3.1356 \(\int (5-x) (3+2 x)^4 \sqrt{2+3 x^2} \, dx\)

Optimal. Leaf size=122 \[ -\frac{1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac{29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac{2341}{18} x \sqrt{3 x^2+2}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(2341*x*Sqrt[2 + 3*x^2])/18 + (923*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/315 + (29*(3 + 2*x)^3*(2 + 3*x^2)^(3/2))/63
- ((3 + 2*x)^4*(2 + 3*x^2)^(3/2))/21 + (2*(13781 + 4599*x)*(2 + 3*x^2)^(3/2))/405 + (2341*ArcSinh[Sqrt[3/2]*x]
)/(9*Sqrt[3])

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Rubi [A]  time = 0.0693347, antiderivative size = 122, 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} \[ -\frac{1}{21} \left (3 x^2+2\right )^{3/2} (2 x+3)^4+\frac{29}{63} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac{923}{315} \left (3 x^2+2\right )^{3/2} (2 x+3)^2+\frac{2}{405} (4599 x+13781) \left (3 x^2+2\right )^{3/2}+\frac{2341}{18} x \sqrt{3 x^2+2}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 3*x^2],x]

[Out]

(2341*x*Sqrt[2 + 3*x^2])/18 + (923*(3 + 2*x)^2*(2 + 3*x^2)^(3/2))/315 + (29*(3 + 2*x)^3*(2 + 3*x^2)^(3/2))/63
- ((3 + 2*x)^4*(2 + 3*x^2)^(3/2))/21 + (2*(13781 + 4599*x)*(2 + 3*x^2)^(3/2))/405 + (2341*ArcSinh[Sqrt[3/2]*x]
)/(9*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)^4 \sqrt{2+3 x^2} \, dx &=-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{1}{21} \int (3+2 x)^3 (331+174 x) \sqrt{2+3 x^2} \, dx\\ &=\frac{29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{1}{378} \int (3+2 x)^2 (15786+16614 x) \sqrt{2+3 x^2} \, dx\\ &=\frac{923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{\int (3+2 x) (577458+772632 x) \sqrt{2+3 x^2} \, dx}{5670}\\ &=\frac{923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac{2341}{9} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{2341}{18} x \sqrt{2+3 x^2}+\frac{923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac{2341}{9} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2341}{18} x \sqrt{2+3 x^2}+\frac{923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac{29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac{2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0792069, size = 65, normalized size = 0.53 \[ \frac{\sqrt{3 x^2+2} \left (-12960 x^6-15120 x^5+297648 x^4+1222200 x^3+1956174 x^2+1558935 x+1167988\right )}{5670}+\frac{2341 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^4*Sqrt[2 + 3*x^2],x]

[Out]

(Sqrt[2 + 3*x^2]*(1167988 + 1558935*x + 1956174*x^2 + 1222200*x^3 + 297648*x^4 - 15120*x^5 - 12960*x^6))/5670
+ (2341*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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Maple [A]  time = 0.013, size = 91, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{5672\,{x}^{2}}{315} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{291997}{2835} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{652\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2341\,x}{18}\sqrt{3\,{x}^{2}+2}}+{\frac{2341\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^4*(3*x^2+2)^(1/2),x)

[Out]

-16/21*x^4*(3*x^2+2)^(3/2)+5672/315*x^2*(3*x^2+2)^(3/2)+291997/2835*(3*x^2+2)^(3/2)-8/9*x^3*(3*x^2+2)^(3/2)+65
2/9*x*(3*x^2+2)^(3/2)+2341/18*x*(3*x^2+2)^(1/2)+2341/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 1.47699, size = 122, normalized size = 1. \begin{align*} -\frac{16}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{4} - \frac{8}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{3} + \frac{5672}{315} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{652}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{291997}{2835} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{2341}{18} \, \sqrt{3 \, x^{2} + 2} x + \frac{2341}{27} \, \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)^4*(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-16/21*(3*x^2 + 2)^(3/2)*x^4 - 8/9*(3*x^2 + 2)^(3/2)*x^3 + 5672/315*(3*x^2 + 2)^(3/2)*x^2 + 652/9*(3*x^2 + 2)^
(3/2)*x + 291997/2835*(3*x^2 + 2)^(3/2) + 2341/18*sqrt(3*x^2 + 2)*x + 2341/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.63265, size = 235, normalized size = 1.93 \begin{align*} -\frac{1}{5670} \,{\left (12960 \, x^{6} + 15120 \, x^{5} - 297648 \, x^{4} - 1222200 \, x^{3} - 1956174 \, x^{2} - 1558935 \, x - 1167988\right )} \sqrt{3 \, x^{2} + 2} + \frac{2341}{54} \, \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)^4*(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/5670*(12960*x^6 + 15120*x^5 - 297648*x^4 - 1222200*x^3 - 1956174*x^2 - 1558935*x - 1167988)*sqrt(3*x^2 + 2)
+ 2341/54*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 6.44113, size = 131, normalized size = 1.07 \begin{align*} - \frac{16 x^{6} \sqrt{3 x^{2} + 2}}{7} - \frac{8 x^{5} \sqrt{3 x^{2} + 2}}{3} + \frac{5512 x^{4} \sqrt{3 x^{2} + 2}}{105} + \frac{1940 x^{3} \sqrt{3 x^{2} + 2}}{9} + \frac{326029 x^{2} \sqrt{3 x^{2} + 2}}{945} + \frac{4949 x \sqrt{3 x^{2} + 2}}{18} + \frac{583994 \sqrt{3 x^{2} + 2}}{2835} + \frac{2341 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**4*(3*x**2+2)**(1/2),x)

[Out]

-16*x**6*sqrt(3*x**2 + 2)/7 - 8*x**5*sqrt(3*x**2 + 2)/3 + 5512*x**4*sqrt(3*x**2 + 2)/105 + 1940*x**3*sqrt(3*x*
*2 + 2)/9 + 326029*x**2*sqrt(3*x**2 + 2)/945 + 4949*x*sqrt(3*x**2 + 2)/18 + 583994*sqrt(3*x**2 + 2)/2835 + 234
1*sqrt(3)*asinh(sqrt(6)*x/2)/27

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Giac [A]  time = 1.11454, size = 86, normalized size = 0.7 \begin{align*} -\frac{1}{5670} \,{\left (3 \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (5 \,{\left (6 \, x + 7\right )} x - 689\right )} x - 16975\right )} x - 326029\right )} x - 519645\right )} x - 1167988\right )} \sqrt{3 \, x^{2} + 2} - \frac{2341}{27} \, \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)^4*(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

-1/5670*(3*(2*(12*(6*(5*(6*x + 7)*x - 689)*x - 16975)*x - 326029)*x - 519645)*x - 1167988)*sqrt(3*x^2 + 2) - 2
341/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))