3.1369 \(\int (5-x) (3+2 x)^3 (2+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{24} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{71}{168} \left (3 x^2+2\right )^{5/2} (2 x+3)^2+\frac{(5405 x+16973) \left (3 x^2+2\right )^{5/2}}{1260}+\frac{1087}{36} x \left (3 x^2+2\right )^{3/2}+\frac{1087}{12} x \sqrt{3 x^2+2}+\frac{1087 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(1087*x*Sqrt[2 + 3*x^2])/12 + (1087*x*(2 + 3*x^2)^(3/2))/36 + (71*(3 + 2*x)^2*(2 + 3*x^2)^(5/2))/168 - ((3 + 2
*x)^3*(2 + 3*x^2)^(5/2))/24 + ((16973 + 5405*x)*(2 + 3*x^2)^(5/2))/1260 + (1087*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[
3])

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Rubi [A]  time = 0.0535518, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{24} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{71}{168} \left (3 x^2+2\right )^{5/2} (2 x+3)^2+\frac{(5405 x+16973) \left (3 x^2+2\right )^{5/2}}{1260}+\frac{1087}{36} x \left (3 x^2+2\right )^{3/2}+\frac{1087}{12} x \sqrt{3 x^2+2}+\frac{1087 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1087*x*Sqrt[2 + 3*x^2])/12 + (1087*x*(2 + 3*x^2)^(3/2))/36 + (71*(3 + 2*x)^2*(2 + 3*x^2)^(5/2))/168 - ((3 + 2
*x)^3*(2 + 3*x^2)^(5/2))/24 + ((16973 + 5405*x)*(2 + 3*x^2)^(5/2))/1260 + (1087*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[
3])

Rule 833

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

Rubi steps

\begin{align*} \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx &=-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{1}{24} \int (3+2 x)^2 (372+213 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{1}{504} \int (3+2 x) (21732+19458 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac{1087}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac{71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac{1087}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{1087}{12} x \sqrt{2+3 x^2}+\frac{1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac{71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac{1087}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{1087}{12} x \sqrt{2+3 x^2}+\frac{1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac{71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac{1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac{(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac{1087 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0701494, size = 70, normalized size = 0.6 \[ \frac{76090 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (3780 x^7-2160 x^6-75600 x^5-186012 x^4-219975 x^3-245136 x^2-226065 x-81392\right )}{1260} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-81392 - 226065*x - 245136*x^2 - 219975*x^3 - 186012*x^4 - 75600*x^5 - 2160*x^6 + 3780*x^7
)) + 76090*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/1260

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Maple [A]  time = 0.006, size = 89, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{64\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1087\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1087\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{1087\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{5087}{315} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^3*(3*x^2+2)^(5/2)+64/9*x*(3*x^2+2)^(5/2)+1087/36*x*(3*x^2+2)^(3/2)+1087/12*x*(3*x^2+2)^(1/2)+1087/18*ar
csinh(1/2*x*6^(1/2))*3^(1/2)+4/21*x^2*(3*x^2+2)^(5/2)+5087/315*(3*x^2+2)^(5/2)

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Maxima [A]  time = 1.51074, size = 119, normalized size = 1.03 \begin{align*} -\frac{1}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{3} + \frac{4}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{2} + \frac{64}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{5087}{315} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{1087}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{1087}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{1087}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}

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]

-1/3*(3*x^2 + 2)^(5/2)*x^3 + 4/21*(3*x^2 + 2)^(5/2)*x^2 + 64/9*(3*x^2 + 2)^(5/2)*x + 5087/315*(3*x^2 + 2)^(5/2
) + 1087/36*(3*x^2 + 2)^(3/2)*x + 1087/12*sqrt(3*x^2 + 2)*x + 1087/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.12999, size = 242, normalized size = 2.09 \begin{align*} -\frac{1}{1260} \,{\left (3780 \, x^{7} - 2160 \, x^{6} - 75600 \, x^{5} - 186012 \, x^{4} - 219975 \, x^{3} - 245136 \, x^{2} - 226065 \, x - 81392\right )} \sqrt{3 \, x^{2} + 2} + \frac{1087}{36} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end{align*}

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]

-1/1260*(3780*x^7 - 2160*x^6 - 75600*x^5 - 186012*x^4 - 219975*x^3 - 245136*x^2 - 226065*x - 81392)*sqrt(3*x^2
+ 2) + 1087/36*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 20.3507, size = 144, normalized size = 1.24 \begin{align*} - 3 x^{7} \sqrt{3 x^{2} + 2} + \frac{12 x^{6} \sqrt{3 x^{2} + 2}}{7} + 60 x^{5} \sqrt{3 x^{2} + 2} + \frac{5167 x^{4} \sqrt{3 x^{2} + 2}}{35} + \frac{2095 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{20428 x^{2} \sqrt{3 x^{2} + 2}}{105} + \frac{2153 x \sqrt{3 x^{2} + 2}}{12} + \frac{20348 \sqrt{3 x^{2} + 2}}{315} + \frac{1087 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \end{align*}

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]

-3*x**7*sqrt(3*x**2 + 2) + 12*x**6*sqrt(3*x**2 + 2)/7 + 60*x**5*sqrt(3*x**2 + 2) + 5167*x**4*sqrt(3*x**2 + 2)/
35 + 2095*x**3*sqrt(3*x**2 + 2)/12 + 20428*x**2*sqrt(3*x**2 + 2)/105 + 2153*x*sqrt(3*x**2 + 2)/12 + 20348*sqrt
(3*x**2 + 2)/315 + 1087*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Giac [A]  time = 1.1514, size = 89, normalized size = 0.77 \begin{align*} -\frac{1}{1260} \,{\left (3 \,{\left ({\left ({\left (12 \,{\left (15 \,{\left ({\left (7 \, x - 4\right )} x - 140\right )} x - 5167\right )} x - 73325\right )} x - 81712\right )} x - 75355\right )} x - 81392\right )} \sqrt{3 \, x^{2} + 2} - \frac{1087}{18} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \end{align*}

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]

-1/1260*(3*(((12*(15*((7*x - 4)*x - 140)*x - 5167)*x - 73325)*x - 81712)*x - 75355)*x - 81392)*sqrt(3*x^2 + 2)
- 1087/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))