3.2413 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{\sqrt{3 x^2+5 x+2} (124 x+121)}{40 (2 x+3)^2}-\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{27 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{80 \sqrt{5}} \]

[Out]

((121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])])/8 + (27*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(80*Sqrt[5])

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Rubi [A]  time = 0.060585, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {810, 843, 621, 206, 724} \[ \frac{\sqrt{3 x^2+5 x+2} (124 x+121)}{40 (2 x+3)^2}-\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{27 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{80 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])])/8 + (27*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(80*Sqrt[5])

Rule 810

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*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx &=\frac{(121+124 x) \sqrt{2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac{1}{80} \int \frac{63+60 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{(121+124 x) \sqrt{2+5 x+3 x^2}}{40 (3+2 x)^2}+\frac{27}{80} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx-\frac{3}{8} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{(121+124 x) \sqrt{2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac{27}{40} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{(121+124 x) \sqrt{2+5 x+3 x^2}}{40 (3+2 x)^2}-\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )+\frac{27 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{80 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0850947, size = 100, normalized size = 0.93 \[ \frac{1}{400} \left (\frac{10 \sqrt{3 x^2+5 x+2} (124 x+121)}{(2 x+3)^2}-27 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-50 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((10*(121 + 124*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 - 27*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x
+ 3*x^2])] - 50*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/400

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Maple [A]  time = 0.01, size = 142, normalized size = 1.3 \begin{align*} -{\frac{13}{40} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{21}{50} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{27}{400}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{27\,\sqrt{5}}{400}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{105+126\,x}{100}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{\sqrt{3}}{8}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3,x)

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(3/2)-21/50/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+27/400*(12*(x+3/2)^2-
16*x-19)^(1/2)-27/400*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+21/100*(5+6*x)*(3*(
x+3/2)^2-4*x-19/4)^(1/2)-1/8*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.55096, size = 177, normalized size = 1.65 \begin{align*} -\frac{1}{8} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{27}{400} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{39}{40} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{21 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{20 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3,x, algorithm="maxima")

[Out]

-1/8*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 27/400*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)
/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 39/40*sqrt(3*x^2 + 5*x + 2) - 13/10*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 1
2*x + 9) - 21/20*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 1.32364, size = 394, normalized size = 3.68 \begin{align*} \frac{50 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\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 ) + 27 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (124 \, x + 121\right )}}{800 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3,x, algorithm="fricas")

[Out]

1/800*(50*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 2
7*sqrt(5)*(4*x^2 + 12*x + 9)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 1
2*x + 9)) + 20*sqrt(3*x^2 + 5*x + 2)*(124*x + 121))/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**3,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(8*
x**3 + 36*x**2 + 54*x + 27), x)

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Giac [B]  time = 1.22945, size = 324, normalized size = 3.03 \begin{align*} \frac{27}{400} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{1}{8} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{886 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 2897 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 9039 \, \sqrt{3} x + 3037 \, \sqrt{3} - 9039 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{40 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3,x, algorithm="giac")

[Out]

27/400*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sq
rt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 1/8*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2)) - 5)) + 1/40*(886*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 2897*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^2 + 9039*sqrt(3)*x + 3037*sqrt(3) - 9039*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6
*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^2