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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 ) \end {gather*}

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 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]

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]

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 = 0.02, size = 55, normalized size = 0.82 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(Sqrt[2 + 3*x^2]*(16 - 375*x + 48*x^2 - 225*x^3 + 36*x^4)) + (5*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/2

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.19, size = 66, normalized size = 0.99 \begin {gather*} \frac {1}{60} \sqrt {3 x^2+2} \left (-36 x^4+225 x^3-48 x^2+375 x-16\right )-\frac {5}{2} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]
])/2

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 60, normalized size = 0.90 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 53, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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
))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 49, normalized size = 0.73 \begin {gather*} \frac {5 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{4}+\frac {15 \sqrt {3 x^{2}+2}\, x}{4}+\frac {5 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{2}-\frac {\left (3 x^{2}+2\right )^{\frac {5}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 48, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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
)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 45, normalized size = 0.67 \begin {gather*} \frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{2}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^4}{5}-\frac {45\,x^3}{4}+\frac {12\,x^2}{5}-\frac {75\,x}{4}+\frac {4}{5}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 + 2)^(3/2)*(x - 5),x)

[Out]

(5*3^(1/2)*asinh((6^(1/2)*x)/2))/2 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((12*x^2)/5 - (75*x)/4 - (45*x^3)/4 + (9*x^4)/
5 + 4/5))/3

________________________________________________________________________________________

sympy [A]  time = 2.76, size = 97, normalized size = 1.45 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________