3.1368 \(\int (5-x) (3+2 x)^4 (2+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac{13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac{(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac{2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac{2777}{12} x \sqrt{3 x^2+2}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(2777*x*Sqrt[2 + 3*x^2])/12 + (2777*x*(2 + 3*x^2)^(3/2))/36 + (4421*(3 + 2*x)^2*(2 + 3*x^2)^(5/2))/2268 + (13*
(3 + 2*x)^3*(2 + 3*x^2)^(5/2))/36 - ((3 + 2*x)^4*(2 + 3*x^2)^(5/2))/27 + ((661583 + 226755*x)*(2 + 3*x^2)^(5/2
))/17010 + (2777*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.0734341, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac{13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac{(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac{2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac{2777}{12} x \sqrt{3 x^2+2}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2777*x*Sqrt[2 + 3*x^2])/12 + (2777*x*(2 + 3*x^2)^(3/2))/36 + (4421*(3 + 2*x)^2*(2 + 3*x^2)^(5/2))/2268 + (13*
(3 + 2*x)^3*(2 + 3*x^2)^(5/2))/36 - ((3 + 2*x)^4*(2 + 3*x^2)^(5/2))/27 + ((661583 + 226755*x)*(2 + 3*x^2)^(5/2
))/17010 + (2777*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)^4 \left (2+3 x^2\right )^{3/2} \, dx &=-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{1}{27} \int (3+2 x)^3 (421+234 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{1}{648} \int (3+2 x)^2 (27504+26526 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{\int (3+2 x) (1520544+1632636 x) \left (2+3 x^2\right )^{3/2} \, dx}{13608}\\ &=\frac{4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac{2777}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac{2777}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{2777}{12} x \sqrt{2+3 x^2}+\frac{2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac{2777}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{2777}{12} x \sqrt{2+3 x^2}+\frac{2777}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4421 (3+2 x)^2 \left (2+3 x^2\right )^{5/2}}{2268}+\frac{13}{36} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^4 \left (2+3 x^2\right )^{5/2}+\frac{(661583+226755 x) \left (2+3 x^2\right )^{5/2}}{17010}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.081682, size = 75, normalized size = 0.54 \[ \frac{\sqrt{3 x^2+2} \left (-181440 x^8-204120 x^7+3676320 x^6+14492520 x^5+24490404 x^4+27468315 x^3+27537072 x^2+19683405 x+8598544\right )}{34020}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(8598544 + 19683405*x + 27537072*x^2 + 27468315*x^3 + 24490404*x^4 + 14492520*x^5 + 3676320*x
^6 - 204120*x^7 - 181440*x^8))/34020 + (2777*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Maple [A]  time = 0.013, size = 103, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{7256\,{x}^{2}}{567} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{537409}{8505} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{x}^{3}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{434\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{2777\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2777\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{2777\,\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)^4*(3*x^2+2)^(3/2),x)

[Out]

-16/27*x^4*(3*x^2+2)^(5/2)+7256/567*x^2*(3*x^2+2)^(5/2)+537409/8505*(3*x^2+2)^(5/2)-2/3*x^3*(3*x^2+2)^(5/2)+43
4/9*x*(3*x^2+2)^(5/2)+2777/36*x*(3*x^2+2)^(3/2)+2777/12*x*(3*x^2+2)^(1/2)+2777/18*arcsinh(1/2*x*6^(1/2))*3^(1/
2)

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Maxima [A]  time = 1.50713, size = 138, normalized size = 1. \begin{align*} -\frac{16}{27} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{4} - \frac{2}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{3} + \frac{7256}{567} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{2} + \frac{434}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{537409}{8505} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{2777}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{2777}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{2777}{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)^4*(3*x^2+2)^(3/2),x, algorithm="maxima")

[Out]

-16/27*(3*x^2 + 2)^(5/2)*x^4 - 2/3*(3*x^2 + 2)^(5/2)*x^3 + 7256/567*(3*x^2 + 2)^(5/2)*x^2 + 434/9*(3*x^2 + 2)^
(5/2)*x + 537409/8505*(3*x^2 + 2)^(5/2) + 2777/36*(3*x^2 + 2)^(3/2)*x + 2777/12*sqrt(3*x^2 + 2)*x + 2777/18*sq
rt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.2066, size = 285, normalized size = 2.07 \begin{align*} -\frac{1}{34020} \,{\left (181440 \, x^{8} + 204120 \, x^{7} - 3676320 \, x^{6} - 14492520 \, x^{5} - 24490404 \, x^{4} - 27468315 \, x^{3} - 27537072 \, x^{2} - 19683405 \, x - 8598544\right )} \sqrt{3 \, x^{2} + 2} + \frac{2777}{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)^4*(3*x^2+2)^(3/2),x, algorithm="fricas")

[Out]

-1/34020*(181440*x^8 + 204120*x^7 - 3676320*x^6 - 14492520*x^5 - 24490404*x^4 - 27468315*x^3 - 27537072*x^2 -
19683405*x - 8598544)*sqrt(3*x^2 + 2) + 2777/36*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 36.2683, size = 162, normalized size = 1.17 \begin{align*} - \frac{16 x^{8} \sqrt{3 x^{2} + 2}}{3} - 6 x^{7} \sqrt{3 x^{2} + 2} + \frac{6808 x^{6} \sqrt{3 x^{2} + 2}}{63} + 426 x^{5} \sqrt{3 x^{2} + 2} + \frac{226763 x^{4} \sqrt{3 x^{2} + 2}}{315} + \frac{9689 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{2294756 x^{2} \sqrt{3 x^{2} + 2}}{2835} + \frac{6943 x \sqrt{3 x^{2} + 2}}{12} + \frac{2149636 \sqrt{3 x^{2} + 2}}{8505} + \frac{2777 \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)**4*(3*x**2+2)**(3/2),x)

[Out]

-16*x**8*sqrt(3*x**2 + 2)/3 - 6*x**7*sqrt(3*x**2 + 2) + 6808*x**6*sqrt(3*x**2 + 2)/63 + 426*x**5*sqrt(3*x**2 +
2) + 226763*x**4*sqrt(3*x**2 + 2)/315 + 9689*x**3*sqrt(3*x**2 + 2)/12 + 2294756*x**2*sqrt(3*x**2 + 2)/2835 +
6943*x*sqrt(3*x**2 + 2)/12 + 2149636*sqrt(3*x**2 + 2)/8505 + 2777*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Giac [A]  time = 1.15886, size = 97, normalized size = 0.7 \begin{align*} -\frac{1}{34020} \,{\left (3 \,{\left ({\left (9 \,{\left (4 \,{\left (10 \,{\left ({\left (21 \,{\left (8 \, x + 9\right )} x - 3404\right )} x - 13419\right )} x - 226763\right )} x - 1017345\right )} x - 9179024\right )} x - 6561135\right )} x - 8598544\right )} \sqrt{3 \, x^{2} + 2} - \frac{2777}{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)^4*(3*x^2+2)^(3/2),x, algorithm="giac")

[Out]

-1/34020*(3*((9*(4*(10*((21*(8*x + 9)*x - 3404)*x - 13419)*x - 226763)*x - 1017345)*x - 9179024)*x - 6561135)*
x - 8598544)*sqrt(3*x^2 + 2) - 2777/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))