3.2407 \(\int (5-x) (3+2 x)^3 \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=135 \[ -\frac{1}{18} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac{11}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}}{3240}+\frac{6221 (6 x+5) \sqrt{3 x^2+5 x+2}}{5184}-\frac{6221 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10368 \sqrt{3}} \]

[Out]

(6221*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5184 + (11*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/15 - ((3 + 2*x)^3*(2 +
5*x + 3*x^2)^(3/2))/18 + ((27487 + 11538*x)*(2 + 5*x + 3*x^2)^(3/2))/3240 - (6221*ArcTanh[(5 + 6*x)/(2*Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])])/(10368*Sqrt[3])

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Rubi [A]  time = 0.0736717, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \[ -\frac{1}{18} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^3+\frac{11}{15} \left (3 x^2+5 x+2\right )^{3/2} (2 x+3)^2+\frac{(11538 x+27487) \left (3 x^2+5 x+2\right )^{3/2}}{3240}+\frac{6221 (6 x+5) \sqrt{3 x^2+5 x+2}}{5184}-\frac{6221 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10368 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6221*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5184 + (11*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(3/2))/15 - ((3 + 2*x)^3*(2 +
5*x + 3*x^2)^(3/2))/18 + ((27487 + 11538*x)*(2 + 5*x + 3*x^2)^(3/2))/3240 - (6221*ArcTanh[(5 + 6*x)/(2*Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])])/(10368*Sqrt[3])

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + 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 + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + 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 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (5-x) (3+2 x)^3 \sqrt{2+5 x+3 x^2} \, dx &=-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{1}{18} \int (3+2 x)^2 \left (\frac{609}{2}+198 x\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{1}{270} \int (3+2 x) \left (\frac{15327}{2}+5769 x\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}+\frac{6221}{432} \int \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{6221 (5+6 x) \sqrt{2+5 x+3 x^2}}{5184}+\frac{11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac{6221 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{10368}\\ &=\frac{6221 (5+6 x) \sqrt{2+5 x+3 x^2}}{5184}+\frac{11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac{6221 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{5184}\\ &=\frac{6221 (5+6 x) \sqrt{2+5 x+3 x^2}}{5184}+\frac{11}{15} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}-\frac{1}{18} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}+\frac{(27487+11538 x) \left (2+5 x+3 x^2\right )^{3/2}}{3240}-\frac{6221 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{10368 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0453462, size = 77, normalized size = 0.57 \[ \frac{-6 \sqrt{3 x^2+5 x+2} \left (34560 x^5-14976 x^4-825840 x^3-2317848 x^2-2432350 x-859701\right )-31105 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{155520} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-859701 - 2432350*x - 2317848*x^2 - 825840*x^3 - 14976*x^4 + 34560*x^5) - 31105*Sqr
t[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/155520

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Maple [A]  time = 0.006, size = 113, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{3}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{14\,{x}^{2}}{15} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{337\,x}{36} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{44011}{3240} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{31105+37326\,x}{5184}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{6221\,\sqrt{3}}{31104}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*x^3*(3*x^2+5*x+2)^(3/2)+14/15*x^2*(3*x^2+5*x+2)^(3/2)+337/36*x*(3*x^2+5*x+2)^(3/2)+44011/3240*(3*x^2+5*x+
2)^(3/2)+6221/5184*(5+6*x)*(3*x^2+5*x+2)^(1/2)-6221/31104*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2
)

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Maxima [A]  time = 1.50976, size = 163, normalized size = 1.21 \begin{align*} -\frac{4}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{3} + \frac{14}{15} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{337}{36} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{44011}{3240} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{6221}{864} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{6221}{31104} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{31105}{5184} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-4/9*(3*x^2 + 5*x + 2)^(3/2)*x^3 + 14/15*(3*x^2 + 5*x + 2)^(3/2)*x^2 + 337/36*(3*x^2 + 5*x + 2)^(3/2)*x + 4401
1/3240*(3*x^2 + 5*x + 2)^(3/2) + 6221/864*sqrt(3*x^2 + 5*x + 2)*x - 6221/31104*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^
2 + 5*x + 2) + 6*x + 5) + 31105/5184*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.31439, size = 263, normalized size = 1.95 \begin{align*} -\frac{1}{25920} \,{\left (34560 \, x^{5} - 14976 \, x^{4} - 825840 \, x^{3} - 2317848 \, x^{2} - 2432350 \, x - 859701\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{6221}{62208} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

-1/25920*(34560*x^5 - 14976*x^4 - 825840*x^3 - 2317848*x^2 - 2432350*x - 859701)*sqrt(3*x^2 + 5*x + 2) + 6221/
62208*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 243 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 126 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 4 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 8 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 135 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-243*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-126*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-4*x**3
*sqrt(3*x**2 + 5*x + 2), x) - Integral(8*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-135*sqrt(3*x**2 + 5*x + 2
), x)

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Giac [A]  time = 1.17293, size = 100, normalized size = 0.74 \begin{align*} -\frac{1}{25920} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (30 \, x - 13\right )} x - 5735\right )} x - 96577\right )} x - 1216175\right )} x - 859701\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{6221}{31104} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^3*(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/25920*(2*(12*(6*(8*(30*x - 13)*x - 5735)*x - 96577)*x - 1216175)*x - 859701)*sqrt(3*x^2 + 5*x + 2) + 6221/3
1104*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))