3.1371 \(\int (5-x) (3+2 x) (2+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac{137}{36} x \left (3 x^2+2\right )^{3/2}+\frac{137}{12} x \sqrt{3 x^2+2}+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(137*x*Sqrt[2 + 3*x^2])/12 + (137*x*(2 + 3*x^2)^(3/2))/36 + ((21 - 5*x)*(2 + 3*x^2)^(5/2))/45 + (137*ArcSinh[S
qrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.0190066, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {780, 195, 215} \[ \frac{1}{45} (21-5 x) \left (3 x^2+2\right )^{5/2}+\frac{137}{36} x \left (3 x^2+2\right )^{3/2}+\frac{137}{12} x \sqrt{3 x^2+2}+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(137*x*Sqrt[2 + 3*x^2])/12 + (137*x*(2 + 3*x^2)^(3/2))/36 + ((21 - 5*x)*(2 + 3*x^2)^(5/2))/45 + (137*ArcSinh[S
qrt[3/2]*x])/(6*Sqrt[3])

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) \left (2+3 x^2\right )^{3/2} \, dx &=\frac{1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac{137}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{137}{36} x \left (2+3 x^2\right )^{3/2}+\frac{1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac{137}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{137}{12} x \sqrt{2+3 x^2}+\frac{137}{36} x \left (2+3 x^2\right )^{3/2}+\frac{1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac{137}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{137}{12} x \sqrt{2+3 x^2}+\frac{137}{36} x \left (2+3 x^2\right )^{3/2}+\frac{1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0582691, size = 60, normalized size = 0.83 \[ \frac{1}{60} \sqrt{3 x^2+2} \left (-60 x^5+252 x^4+605 x^3+336 x^2+1115 x+112\right )+\frac{137 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(112 + 1115*x + 336*x^2 + 605*x^3 + 252*x^4 - 60*x^5))/60 + (137*ArcSinh[Sqrt[3/2]*x])/(6*Sqr
t[3])

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Maple [A]  time = 0.006, size = 61, normalized size = 0.9 \begin{align*} -{\frac{x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{137\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{137\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{137\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{7}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*x*(3*x^2+2)^(5/2)+137/36*x*(3*x^2+2)^(3/2)+137/12*x*(3*x^2+2)^(1/2)+137/18*arcsinh(1/2*x*6^(1/2))*3^(1/2)
+7/15*(3*x^2+2)^(5/2)

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Maxima [A]  time = 1.46995, size = 81, normalized size = 1.12 \begin{align*} -\frac{1}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{7}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{137}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{137}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{137}{18} \, \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*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-1/9*(3*x^2 + 2)^(5/2)*x + 7/15*(3*x^2 + 2)^(5/2) + 137/36*(3*x^2 + 2)^(3/2)*x + 137/12*sqrt(3*x^2 + 2)*x + 13
7/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.08861, size = 186, normalized size = 2.58 \begin{align*} -\frac{1}{60} \,{\left (60 \, x^{5} - 252 \, x^{4} - 605 \, x^{3} - 336 \, x^{2} - 1115 \, x - 112\right )} \sqrt{3 \, x^{2} + 2} + \frac{137}{36} \, \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*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

-1/60*(60*x^5 - 252*x^4 - 605*x^3 - 336*x^2 - 1115*x - 112)*sqrt(3*x^2 + 2) + 137/36*sqrt(3)*log(-sqrt(3)*sqrt
(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 6.84533, size = 110, normalized size = 1.53 \begin{align*} - x^{5} \sqrt{3 x^{2} + 2} + \frac{21 x^{4} \sqrt{3 x^{2} + 2}}{5} + \frac{121 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{28 x^{2} \sqrt{3 x^{2} + 2}}{5} + \frac{223 x \sqrt{3 x^{2} + 2}}{12} + \frac{28 \sqrt{3 x^{2} + 2}}{15} + \frac{137 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**5*sqrt(3*x**2 + 2) + 21*x**4*sqrt(3*x**2 + 2)/5 + 121*x**3*sqrt(3*x**2 + 2)/12 + 28*x**2*sqrt(3*x**2 + 2)/
5 + 223*x*sqrt(3*x**2 + 2)/12 + 28*sqrt(3*x**2 + 2)/15 + 137*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Giac [A]  time = 1.13768, size = 76, normalized size = 1.06 \begin{align*} -\frac{1}{60} \,{\left ({\left ({\left ({\left (12 \,{\left (5 \, x - 21\right )} x - 605\right )} x - 336\right )} x - 1115\right )} x - 112\right )} \sqrt{3 \, x^{2} + 2} - \frac{137}{18} \, \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*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-1/60*((((12*(5*x - 21)*x - 605)*x - 336)*x - 1115)*x - 112)*sqrt(3*x^2 + 2) - 137/18*sqrt(3)*log(-sqrt(3)*x +
sqrt(3*x^2 + 2))