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

Optimal. Leaf size=67 \[ -\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {3 x^2+2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]

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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, 778, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {3 x^2+2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^2)/(18*(2 + 3*x^2)^(3/2)) - (556 - 1461*x)/(54*Sqrt[2 + 3*x^2]) - (8*ArcSinh[Sqrt[3/2]
*x])/(9*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]

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 )^{5/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}+\frac {1}{18} \int \frac {(314-24 x) (3+2 x)}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8}{9} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 58, normalized size = 0.87 \begin {gather*} -\frac {-4971 x^3+72 x^2+16 \sqrt {3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-3741 x+1490}{54 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(1490 - 3741*x + 72*x^2 - 4971*x^3 + 16*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(2 + 3*x^2)^(3/2
)

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IntegrateAlgebraic [A]  time = 0.42, size = 61, normalized size = 0.91 \begin {gather*} \frac {8 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{9 \sqrt {3}}+\frac {4971 x^3-72 x^2+3741 x-1490}{54 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1490 + 3741*x - 72*x^2 + 4971*x^3)/(54*(2 + 3*x^2)^(3/2)) + (8*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(9*Sqrt[
3])

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fricas [A]  time = 0.42, size = 81, normalized size = 1.21 \begin {gather*} \frac {8 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (4971 \, x^{3} - 72 \, x^{2} + 3741 \, x - 1490\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\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)^(5/2),x, algorithm="fricas")

[Out]

1/54*(8*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + (4971*x^3 - 72*x^2 + 3741*x
- 1490)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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giac [A]  time = 0.20, size = 48, normalized size = 0.72 \begin {gather*} \frac {8}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left ({\left (1657 \, x - 24\right )} x + 1247\right )} x - 1490}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{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)^(5/2),x, algorithm="giac")

[Out]

8/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/54*(3*((1657*x - 24)*x + 1247)*x - 1490)/(3*x^2 + 2)^(3/2)

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maple [A]  time = 0.05, size = 77, normalized size = 1.15 \begin {gather*} \frac {8 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {4 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {547 x}{18 \sqrt {3 x^{2}+2}}+\frac {17 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{27}-\frac {745}{27 \left (3 x^{2}+2\right )^{\frac {3}{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)^(5/2),x)

[Out]

8/9/(3*x^2+2)^(3/2)*x^3+547/18/(3*x^2+2)^(1/2)*x-8/27*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-4/3/(3*x^2+2)^(3/2)*x^2-7
45/27/(3*x^2+2)^(3/2)+17/2/(3*x^2+2)^(3/2)*x

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maxima [A]  time = 1.18, size = 91, normalized size = 1.36 \begin {gather*} \frac {8}{27} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} - \frac {8}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1609 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {17 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {745}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{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)^(5/2),x, algorithm="maxima")

[Out]

8/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) - 8/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1609/54*x/sqrt(
3*x^2 + 2) - 4/3*x^2/(3*x^2 + 2)^(3/2) + 17/2*x/(3*x^2 + 2)^(3/2) - 745/27/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 1.70, size = 200, normalized size = 2.99 \begin {gather*} -\frac {8\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {427}{72}+\frac {\sqrt {6}\,721{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {427}{72}+\frac {\sqrt {6}\,721{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-96+\sqrt {6}\,2067{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (96+\sqrt {6}\,2067{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\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)^(5/2),x)

[Out]

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*721i)/48 + 427/48)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*721i)/72 +
427/72)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*721i)/48 - 427/48)/(x + (6
^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*721i)/72 - 427/72)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (8*3^(1/2)*asinh((
2^(1/2)*3^(1/2)*x)/2))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*2067i - 96)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2)*1i
)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*2067i + 96)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x + (6^(1/2)*1i)/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {243 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {8 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {135}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \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)**(5/2),x)

[Out]

-Integral(-243*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-126
*x**2/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-4*x**3/(9*x**4
*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(8*x**4/(9*x**4*sqrt(3*x**2 +
2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-135/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sq
rt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)

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