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

Optimal. Leaf size=40 \[ -\frac {7 (2-7 x)}{6 \sqrt {3 x^2+2}}-\frac {2 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {778, 215} \begin {gather*} -\frac {7 (2-7 x)}{6 \sqrt {3 x^2+2}}-\frac {2 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x))/(6*Sqrt[2 + 3*x^2]) - (2*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 43, normalized size = 1.08 \begin {gather*} -\frac {4 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-147 x+42}{18 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*(42 - 147*x + 4*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/Sqrt[2 + 3*x^2]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.26, size = 51, normalized size = 1.28 \begin {gather*} \frac {7 (7 x-2)}{6 \sqrt {3 x^2+2}}+\frac {2 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(-2 + 7*x))/(6*Sqrt[2 + 3*x^2]) + (2*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 62, normalized size = 1.55 \begin {gather*} \frac {2 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 21 \, \sqrt {3 \, x^{2} + 2} {\left (7 \, x - 2\right )}}{18 \, {\left (3 \, x^{2} + 2\right )}} \end {gather*}

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]

1/18*(2*sqrt(3)*(3*x^2 + 2)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 21*sqrt(3*x^2 + 2)*(7*x - 2))/(3*x^2
+ 2)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 39, normalized size = 0.98 \begin {gather*} \frac {2}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {7 \, {\left (7 \, x - 2\right )}}{6 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

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]

2/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 7/6*(7*x - 2)/sqrt(3*x^2 + 2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 37, normalized size = 0.92 \begin {gather*} \frac {49 x}{6 \sqrt {3 x^{2}+2}}-\frac {2 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}-\frac {7}{3 \sqrt {3 x^{2}+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/6/(3*x^2+2)^(1/2)*x-2/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-7/3/(3*x^2+2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.24, size = 36, normalized size = 0.90 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {49 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {7}{3 \, \sqrt {3 \, x^{2} + 2}} \end {gather*}

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]

-2/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 49/6*x/sqrt(3*x^2 + 2) - 7/3/sqrt(3*x^2 + 2)

________________________________________________________________________________________

mupad [B]  time = 1.78, size = 88, normalized size = 2.20 \begin {gather*} -\frac {2\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-126+\sqrt {6}\,147{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (126+\sqrt {6}\,147{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*6^(1/2)*(6^(1/2)*147i - 126)*(x^2 + 2/3)^(1/2)*1i)/(64
8*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*147i + 126)*(x^2 + 2/3)^(1/2)*1i)/(648*(x + (6^(1/2)*1i)/3
))

________________________________________________________________________________________

sympy [B]  time = 14.98, size = 99, normalized size = 2.48 \begin {gather*} - \frac {6 \sqrt {3} x^{2} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} + \frac {6 x \sqrt {3 x^{2} + 2}}{27 x^{2} + 18} + \frac {15 x}{2 \sqrt {3 x^{2} + 2}} - \frac {4 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} - \frac {7}{3 \sqrt {3 x^{2} + 2}} \end {gather*}

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]

-6*sqrt(3)*x**2*asinh(sqrt(6)*x/2)/(27*x**2 + 18) + 6*x*sqrt(3*x**2 + 2)/(27*x**2 + 18) + 15*x/(2*sqrt(3*x**2
+ 2)) - 4*sqrt(3)*asinh(sqrt(6)*x/2)/(27*x**2 + 18) - 7/(3*sqrt(3*x**2 + 2))

________________________________________________________________________________________