3.1357 \(\int (5-x) (3+2 x)^3 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(511*x*Sqrt[2 + 3*x^2])/9 + (17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/30 - ((3 + 2*x)^3*(2 + 3*x^2)^(3/2))/18 + (7*(8
98 + 267*x)*(2 + 3*x^2)^(3/2))/270 + (1022*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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Rubi [A]  time = 0.0466594, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{18} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{17}{30} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{7}{270} (267 x+898) \left (3 x^2+2\right )^{3/2}+\frac{511}{9} x \sqrt{3 x^2+2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(511*x*Sqrt[2 + 3*x^2])/9 + (17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/30 - ((3 + 2*x)^3*(2 + 3*x^2)^(3/2))/18 + (7*(8
98 + 267*x)*(2 + 3*x^2)^(3/2))/270 + (1022*ArcSinh[Sqrt[3/2]*x])/(9*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)^3 \sqrt{2+3 x^2} \, dx &=-\frac{1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac{1}{18} \int (3+2 x)^2 (282+153 x) \sqrt{2+3 x^2} \, dx\\ &=\frac{17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac{1}{270} \int (3+2 x) (11466+11214 x) \sqrt{2+3 x^2} \, dx\\ &=\frac{17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac{7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac{1022}{9} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{511}{9} x \sqrt{2+3 x^2}+\frac{17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac{7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac{1022}{9} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{511}{9} x \sqrt{2+3 x^2}+\frac{17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac{7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac{1022 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0512709, size = 60, normalized size = 0.6 \[ \frac{1}{270} \left (10220 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (360 x^5-216 x^4-8445 x^3-21918 x^2-21120 x-14516\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-14516 - 21120*x - 21918*x^2 - 8445*x^3 - 216*x^4 + 360*x^5)) + 10220*Sqrt[3]*ArcSinh[Sqrt
[3/2]*x])/270

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Maple [A]  time = 0.004, size = 77, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{193\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{511\,x}{9}\sqrt{3\,{x}^{2}+2}}+{\frac{1022\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{3629}{135} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*x^3*(3*x^2+2)^(3/2)+193/18*x*(3*x^2+2)^(3/2)+511/9*x*(3*x^2+2)^(1/2)+1022/27*arcsinh(1/2*x*6^(1/2))*3^(1/
2)+4/15*x^2*(3*x^2+2)^(3/2)+3629/135*(3*x^2+2)^(3/2)

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Maxima [A]  time = 1.48563, size = 103, normalized size = 1.03 \begin{align*} -\frac{4}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + \frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{193}{18} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{3629}{135} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{511}{9} \, \sqrt{3 \, x^{2} + 2} x + \frac{1022}{27} \, \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)^3*(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-4/9*(3*x^2 + 2)^(3/2)*x^3 + 4/15*(3*x^2 + 2)^(3/2)*x^2 + 193/18*(3*x^2 + 2)^(3/2)*x + 3629/135*(3*x^2 + 2)^(3
/2) + 511/9*sqrt(3*x^2 + 2)*x + 1022/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.82566, size = 197, normalized size = 1.97 \begin{align*} -\frac{1}{270} \,{\left (360 \, x^{5} - 216 \, x^{4} - 8445 \, x^{3} - 21918 \, x^{2} - 21120 \, x - 14516\right )} \sqrt{3 \, x^{2} + 2} + \frac{511}{27} \, \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)^3*(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/270*(360*x^5 - 216*x^4 - 8445*x^3 - 21918*x^2 - 21120*x - 14516)*sqrt(3*x^2 + 2) + 511/27*sqrt(3)*log(-sqrt
(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 32.9304, size = 150, normalized size = 1.5 \begin{align*} - \frac{4 x^{7}}{\sqrt{3 x^{2} + 2}} + \frac{547 x^{5}}{6 \sqrt{3 x^{2} + 2}} + \frac{1705 x^{3}}{18 \sqrt{3 x^{2} + 2}} + \frac{135 x \sqrt{3 x^{2} + 2}}{2} + \frac{193 x}{9 \sqrt{3 x^{2} + 2}} + \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{5}{2}}}{45} - \frac{16 \sqrt{2} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{2}}}{27} + 27 \left (3 x^{2} + 2\right )^{\frac{3}{2}} + \frac{1022 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x**7/sqrt(3*x**2 + 2) + 547*x**5/(6*sqrt(3*x**2 + 2)) + 1705*x**3/(18*sqrt(3*x**2 + 2)) + 135*x*sqrt(3*x**2
+ 2)/2 + 193*x/(9*sqrt(3*x**2 + 2)) + 16*sqrt(2)*(3*x**2/2 + 1)**(5/2)/45 - 16*sqrt(2)*(3*x**2/2 + 1)**(3/2)/
27 + 27*(3*x**2 + 2)**(3/2) + 1022*sqrt(3)*asinh(sqrt(6)*x/2)/27

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Giac [A]  time = 1.21564, size = 77, normalized size = 0.77 \begin{align*} -\frac{1}{270} \,{\left (3 \,{\left ({\left ({\left (24 \,{\left (5 \, x - 3\right )} x - 2815\right )} x - 7306\right )} x - 7040\right )} x - 14516\right )} \sqrt{3 \, x^{2} + 2} - \frac{1022}{27} \, \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)^3*(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-1/270*(3*(((24*(5*x - 3)*x - 2815)*x - 7306)*x - 7040)*x - 14516)*sqrt(3*x^2 + 2) - 1022/27*sqrt(3)*log(-sqrt
(3)*x + sqrt(3*x^2 + 2))