3.2497 \(\int \frac{(5-x) (3+2 x)^2}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{1}{9} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{54} (194 x+699) \sqrt{3 x^2+5 x+2}+\frac{1147 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{108 \sqrt{3}} \]

[Out]

-((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/9 + ((699 + 194*x)*Sqrt[2 + 5*x + 3*x^2])/54 + (1147*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(108*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0452249, antiderivative size = 87, 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 = {832, 779, 621, 206} \[ -\frac{1}{9} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{1}{54} (194 x+699) \sqrt{3 x^2+5 x+2}+\frac{1147 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{108 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/9 + ((699 + 194*x)*Sqrt[2 + 5*x + 3*x^2])/54 + (1147*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(108*Sqrt[3])

Rule 832

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

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), 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))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{1}{9} \int \frac{(3+2 x) \left (\frac{301}{2}+97 x\right )}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{1}{54} (699+194 x) \sqrt{2+5 x+3 x^2}+\frac{1147}{108} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{1}{54} (699+194 x) \sqrt{2+5 x+3 x^2}+\frac{1147}{54} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{1}{9} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{1}{54} (699+194 x) \sqrt{2+5 x+3 x^2}+\frac{1147 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{108 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0280469, size = 62, normalized size = 0.71 \[ \frac{1}{324} \left (1147 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )-6 \sqrt{3 x^2+5 x+2} \left (24 x^2-122 x-645\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-645 - 122*x + 24*x^2) + 1147*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]
)/324

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 77, normalized size = 0.9 \begin{align*} -{\frac{4\,{x}^{2}}{9}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{61\,x}{27}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{215}{18}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{1147\,\sqrt{3}}{324}\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)^2/(3*x^2+5*x+2)^(1/2),x)

[Out]

-4/9*x^2*(3*x^2+5*x+2)^(1/2)+61/27*x*(3*x^2+5*x+2)^(1/2)+215/18*(3*x^2+5*x+2)^(1/2)+1147/324*ln(1/3*(5/2+3*x)*
3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.65423, size = 101, normalized size = 1.16 \begin{align*} -\frac{4}{9} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + \frac{61}{27} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{1147}{324} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{215}{18} \, \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)^(1/2),x, algorithm="maxima")

[Out]

-4/9*sqrt(3*x^2 + 5*x + 2)*x^2 + 61/27*sqrt(3*x^2 + 5*x + 2)*x + 1147/324*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5
*x + 2) + 6*x + 5) + 215/18*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 2.14731, size = 189, normalized size = 2.17 \begin{align*} -\frac{1}{54} \,{\left (24 \, x^{2} - 122 \, x - 645\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{1147}{648} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\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)^(1/2),x, algorithm="fricas")

[Out]

-1/54*(24*x^2 - 122*x - 645)*sqrt(3*x^2 + 5*x + 2) + 1147/648*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x
+ 5) + 72*x^2 + 120*x + 49)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{51 x}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{8 x^{2}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{4 x^{3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{45}{\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)**(1/2),x)

[Out]

-Integral(-51*x/sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2/sqrt(3*x**2 + 5*x + 2), x) - Integral(4*x**3/sqr
t(3*x**2 + 5*x + 2), x) - Integral(-45/sqrt(3*x**2 + 5*x + 2), x)

________________________________________________________________________________________

Giac [A]  time = 1.12411, size = 80, normalized size = 0.92 \begin{align*} -\frac{1}{54} \,{\left (2 \,{\left (12 \, x - 61\right )} x - 645\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1147}{324} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\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)^(1/2),x, algorithm="giac")

[Out]

-1/54*(2*(12*x - 61)*x - 645)*sqrt(3*x^2 + 5*x + 2) - 1147/324*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2)) - 5))