} \int (3+2 x) (233+132 x) \sqrt{2+3 x^2} \, dx\\ &=-\frac{1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac{131}{3} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{131}{6} x \sqrt{2+3 x^2}-\frac{1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac{131}{3} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{131}{6} x \sqrt{2+3 x^2}-\frac{1}{15} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{2}{135} (431+99 x) \left (2+3 x^2\right )^{3/2}+\frac{131 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0477739, size = 55, normalized size = 0.71 \[ \frac{1}{270} \left (3930 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (216 x^4-540 x^3-4542 x^2-6255 x-3124\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-3124 - 6255*x - 4542*x^2 - 540*x^3 + 216*x^4)) + 3930*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/270

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Maple [A]  time = 0.004, size = 63, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{781}{135} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2\,x}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{131\,x}{6}\sqrt{3\,{x}^{2}+2}}+{\frac{131\,\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]

-4/15*x^2*(3*x^2+2)^(3/2)+781/135*(3*x^2+2)^(3/2)+2/3*x*(3*x^2+2)^(3/2)+131/6*x*(3*x^2+2)^(1/2)+131/9*arcsinh(
1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 1.47198, size = 84, normalized size = 1.08 \begin{align*} -\frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{2}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{781}{135} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{131}{6} \, \sqrt{3 \, x^{2} + 2} x + \frac{131}{9} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\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="maxima")

[Out]

-4/15*(3*x^2 + 2)^(3/2)*x^2 + 2/3*(3*x^2 + 2)^(3/2)*x + 781/135*(3*x^2 + 2)^(3/2) + 131/6*sqrt(3*x^2 + 2)*x +
131/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.3862, size = 178, normalized size = 2.28 \begin{align*} -\frac{1}{270} \,{\left (216 \, x^{4} - 540 \, x^{3} - 4542 \, x^{2} - 6255 \, x - 3124\right )} \sqrt{3 \, x^{2} + 2} + \frac{131}{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]

-1/270*(216*x^4 - 540*x^3 - 4542*x^2 - 6255*x - 3124)*sqrt(3*x^2 + 2) + 131/18*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2
+ 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 1.49095, size = 95, normalized size = 1.22 \begin{align*} - \frac{4 x^{4} \sqrt{3 x^{2} + 2}}{5} + 2 x^{3} \sqrt{3 x^{2} + 2} + \frac{757 x^{2} \sqrt{3 x^{2} + 2}}{45} + \frac{139 x \sqrt{3 x^{2} + 2}}{6} + \frac{1562 \sqrt{3 x^{2} + 2}}{135} + \frac{131 \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**4*sqrt(3*x**2 + 2)/5 + 2*x**3*sqrt(3*x**2 + 2) + 757*x**2*sqrt(3*x**2 + 2)/45 + 139*x*sqrt(3*x**2 + 2)/6
+ 1562*sqrt(3*x**2 + 2)/135 + 131*sqrt(3)*asinh(sqrt(6)*x/2)/9

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Giac [A]  time = 1.15751, size = 73, normalized size = 0.94 \begin{align*} -\frac{1}{270} \,{\left (3 \,{\left (2 \,{\left (18 \,{\left (2 \, x - 5\right )} x - 757\right )} x - 2085\right )} x - 3124\right )} \sqrt{3 \, x^{2} + 2} - \frac{131}{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]

-1/270*(3*(2*(18*(2*x - 5)*x - 757)*x - 2085)*x - 3124)*sqrt(3*x^2 + 2) - 131/9*sqrt(3)*log(-sqrt(3)*x + sqrt(
3*x^2 + 2))