3.2506 \(\int \frac{(5-x) (3+2 x)^4}{(2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{2 (139 x+121) (2 x+3)^3}{3 \sqrt{3 x^2+5 x+2}}+\frac{1664}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{10}{81} (1438 x+3369) \sqrt{3 x^2+5 x+2}+\frac{6265 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{81 \sqrt{3}} \]

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (1664*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/27 + (10*(
3369 + 1438*x)*Sqrt[2 + 5*x + 3*x^2])/81 + (6265*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(81*Sqr
t[3])

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Rubi [A]  time = 0.0742198, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {818, 832, 779, 621, 206} \[ -\frac{2 (139 x+121) (2 x+3)^3}{3 \sqrt{3 x^2+5 x+2}}+\frac{1664}{27} \sqrt{3 x^2+5 x+2} (2 x+3)^2+\frac{10}{81} (1438 x+3369) \sqrt{3 x^2+5 x+2}+\frac{6265 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{81 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)^3*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (1664*(3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2])/27 + (10*(
3369 + 1438*x)*Sqrt[2 + 5*x + 3*x^2])/81 + (6265*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(81*Sqr
t[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 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)^4}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (3+2 x)^3 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{2}{3} \int \frac{(3+2 x)^2 (723+832 x)}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{1664}{27} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{2}{27} \int \frac{(3+2 x) (6625+7190 x)}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{1664}{27} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{10}{81} (3369+1438 x) \sqrt{2+5 x+3 x^2}+\frac{6265}{81} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (3+2 x)^3 (121+139 x)}{3 \sqrt{2+5 x+3 x^2}}+\frac{1664}{27} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{10}{81} (3369+1438 x) \sqrt{2+5 x+3 x^2}+\frac{12530}{81} \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)}{3 \sqrt{2+5 x+3 x^2}}+\frac{1664}{27} (3+2 x)^2 \sqrt{2+5 x+3 x^2}+\frac{10}{81} (3369+1438 x) \sqrt{2+5 x+3 x^2}+\frac{6265 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{81 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0500736, size = 81, normalized size = 0.69 \[ -\frac{6 \left (72 x^4-102 x^3-3331 x^2+6920 x+9591\right )-6265 \sqrt{9 x^2+15 x+6} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{243 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*(9591 + 6920*x - 3331*x^2 - 102*x^3 + 72*x^4) - 6265*Sqrt[6 + 15*x + 9*x^2]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 +
15*x + 9*x^2])])/(243*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.01, size = 130, normalized size = 1.1 \begin{align*}{\frac{6265\,\sqrt{3}}{243}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{12625+15150\,x}{162}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{16\,{x}^{4}}{9}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{68\,{x}^{3}}{27}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{6662\,{x}^{2}}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{6265\,x}{81}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}-{\frac{25739}{162}{\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)^(3/2),x)

[Out]

6265/243*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-2525/162*(5+6*x)/(3*x^2+5*x+2)^(1/2)-16/9*x^4/(
3*x^2+5*x+2)^(1/2)+68/27*x^3/(3*x^2+5*x+2)^(1/2)+6662/81*x^2/(3*x^2+5*x+2)^(1/2)-6265/81*x/(3*x^2+5*x+2)^(1/2)
-25739/162/(3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 1.90761, size = 147, normalized size = 1.26 \begin{align*} -\frac{16 \, x^{4}}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{68 \, x^{3}}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{6662 \, x^{2}}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{6265}{243} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{13840 \, x}{81 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{6394}{27 \, \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)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-16/9*x^4/sqrt(3*x^2 + 5*x + 2) + 68/27*x^3/sqrt(3*x^2 + 5*x + 2) + 6662/81*x^2/sqrt(3*x^2 + 5*x + 2) + 6265/2
43*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 13840/81*x/sqrt(3*x^2 + 5*x + 2) - 6394/27/sqrt(3*
x^2 + 5*x + 2)

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Fricas [A]  time = 2.03327, size = 270, normalized size = 2.31 \begin{align*} \frac{6265 \, \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 ) - 12 \,{\left (72 \, x^{4} - 102 \, x^{3} - 3331 \, x^{2} + 6920 \, x + 9591\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{486 \,{\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)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/486*(6265*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) - 1
2*(72*x^4 - 102*x^3 - 3331*x^2 + 6920*x + 9591)*sqrt(3*x^2 + 5*x + 2))/(3*x^2 + 5*x + 2)

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

[Out]

-Integral(-999*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) -
Integral(-864*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
) - Integral(-264*x**3/(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) - Integral(16*x**4/(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) - Integral(16*x**5/(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) - Integral(-405/(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 = 1.11249, size = 90, normalized size = 0.77 \begin{align*} -\frac{6265}{243} \, \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 ({\left (6 \,{\left (12 \, x - 17\right )} x - 3331\right )} x + 6920\right )} x + 9591\right )}}{81 \, \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)^4/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

-6265/243*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/81*(((6*(12*x - 17)*x - 333
1)*x + 6920)*x + 9591)/sqrt(3*x^2 + 5*x + 2)