3.2508 \(\int \frac{(5-x) (3+2 x)^2}{(2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (2 x+3) (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}+\frac{184}{3} \sqrt{3 x^2+5 x+2}+2 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

(-2*(3 + 2*x)*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (184*Sqrt[2 + 5*x + 3*x^2])/3 + 2*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]

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Rubi [A]  time = 0.0388993, antiderivative size = 83, 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, 640, 621, 206} \[ -\frac{2 (2 x+3) (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}+\frac{184}{3} \sqrt{3 x^2+5 x+2}+2 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (184*Sqrt[2 + 5*x + 3*x^2])/3 + 2*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]

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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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)^2}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (3+2 x) (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{239+276 x}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x) (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{184}{3} \sqrt{2+5 x+3 x^2}+6 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x) (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{184}{3} \sqrt{2+5 x+3 x^2}+12 \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) (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{184}{3} \sqrt{2+5 x+3 x^2}+2 \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0299438, size = 68, normalized size = 0.82 \[ -\frac{4 x^2-6 \sqrt{9 x^2+15 x+6} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )+398 x+358}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(358 + 398*x + 4*x^2 - 6*Sqrt[6 + 15*x + 9*x^2]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/(3*Sqrt[2 + 5*
x + 3*x^2])

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Maple [A]  time = 0.005, size = 96, normalized size = 1.2 \begin{align*} -{\frac{4\,{x}^{2}}{3}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-6\,{\frac{x}{\sqrt{3\,{x}^{2}+5\,x+2}}}-{\frac{124}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{950+1140\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+2\,\ln \left ( 1/3\, \left ( 5/2+3\,x \right ) \sqrt{3}+\sqrt{3\,{x}^{2}+5\,x+2} \right ) \sqrt{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*x^2/(3*x^2+5*x+2)^(1/2)-6*x/(3*x^2+5*x+2)^(1/2)-124/9/(3*x^2+5*x+2)^(1/2)-190/9*(5+6*x)/(3*x^2+5*x+2)^(1/
2)+2*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 = 1.83455, size = 101, normalized size = 1.22 \begin{align*} -\frac{4 \, x^{2}}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + 2 \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{398 \, x}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{358}{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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^2/sqrt(3*x^2 + 5*x + 2) + 2*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 398/3*x/sqrt(3*x^2
+ 5*x + 2) - 358/3/sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.87636, size = 230, normalized size = 2.77 \begin{align*} \frac{3 \, \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 ) - 2 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x^{2} + 199 \, x + 179\right )}}{3 \,{\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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(3*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) - 2*sqrt
(3*x^2 + 5*x + 2)*(2*x^2 + 199*x + 179))/(3*x^2 + 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{51 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{8 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{45}{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)**2/(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-51*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(-8*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(-45/(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 = 1.09832, size = 78, normalized size = 0.94 \begin{align*} -2 \, \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 (2 \, x + 199\right )} x + 179\right )}}{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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*((2*x + 199)*x + 179)/sqrt(3*x^2
+ 5*x + 2)