3.1372 \(\int (5-x) (2+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{15} \left (3 x^2+2\right )^{5/2}+\frac{5}{4} x \left (3 x^2+2\right )^{3/2}+\frac{15}{4} x \sqrt{3 x^2+2}+\frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(15*x*Sqrt[2 + 3*x^2])/4 + (5*x*(2 + 3*x^2)^(3/2))/4 - (2 + 3*x^2)^(5/2)/15 + (5*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])
/2

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Rubi [A]  time = 0.014464, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {641, 195, 215} \[ -\frac{1}{15} \left (3 x^2+2\right )^{5/2}+\frac{5}{4} x \left (3 x^2+2\right )^{3/2}+\frac{15}{4} x \sqrt{3 x^2+2}+\frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(2 + 3*x^2)^(3/2),x]

[Out]

(15*x*Sqrt[2 + 3*x^2])/4 + (5*x*(2 + 3*x^2)^(3/2))/4 - (2 + 3*x^2)^(5/2)/15 + (5*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])
/2

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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) \left (2+3 x^2\right )^{3/2} \, dx &=-\frac{1}{15} \left (2+3 x^2\right )^{5/2}+5 \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{5}{4} x \left (2+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+3 x^2\right )^{5/2}+\frac{15}{2} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{15}{4} x \sqrt{2+3 x^2}+\frac{5}{4} x \left (2+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+3 x^2\right )^{5/2}+\frac{15}{2} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{15}{4} x \sqrt{2+3 x^2}+\frac{5}{4} x \left (2+3 x^2\right )^{3/2}-\frac{1}{15} \left (2+3 x^2\right )^{5/2}+\frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\\ \end{align*}

Mathematica [A]  time = 0.0262313, size = 55, normalized size = 0.82 \[ \frac{5}{2} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{1}{60} \sqrt{3 x^2+2} \left (36 x^4-225 x^3+48 x^2-375 x+16\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2 + 3*x^2]*(16 - 375*x + 48*x^2 - 225*x^3 + 36*x^4))/60 + (5*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2

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Maple [A]  time = 0.004, size = 49, normalized size = 0.7 \begin{align*}{\frac{5\,x}{4} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5\,\sqrt{3}}{2}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{15\,x}{4}\sqrt{3\,{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x*(3*x^2+2)^(3/2)-1/15*(3*x^2+2)^(5/2)+5/2*arcsinh(1/2*x*6^(1/2))*3^(1/2)+15/4*x*(3*x^2+2)^(1/2)

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Maxima [A]  time = 1.49432, size = 65, normalized size = 0.97 \begin{align*} -\frac{1}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{5}{4} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{15}{4} \, \sqrt{3 \, x^{2} + 2} x + \frac{5}{2} \, \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*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-1/15*(3*x^2 + 2)^(5/2) + 5/4*(3*x^2 + 2)^(3/2)*x + 15/4*sqrt(3*x^2 + 2)*x + 5/2*sqrt(3)*arcsinh(1/2*sqrt(6)*x
)

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Fricas [A]  time = 2.16595, size = 165, normalized size = 2.46 \begin{align*} -\frac{1}{60} \,{\left (36 \, x^{4} - 225 \, x^{3} + 48 \, x^{2} - 375 \, x + 16\right )} \sqrt{3 \, x^{2} + 2} + \frac{5}{4} \, \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*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

-1/60*(36*x^4 - 225*x^3 + 48*x^2 - 375*x + 16)*sqrt(3*x^2 + 2) + 5/4*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x -
3*x^2 - 1)

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Sympy [A]  time = 3.39488, size = 97, normalized size = 1.45 \begin{align*} - \frac{3 x^{4} \sqrt{3 x^{2} + 2}}{5} + \frac{15 x^{3} \sqrt{3 x^{2} + 2}}{4} - \frac{4 x^{2} \sqrt{3 x^{2} + 2}}{5} + \frac{25 x \sqrt{3 x^{2} + 2}}{4} - \frac{4 \sqrt{3 x^{2} + 2}}{15} + \frac{5 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x**4*sqrt(3*x**2 + 2)/5 + 15*x**3*sqrt(3*x**2 + 2)/4 - 4*x**2*sqrt(3*x**2 + 2)/5 + 25*x*sqrt(3*x**2 + 2)/4
- 4*sqrt(3*x**2 + 2)/15 + 5*sqrt(3)*asinh(sqrt(6)*x/2)/2

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Giac [A]  time = 1.23425, size = 72, normalized size = 1.07 \begin{align*} -\frac{1}{60} \,{\left (3 \,{\left ({\left (3 \,{\left (4 \, x - 25\right )} x + 16\right )} x - 125\right )} x + 16\right )} \sqrt{3 \, x^{2} + 2} - \frac{5}{2} \, \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*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-1/60*(3*((3*(4*x - 25)*x + 16)*x - 125)*x + 16)*sqrt(3*x^2 + 2) - 5/2*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2
))