3.2517 \(\int \frac{(5-x) (3+2 x)^3}{(2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2834 x+2481)}{9 \sqrt{3 x^2+5 x+2}}-\frac{8 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(2481 + 2834*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (
8*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(9*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0420306, 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, 777, 621, 206} \[ -\frac{2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (2834 x+2481)}{9 \sqrt{3 x^2+5 x+2}}-\frac{8 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)^2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(2481 + 2834*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (
8*ArcTanh[(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 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)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(-359-6 x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2481+2834 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{8}{9} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2481+2834 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{16}{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)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2481+2834 x)}{9 \sqrt{2+5 x+3 x^2}}-\frac{8 \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 = 0.0625904, size = 67, normalized size = 0.73 \[ \frac{2 \left (16448 x^3+41074 x^2+33443 x+8835\right )}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{8 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8835 + 33443*x + 41074*x^2 + 16448*x^3))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (8*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15
*x + 9*x^2])])/(9*Sqrt[3])

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 161, normalized size = 1.8 \begin{align*}{\frac{8\,{x}^{3}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{32\,{x}^{2}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{607\,x}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{10855}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{20165+24198\,x}{486} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{82160+98592\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{8\,x}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{20}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{8\,\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)^(5/2),x)

[Out]

8/9*x^3/(3*x^2+5*x+2)^(3/2)-32/9*x^2/(3*x^2+5*x+2)^(3/2)-607/27*x/(3*x^2+5*x+2)^(3/2)-10855/486/(3*x^2+5*x+2)^
(3/2)-4033/486*(5+6*x)/(3*x^2+5*x+2)^(3/2)+16432/81*(5+6*x)/(3*x^2+5*x+2)^(1/2)+8/9*x/(3*x^2+5*x+2)^(1/2)-20/2
7/(3*x^2+5*x+2)^(1/2)-8/27*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.72173, size = 266, normalized size = 2.89 \begin{align*} \frac{8}{27} \, x{\left (\frac{1410 \, x}{\sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{1175}{\sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\right )} - \frac{8}{27} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{3760}{27} \, \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{42272 \, x}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{11680}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{2318 \, x}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{2030}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="maxima")

[Out]

8/27*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 1175/sqrt(3*x^2 + 5*x + 2) - 55*x/(3*x^
2 + 5*x + 2)^(3/2) - 46/(3*x^2 + 5*x + 2)^(3/2)) - 8/27*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5)
- 3760/27*sqrt(3*x^2 + 5*x + 2) + 42272/27*x/sqrt(3*x^2 + 5*x + 2) - 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2) + 11680/
9/sqrt(3*x^2 + 5*x + 2) - 2318/27*x/(3*x^2 + 5*x + 2)^(3/2) - 2030/27/(3*x^2 + 5*x + 2)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.84992, size = 309, normalized size = 3.36 \begin{align*} \frac{2 \,{\left (2 \, \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\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 ) + 3 \,{\left (16448 \, x^{3} + 41074 \, x^{2} + 33443 \, x + 8835\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{27 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\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)^(5/2),x, algorithm="fricas")

[Out]

2/27*(2*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 +
120*x + 49) + 3*(16448*x^3 + 41074*x^2 + 33443*x + 8835)*sqrt(3*x^2 + 5*x + 2))/(9*x^4 + 30*x^3 + 37*x^2 + 20
*x + 4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{243 x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{126 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{4 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{8 x^{4}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{135}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \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)**(5/2),x)

[Out]

-Integral(-243*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x +
2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-126*x**2/(9*x**4*sqrt(3*x**2 + 5
*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sq
rt(3*x**2 + 5*x + 2)), x) - Integral(-4*x**3/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) +
37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(8*x**
4/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt
(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-135/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sq
rt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)
), x)

________________________________________________________________________________________

Giac [A]  time = 1.0923, size = 85, normalized size = 0.92 \begin{align*} \frac{8}{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 (2 \,{\left (8224 \, x + 20537\right )} x + 33443\right )} x + 8835\right )}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

8/27*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 2/9*((2*(8224*x + 20537)*x + 33443
)*x + 8835)/(3*x^2 + 5*x + 2)^(3/2)