3.2522 \(\int \frac{5-x}{(3+2 x)^2 (2+5 x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4416 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{4 (462 x+401)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}+\frac{408 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(401 + 462*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
+ (4416*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (408*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2
5*Sqrt[5])

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Rubi [A]  time = 0.0726515, antiderivative size = 124, 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 = {822, 806, 724, 206} \[ -\frac{2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4416 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{4 (462 x+401)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}+\frac{408 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(401 + 462*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])
+ (4416*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (408*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(2
5*Sqrt[5])

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 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]

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 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1029+846 x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (401+462 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{12510+13860 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (401+462 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}+\frac{4416 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{408}{25} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (401+462 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}+\frac{4416 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}-\frac{816}{25} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (401+462 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}+\frac{4416 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{408 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{25 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0598588, size = 113, normalized size = 0.91 \[ \frac{10 \left (19872 x^4+80100 x^3+116826 x^2+73215 x+16667\right )-408 \sqrt{5} \sqrt{3 x^2+5 x+2} \left (6 x^3+19 x^2+19 x+6\right ) \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(16667 + 73215*x + 116826*x^2 + 80100*x^3 + 19872*x^4) - 408*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]*(6 + 19*x + 19*
x^2 + 6*x^3)*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.012, size = 127, normalized size = 1. \begin{align*}{\frac{17}{5} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{80+96\,x}{5} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{5520+6624\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{204}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{408\,\sqrt{5}}{125}{\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{13}{10} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

17/5/(3*(x+3/2)^2-4*x-19/4)^(3/2)-16/5*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(3/2)+1104/25*(5+6*x)/(3*(x+3/2)^2-4*x-1
9/4)^(1/2)+204/25/(3*(x+3/2)^2-4*x-19/4)^(1/2)-408/125*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16
*x-19)^(1/2))-13/10/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 1.4863, size = 182, normalized size = 1.47 \begin{align*} -\frac{408}{125} \, \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{6624 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{5724}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{96 \, x}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{5 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{63}{5 \,{\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)^2/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-408/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 6624/25*x/sqrt(3*x^2
+ 5*x + 2) + 5724/25/sqrt(3*x^2 + 5*x + 2) - 96/5*x/(3*x^2 + 5*x + 2)^(3/2) - 13/5/(2*(3*x^2 + 5*x + 2)^(3/2)
*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 63/5/(3*x^2 + 5*x + 2)^(3/2)

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Fricas [A]  time = 1.87732, size = 396, normalized size = 3.19 \begin{align*} \frac{2 \,{\left (102 \, \sqrt{5}{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\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 ) + 5 \,{\left (19872 \, x^{4} + 80100 \, x^{3} + 116826 \, x^{2} + 73215 \, x + 16667\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{125 \,{\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\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)^(5/2),x, algorithm="fricas")

[Out]

2/125*(102*sqrt(5)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x
+ 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 5*(19872*x^4 + 80100*x^3 + 116826*x^2 + 73215*x + 16667)*s
qrt(3*x^2 + 5*x + 2))/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{36 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 228 x \sqrt{3 x^{2} + 5 x + 2} + 36 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{36 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 228 x \sqrt{3 x^{2} + 5 x + 2} + 36 \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)**(5/2),x)

[Out]

-Integral(x/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) + 589*x**4*sqrt(3*x**2 + 5*x + 2
) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x**2 + 5*x + 2) + 36*sqrt
(3*x**2 + 5*x + 2)), x) - Integral(-5/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) + 589*
x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3
*x**2 + 5*x + 2) + 36*sqrt(3*x**2 + 5*x + 2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{2}}\,{d x} \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)^(5/2),x, algorithm="giac")

[Out]

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