3.1370 \(\int (5-x) (3+2 x)^2 (2+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac{2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac{397}{36} x \left (3 x^2+2\right )^{3/2}+\frac{397}{12} x \sqrt{3 x^2+2}+\frac{397 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(397*x*Sqrt[2 + 3*x^2])/12 + (397*x*(2 + 3*x^2)^(3/2))/36 - ((3 + 2*x)^2*(2 + 3*x^2)^(5/2))/21 + (2*(611 + 160
*x)*(2 + 3*x^2)^(5/2))/315 + (397*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.0364657, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac{2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac{397}{36} x \left (3 x^2+2\right )^{3/2}+\frac{397}{12} x \sqrt{3 x^2+2}+\frac{397 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(397*x*Sqrt[2 + 3*x^2])/12 + (397*x*(2 + 3*x^2)^(3/2))/36 - ((3 + 2*x)^2*(2 + 3*x^2)^(5/2))/21 + (2*(611 + 160
*x)*(2 + 3*x^2)^(5/2))/315 + (397*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)^2 \left (2+3 x^2\right )^{3/2} \, dx &=-\frac{1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac{1}{21} \int (3+2 x) (323+192 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=-\frac{1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac{2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac{397}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{397}{36} x \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac{2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac{397}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{397}{12} x \sqrt{2+3 x^2}+\frac{397}{36} x \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac{2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac{397}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{397}{12} x \sqrt{2+3 x^2}+\frac{397}{36} x \left (2+3 x^2\right )^{3/2}-\frac{1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac{2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac{397 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0559905, size = 65, normalized size = 0.69 \[ \frac{27790 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (2160 x^6-5040 x^5-36252 x^4-48405 x^3-51216 x^2-71715 x-17392\right )}{1260} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-17392 - 71715*x - 51216*x^2 - 48405*x^3 - 36252*x^4 - 5040*x^5 + 2160*x^6)) + 27790*Sqrt[
3]*ArcSinh[Sqrt[3/2]*x])/1260

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Maple [A]  time = 0.007, size = 75, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{2}}{21} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1087}{315} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{4\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{397\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{397\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{397\,\sqrt{3}}{18}{\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)^2*(3*x^2+2)^(3/2),x)

[Out]

-4/21*x^2*(3*x^2+2)^(5/2)+1087/315*(3*x^2+2)^(5/2)+4/9*x*(3*x^2+2)^(5/2)+397/36*x*(3*x^2+2)^(3/2)+397/12*x*(3*
x^2+2)^(1/2)+397/18*arcsinh(1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 1.48773, size = 100, normalized size = 1.06 \begin{align*} -\frac{4}{21} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{2} + \frac{4}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{1087}{315} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{397}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{397}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{397}{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)^2*(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-4/21*(3*x^2 + 2)^(5/2)*x^2 + 4/9*(3*x^2 + 2)^(5/2)*x + 1087/315*(3*x^2 + 2)^(5/2) + 397/36*(3*x^2 + 2)^(3/2)*
x + 397/12*sqrt(3*x^2 + 2)*x + 397/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.18167, size = 219, normalized size = 2.33 \begin{align*} -\frac{1}{1260} \,{\left (2160 \, x^{6} - 5040 \, x^{5} - 36252 \, x^{4} - 48405 \, x^{3} - 51216 \, x^{2} - 71715 \, x - 17392\right )} \sqrt{3 \, x^{2} + 2} + \frac{397}{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)^2*(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

-1/1260*(2160*x^6 - 5040*x^5 - 36252*x^4 - 48405*x^3 - 51216*x^2 - 71715*x - 17392)*sqrt(3*x^2 + 2) + 397/36*s
qrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 12.3309, size = 129, normalized size = 1.37 \begin{align*} - \frac{12 x^{6} \sqrt{3 x^{2} + 2}}{7} + 4 x^{5} \sqrt{3 x^{2} + 2} + \frac{1007 x^{4} \sqrt{3 x^{2} + 2}}{35} + \frac{461 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{4268 x^{2} \sqrt{3 x^{2} + 2}}{105} + \frac{683 x \sqrt{3 x^{2} + 2}}{12} + \frac{4348 \sqrt{3 x^{2} + 2}}{315} + \frac{397 \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)**2*(3*x**2+2)**(3/2),x)

[Out]

-12*x**6*sqrt(3*x**2 + 2)/7 + 4*x**5*sqrt(3*x**2 + 2) + 1007*x**4*sqrt(3*x**2 + 2)/35 + 461*x**3*sqrt(3*x**2 +
2)/12 + 4268*x**2*sqrt(3*x**2 + 2)/105 + 683*x*sqrt(3*x**2 + 2)/12 + 4348*sqrt(3*x**2 + 2)/315 + 397*sqrt(3)*
asinh(sqrt(6)*x/2)/18

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Giac [A]  time = 1.16967, size = 84, normalized size = 0.89 \begin{align*} -\frac{1}{1260} \,{\left (3 \,{\left ({\left ({\left (12 \,{\left (20 \,{\left (3 \, x - 7\right )} x - 1007\right )} x - 16135\right )} x - 17072\right )} x - 23905\right )} x - 17392\right )} \sqrt{3 \, x^{2} + 2} - \frac{397}{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)^2*(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-1/1260*(3*(((12*(20*(3*x - 7)*x - 1007)*x - 16135)*x - 17072)*x - 23905)*x - 17392)*sqrt(3*x^2 + 2) - 397/18*
sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))