3.1385 \(\int (5-x) (2+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{1}{21} \left (3 x^2+2\right )^{7/2}+\frac{5}{6} x \left (3 x^2+2\right )^{5/2}+\frac{25}{12} x \left (3 x^2+2\right )^{3/2}+\frac{25}{4} x \sqrt{3 x^2+2}+\frac{25 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{2 \sqrt{3}} \]

[Out]

(25*x*Sqrt[2 + 3*x^2])/4 + (25*x*(2 + 3*x^2)^(3/2))/12 + (5*x*(2 + 3*x^2)^(5/2))/6 - (2 + 3*x^2)^(7/2)/21 + (2
5*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.019657, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {641, 195, 215} \[ -\frac{1}{21} \left (3 x^2+2\right )^{7/2}+\frac{5}{6} x \left (3 x^2+2\right )^{5/2}+\frac{25}{12} x \left (3 x^2+2\right )^{3/2}+\frac{25}{4} x \sqrt{3 x^2+2}+\frac{25 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(2 + 3*x^2)^(5/2),x]

[Out]

(25*x*Sqrt[2 + 3*x^2])/4 + (25*x*(2 + 3*x^2)^(3/2))/12 + (5*x*(2 + 3*x^2)^(5/2))/6 - (2 + 3*x^2)^(7/2)/21 + (2
5*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3])

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 )^{5/2} \, dx &=-\frac{1}{21} \left (2+3 x^2\right )^{7/2}+5 \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{5}{6} x \left (2+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+3 x^2\right )^{7/2}+\frac{25}{3} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{25}{12} x \left (2+3 x^2\right )^{3/2}+\frac{5}{6} x \left (2+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+3 x^2\right )^{7/2}+\frac{25}{2} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{25}{4} x \sqrt{2+3 x^2}+\frac{25}{12} x \left (2+3 x^2\right )^{3/2}+\frac{5}{6} x \left (2+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+3 x^2\right )^{7/2}+\frac{25}{2} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{25}{4} x \sqrt{2+3 x^2}+\frac{25}{12} x \left (2+3 x^2\right )^{3/2}+\frac{5}{6} x \left (2+3 x^2\right )^{5/2}-\frac{1}{21} \left (2+3 x^2\right )^{7/2}+\frac{25 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{2 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0426358, size = 65, normalized size = 0.78 \[ \frac{1}{84} \left (350 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (108 x^6-630 x^5+216 x^4-1365 x^3+144 x^2-1155 x+32\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(32 - 1155*x + 144*x^2 - 1365*x^3 + 216*x^4 - 630*x^5 + 108*x^6)) + 350*Sqrt[3]*ArcSinh[Sqr
t[3/2]*x])/84

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 61, normalized size = 0.7 \begin{align*}{\frac{25\,x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,x}{6} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{25\,\sqrt{3}}{6}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{25\,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)^(5/2),x)

[Out]

25/12*x*(3*x^2+2)^(3/2)+5/6*x*(3*x^2+2)^(5/2)-1/21*(3*x^2+2)^(7/2)+25/6*arcsinh(1/2*x*6^(1/2))*3^(1/2)+25/4*x*
(3*x^2+2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.47365, size = 81, normalized size = 0.98 \begin{align*} -\frac{1}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{5}{6} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{25}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{25}{4} \, \sqrt{3 \, x^{2} + 2} x + \frac{25}{6} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-1/21*(3*x^2 + 2)^(7/2) + 5/6*(3*x^2 + 2)^(5/2)*x + 25/12*(3*x^2 + 2)^(3/2)*x + 25/4*sqrt(3*x^2 + 2)*x + 25/6*
sqrt(3)*arcsinh(1/2*sqrt(6)*x)

________________________________________________________________________________________

Fricas [A]  time = 1.80459, size = 200, normalized size = 2.41 \begin{align*} -\frac{1}{84} \,{\left (108 \, x^{6} - 630 \, x^{5} + 216 \, x^{4} - 1365 \, x^{3} + 144 \, x^{2} - 1155 \, x + 32\right )} \sqrt{3 \, x^{2} + 2} + \frac{25}{12} \, \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)^(5/2),x, algorithm="fricas")

[Out]

-1/84*(108*x^6 - 630*x^5 + 216*x^4 - 1365*x^3 + 144*x^2 - 1155*x + 32)*sqrt(3*x^2 + 2) + 25/12*sqrt(3)*log(-sq
rt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

________________________________________________________________________________________

Sympy [A]  time = 31.9793, size = 131, normalized size = 1.58 \begin{align*} - \frac{9 x^{6} \sqrt{3 x^{2} + 2}}{7} + \frac{15 x^{5} \sqrt{3 x^{2} + 2}}{2} - \frac{18 x^{4} \sqrt{3 x^{2} + 2}}{7} + \frac{65 x^{3} \sqrt{3 x^{2} + 2}}{4} - \frac{12 x^{2} \sqrt{3 x^{2} + 2}}{7} + \frac{55 x \sqrt{3 x^{2} + 2}}{4} - \frac{8 \sqrt{3 x^{2} + 2}}{21} + \frac{25 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*x**6*sqrt(3*x**2 + 2)/7 + 15*x**5*sqrt(3*x**2 + 2)/2 - 18*x**4*sqrt(3*x**2 + 2)/7 + 65*x**3*sqrt(3*x**2 + 2
)/4 - 12*x**2*sqrt(3*x**2 + 2)/7 + 55*x*sqrt(3*x**2 + 2)/4 - 8*sqrt(3*x**2 + 2)/21 + 25*sqrt(3)*asinh(sqrt(6)*
x/2)/6

________________________________________________________________________________________

Giac [A]  time = 1.18812, size = 82, normalized size = 0.99 \begin{align*} -\frac{1}{84} \,{\left (3 \,{\left ({\left ({\left (6 \,{\left ({\left (6 \, x - 35\right )} x + 12\right )} x - 455\right )} x + 48\right )} x - 385\right )} x + 32\right )} \sqrt{3 \, x^{2} + 2} - \frac{25}{6} \, \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)^(5/2),x, algorithm="giac")

[Out]

-1/84*(3*(((6*((6*x - 35)*x + 12)*x - 455)*x + 48)*x - 385)*x + 32)*sqrt(3*x^2 + 2) - 25/6*sqrt(3)*log(-sqrt(3
)*x + sqrt(3*x^2 + 2))