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

Optimal. Leaf size=67 \[ -\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}-\frac {2}{9} (51 x+131) \sqrt {3 x^2+2}+\frac {134 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {819, 780, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}-\frac {2}{9} (51 x+131) \sqrt {3 x^2+2}+\frac {134 \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)^3)/(2 + 3*x^2)^(3/2),x]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) - (2*(131 + 51*x)*Sqrt[2 + 3*x^2])/9 + (134*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 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 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^2}{6 \sqrt {2+3 x^2}}+\frac {1}{6} \int \frac {(44-204 x) (3+2 x)}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{6 \sqrt {2+3 x^2}}-\frac {2}{9} (131+51 x) \sqrt {2+3 x^2}+\frac {134}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{6 \sqrt {2+3 x^2}}-\frac {2}{9} (131+51 x) \sqrt {2+3 x^2}+\frac {134 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 53, normalized size = 0.79 \begin {gather*} -\frac {24 x^3-24 x^2-268 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-411 x+1426}{18 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*(1426 - 411*x - 24*x^2 + 24*x^3 - 268*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/Sqrt[2 + 3*x^2]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.36, size = 61, normalized size = 0.91 \begin {gather*} \frac {-24 x^3+24 x^2+411 x-1426}{18 \sqrt {3 x^2+2}}-\frac {134 \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)^3)/(2 + 3*x^2)^(3/2),x]

[Out]

(-1426 + 411*x + 24*x^2 - 24*x^3)/(18*Sqrt[2 + 3*x^2]) - (134*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 73, normalized size = 1.09 \begin {gather*} \frac {134 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (24 \, x^{3} - 24 \, x^{2} - 411 \, x + 1426\right )} \sqrt {3 \, x^{2} + 2}}{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/(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

1/18*(134*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (24*x^3 - 24*x^2 - 411*x + 1426)*s
qrt(3*x^2 + 2))/(3*x^2 + 2)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 47, normalized size = 0.70 \begin {gather*} -\frac {134}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (8 \, {\left (x - 1\right )} x - 137\right )} x + 1426}{18 \, \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/(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-134/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*(x - 1)*x - 137)*x + 1426)/sqrt(3*x^2 + 2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 65, normalized size = 0.97 \begin {gather*} -\frac {4 x^{3}}{3 \sqrt {3 x^{2}+2}}+\frac {4 x^{2}}{3 \sqrt {3 x^{2}+2}}+\frac {137 x}{6 \sqrt {3 x^{2}+2}}+\frac {134 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}-\frac {713}{9 \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/(3*x^2+2)^(3/2),x)

[Out]

-4/3/(3*x^2+2)^(1/2)*x^3+137/6/(3*x^2+2)^(1/2)*x+134/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+4/3/(3*x^2+2)^(1/2)*x^2-
713/9/(3*x^2+2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 64, normalized size = 0.96 \begin {gather*} -\frac {4 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {4 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {134}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {137 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {713}{9 \, \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/(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^3/sqrt(3*x^2 + 2) + 4/3*x^2/sqrt(3*x^2 + 2) + 134/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 137/6*x/sqrt(3*x^2
+ 2) - 713/9/sqrt(3*x^2 + 2)

________________________________________________________________________________________

mupad [B]  time = 1.76, size = 105, normalized size = 1.57 \begin {gather*} \frac {134\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\left (\frac {4\,x}{3}-\frac {4}{3}\right )\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-12978+\sqrt {6}\,1281{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (12978+\sqrt {6}\,1281{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\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)^3*(x - 5))/(3*x^2 + 2)^(3/2),x)

[Out]

(134*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*((4*x)/3 - 4/3)*(x^2 + 2/3)^(1/2))/3 - (3^(1/2)*6^(1/2
)*(6^(1/2)*1281i - 12978)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*1281i
+ 12978)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {243 x}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {8 x^{4}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {135}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-243*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-126*x**2/(3*x**2*sqrt(3*x**2 +
2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-4*x**3/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral
(8*x**4/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-135/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3
*x**2 + 2)), x)

________________________________________________________________________________________