3.2516 \(\int \frac{(5-x) (3+2 x)^4}{(2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (7976 x+6809) (2 x+3)}{27 \sqrt{3 x^2+5 x+2}}-\frac{6848}{9} \sqrt{3 x^2+5 x+2}+\frac{152 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{27 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)*(6809 + 7976*x))/(27*Sqrt[2 + 5*x +
3*x^2]) - (6848*Sqrt[2 + 5*x + 3*x^2])/9 + (152*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(27*Sqrt
[3])

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Rubi [A]  time = 0.0571135, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 640, 621, 206} \[ -\frac{2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 (7976 x+6809) (2 x+3)}{27 \sqrt{3 x^2+5 x+2}}-\frac{6848}{9} \sqrt{3 x^2+5 x+2}+\frac{152 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)*(6809 + 7976*x))/(27*Sqrt[2 + 5*x +
3*x^2]) - (6848*Sqrt[2 + 5*x + 3*x^2])/9 + (152*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(27*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 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)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{(3+2 x)^2 (-117+272 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x) (6809+7976 x)}{27 \sqrt{2+5 x+3 x^2}}+\frac{4}{27} \int \frac{-12802-15408 x}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x) (6809+7976 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{6848}{9} \sqrt{2+5 x+3 x^2}+\frac{152}{27} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x) (6809+7976 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{6848}{9} \sqrt{2+5 x+3 x^2}+\frac{304}{27} \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)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (3+2 x) (6809+7976 x)}{27 \sqrt{2+5 x+3 x^2}}-\frac{6848}{9} \sqrt{2+5 x+3 x^2}+\frac{152 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{27 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0799541, size = 83, normalized size = 0.72 \[ \frac{2 \left (-216 x^4+176160 x^3+438540 x^2+76 \sqrt{3} \left (3 x^2+5 x+2\right )^{3/2} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )+354459 x+92457\right )}{81 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(2*(92457 + 354459*x + 438540*x^2 + 176160*x^3 - 216*x^4 + 76*Sqrt[3]*(2 + 5*x + 3*x^2)^(3/2)*ArcTanh[(5 + 6*x
)/(2*Sqrt[6 + 15*x + 9*x^2])]))/(81*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.014, size = 178, normalized size = 1.6 \begin{align*}{\frac{152\,\sqrt{3}}{81}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{295120+354144\,x}{243}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{80905+97086\,x}{1458} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{152\,{x}^{3}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{2380\,{x}^{2}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{14639\,x}{81} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{152\,x}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{16\,{x}^{4}}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{145763}{1458} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{380}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^4/(3*x^2+5*x+2)^(5/2),x)

[Out]

152/81*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+59024/243*(5+6*x)/(3*x^2+5*x+2)^(1/2)-16181/1458*
(5+6*x)/(3*x^2+5*x+2)^(3/2)-152/27*x^3/(3*x^2+5*x+2)^(3/2)-2380/27*x^2/(3*x^2+5*x+2)^(3/2)-14639/81*x/(3*x^2+5
*x+2)^(3/2)-152/27*x/(3*x^2+5*x+2)^(1/2)-16/3*x^4/(3*x^2+5*x+2)^(3/2)-145763/1458/(3*x^2+5*x+2)^(3/2)+380/81/(
3*x^2+5*x+2)^(1/2)

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Maxima [B]  time = 2.06319, size = 289, normalized size = 2.51 \begin{align*} -\frac{16 \, x^{4}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{152}{81} \, 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{152}{81} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{71440}{81} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{60704 \, x}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{920 \, x^{2}}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{15680}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13066 \, x}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{6766}{81 \,{\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)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-16/3*x^4/(3*x^2 + 5*x + 2)^(3/2) - 152/81*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 1
175/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)) + 152/81*sqrt(3)*log(2*
sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 71440/81*sqrt(3*x^2 + 5*x + 2) - 60704/81*x/sqrt(3*x^2 + 5*x + 2) -
920/9*x^2/(3*x^2 + 5*x + 2)^(3/2) - 15680/27/sqrt(3*x^2 + 5*x + 2) - 13066/81*x/(3*x^2 + 5*x + 2)^(3/2) - 676
6/81/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 1.94401, size = 325, normalized size = 2.83 \begin{align*} \frac{2 \,{\left (38 \, \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 (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{81 \,{\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)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2/81*(38*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*(72*x^4 - 58720*x^3 - 146180*x^2 - 118153*x - 30819)*sqrt(3*x^2 + 5*x + 2))/(9*x^4 + 30*x^3 +
37*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{999 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{864 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{264 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{16 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{16 x^{5}}{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{405}{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)**4/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-999*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(-864*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(-264*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(16*
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*s
qrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*x**5/(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(-405/(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)

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Giac [A]  time = 1.10438, size = 92, normalized size = 0.8 \begin{align*} -\frac{152}{81} \, \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 (4 \,{\left (2 \,{\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{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)^4/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-152/81*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/27*((4*(2*(9*x - 7340)*x - 36
545)*x - 118153)*x - 30819)/(3*x^2 + 5*x + 2)^(3/2)