3.2507 \(\int \frac{(5-x) (3+2 x)^3}{(2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 (139 x+121) (2 x+3)^2}{3 \sqrt{3 x^2+5 x+2}}+\frac{2}{9} (554 x+1239) \sqrt{3 x^2+5 x+2}+\frac{247 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (2*(1239 + 554*x)*Sqrt[2 + 5*x + 3*x^2])/9 + (247*A
rcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(9*Sqrt[3])

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Rubi [A]  time = 0.0522933, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 779, 621, 206} \[ -\frac{2 (139 x+121) (2 x+3)^2}{3 \sqrt{3 x^2+5 x+2}}+\frac{2}{9} (554 x+1239) \sqrt{3 x^2+5 x+2}+\frac{247 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (2*(1239 + 554*x)*Sqrt[2 + 5*x + 3*x^2])/9 + (247*A
rcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(9*Sqrt[3])

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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 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 \frac{(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (3+2 x)^2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{(3+2 x) (481+554 x)}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{9} (1239+554 x) \sqrt{2+5 x+3 x^2}+\frac{247}{9} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{9} (1239+554 x) \sqrt{2+5 x+3 x^2}+\frac{494}{9} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{2 (3+2 x)^2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{9} (1239+554 x) \sqrt{2+5 x+3 x^2}+\frac{247 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{9 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0379062, size = 76, normalized size = 0.83 \[ -\frac{6 \left (6 x^3-31 x^2+806 x+789\right )-247 \sqrt{9 x^2+15 x+6} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{27 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*(789 + 806*x - 31*x^2 + 6*x^3) - 247*Sqrt[6 + 15*x + 9*x^2]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])
/(27*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.007, size = 113, normalized size = 1.2 \begin{align*} -{\frac{4\,{x}^{3}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{62\,{x}^{2}}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{247\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{881}{18}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{2275+2730\,x}{18}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{247\,\sqrt{3}}{27}\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)^(3/2),x)

[Out]

-4/3*x^3/(3*x^2+5*x+2)^(1/2)+62/9*x^2/(3*x^2+5*x+2)^(1/2)-247/9*x/(3*x^2+5*x+2)^(1/2)-881/18/(3*x^2+5*x+2)^(1/
2)-455/18*(5+6*x)/(3*x^2+5*x+2)^(1/2)+247/27*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 = 1.70909, size = 124, normalized size = 1.35 \begin{align*} -\frac{4 \, x^{3}}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{62 \, x^{2}}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{247}{27} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{1612 \, x}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{526}{3 \, \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)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^3/sqrt(3*x^2 + 5*x + 2) + 62/9*x^2/sqrt(3*x^2 + 5*x + 2) + 247/27*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*
x + 2) + 6*x + 5) - 1612/9*x/sqrt(3*x^2 + 5*x + 2) - 526/3/sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.83571, size = 247, normalized size = 2.68 \begin{align*} \frac{247 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 12 \,{\left (6 \, x^{3} - 31 \, x^{2} + 806 \, x + 789\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{54 \,{\left (3 \, x^{2} + 5 \, x + 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+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/54*(247*sqrt(3)*(3*x^2 + 5*x + 2)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 12*
(6*x^3 - 31*x^2 + 806*x + 789)*sqrt(3*x^2 + 5*x + 2))/(3*x^2 + 5*x + 2)

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

[Out]

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

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Giac [A]  time = 1.10305, size = 84, normalized size = 0.91 \begin{align*} -\frac{247}{27} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{2 \,{\left ({\left ({\left (6 \, x - 31\right )} x + 806\right )} x + 789\right )}}{9 \, \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)^(3/2),x, algorithm="giac")

[Out]

-247/27*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/9*(((6*x - 31)*x + 806)*x + 7
89)/sqrt(3*x^2 + 5*x + 2)