3.1419 \(\int \frac{(5-x) (3+2 x)^5}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac{5 (16-421 x) (2 x+3)^2}{54 \sqrt{3 x^2+2}}-\frac{50}{81} (93 x+299) \sqrt{3 x^2+2}+\frac{1600 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) - (5*(16 - 421*x)*(3 + 2*x)^2)/(54*Sqrt[2 + 3*x^2]) - (50*(2
99 + 93*x)*Sqrt[2 + 3*x^2])/81 + (1600*ArcSinh[Sqrt[3/2]*x])/(27*Sqrt[3])

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Rubi [A]  time = 0.0437378, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {819, 780, 215} \[ -\frac{7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac{5 (16-421 x) (2 x+3)^2}{54 \sqrt{3 x^2+2}}-\frac{50}{81} (93 x+299) \sqrt{3 x^2+2}+\frac{1600 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) - (5*(16 - 421*x)*(3 + 2*x)^2)/(54*Sqrt[2 + 3*x^2]) - (50*(2
99 + 93*x)*Sqrt[2 + 3*x^2])/81 + (1600*ArcSinh[Sqrt[3/2]*x])/(27*Sqrt[3])

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])

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 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 \frac{(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}+\frac{1}{18} \int \frac{(370-220 x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac{5 (16-421 x) (3+2 x)^2}{54 \sqrt{2+3 x^2}}+\frac{1}{108} \int \frac{(-2000-18600 x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac{5 (16-421 x) (3+2 x)^2}{54 \sqrt{2+3 x^2}}-\frac{50}{81} (299+93 x) \sqrt{2+3 x^2}+\frac{1600}{27} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac{5 (16-421 x) (3+2 x)^2}{54 \sqrt{2+3 x^2}}-\frac{50}{81} (299+93 x) \sqrt{2+3 x^2}+\frac{1600 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0698545, size = 68, normalized size = 0.72 \[ -\frac{864 x^5+4320 x^4-183945 x^3+147600 x^2-3200 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-79215 x+134126}{162 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(134126 - 79215*x + 147600*x^2 - 183945*x^3 + 4320*x^4 + 864*x^5 - 3200*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqr
t[3/2]*x])/(162*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 105, normalized size = 1.1 \begin{align*} -{\frac{16\,{x}^{5}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{1600\,{x}^{3}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{21505\,x}{54}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{1600\,\sqrt{3}}{81}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{80\,{x}^{4}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{8200\,{x}^{2}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{67063}{81} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{615\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/3*x^5/(3*x^2+2)^(3/2)-1600/27*x^3/(3*x^2+2)^(3/2)+21505/54*x/(3*x^2+2)^(1/2)+1600/81*arcsinh(1/2*x*6^(1/2)
)*3^(1/2)-80/3*x^4/(3*x^2+2)^(3/2)-8200/9*x^2/(3*x^2+2)^(3/2)-67063/81/(3*x^2+2)^(3/2)-615/2*x/(3*x^2+2)^(3/2)

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Maxima [A]  time = 1.48205, size = 161, normalized size = 1.71 \begin{align*} -\frac{16 \, x^{5}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{80 \, x^{4}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{1600}{81} \, 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{1600}{81} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{70915 \, x}{162 \, \sqrt{3 \, x^{2} + 2}} - \frac{8200 \, x^{2}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{615 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{67063}{81 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-16/3*x^5/(3*x^2 + 2)^(3/2) - 80/3*x^4/(3*x^2 + 2)^(3/2) - 1600/81*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^
(3/2)) + 1600/81*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 70915/162*x/sqrt(3*x^2 + 2) - 8200/9*x^2/(3*x^2 + 2)^(3/2) -
615/2*x/(3*x^2 + 2)^(3/2) - 67063/81/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.57608, size = 259, normalized size = 2.76 \begin{align*} \frac{1600 \, \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 (864 \, x^{5} + 4320 \, x^{4} - 183945 \, x^{3} + 147600 \, x^{2} - 79215 \, x + 134126\right )} \sqrt{3 \, x^{2} + 2}}{162 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/162*(1600*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (864*x^5 + 4320*x^4 - 1
83945*x^3 + 147600*x^2 - 79215*x + 134126)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15745, size = 74, normalized size = 0.79 \begin{align*} -\frac{1600}{81} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left ({\left ({\left (288 \,{\left (x + 5\right )} x - 61315\right )} x + 49200\right )} x - 26405\right )} x + 134126}{162 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-1600/81*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(((288*(x + 5)*x - 61315)*x + 49200)*x - 26405)*
x + 134126)/(3*x^2 + 2)^(3/2)