3.2509 \(\int \frac{(5-x) (3+2 x)}{(2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0236256, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {777, 621, 206} \[ -\frac{2 (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), 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))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[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)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{2}{3} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{4}{3} \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 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}-\frac{2 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.101289, size = 53, normalized size = 0.85 \[ -\frac{2}{9} \left (\frac{417 x+363}{\sqrt{3 x^2+5 x+2}}+\sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((363 + 417*x)/Sqrt[2 + 5*x + 3*x^2] + Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]]))/9

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Maple [A]  time = 0.005, size = 79, normalized size = 1.3 \begin{align*}{\frac{2\,x}{3}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{26}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{700+840\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{2\,\sqrt{3}}{9}\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*x^2+5*x+2)^(3/2),x)

[Out]

2/3*x/(3*x^2+5*x+2)^(1/2)-26/9/(3*x^2+5*x+2)^(1/2)-140/9*(5+6*x)/(3*x^2+5*x+2)^(1/2)-2/9*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 = 1.76025, size = 78, normalized size = 1.26 \begin{align*} -\frac{2}{9} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{278 \, x}{3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{242}{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*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 278/3*x/sqrt(3*x^2 + 5*x + 2) - 242/3/sqrt(3*x^2
+ 5*x + 2)

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Fricas [A]  time = 1.84312, size = 217, normalized size = 3.5 \begin{align*} \frac{\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 ) - 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (139 \, x + 121\right )}}{9 \,{\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*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/9*(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) - 6*sqrt(
3*x^2 + 5*x + 2)*(139*x + 121))/(3*x^2 + 5*x + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{7 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{2 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{15}{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*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-7*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) - I
ntegral(2*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) - I
ntegral(-15/(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 = 1.1096, size = 73, normalized size = 1.18 \begin{align*} \frac{2}{9} \, \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 (139 \, x + 121\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)/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*(139*x + 121)/sqrt(3*x^2 + 5*x
+ 2)