3.2514 \(\int \frac{5-x}{(3+2 x)^4 (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{6 (47 x+37)}{5 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}-\frac{4632 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}-\frac{478 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac{2464 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac{3289 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - (2464*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (478*
Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) - (4632*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (3289*ArcTanh[(7 + 8*
x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*Sqrt[5])

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Rubi [A]  time = 0.102739, antiderivative size = 144, 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 = {822, 834, 806, 724, 206} \[ -\frac{6 (47 x+37)}{5 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}-\frac{4632 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}-\frac{478 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}-\frac{2464 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}+\frac{3289 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - (2464*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (478*
Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^2) - (4632*Sqrt[2 + 5*x + 3*x^2])/(125*(3 + 2*x)) + (3289*ArcTanh[(7 + 8*
x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(125*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{653+846 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2464 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac{2}{75} \int \frac{-5113-7392 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2464 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{478 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac{1}{375} \int \frac{19035+35850 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2464 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{478 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac{4632 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}+\frac{3289}{125} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2464 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{478 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac{4632 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}-\frac{6578}{125} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}-\frac{2464 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{478 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}-\frac{4632 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}+\frac{3289 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{125 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0607025, size = 84, normalized size = 0.58 \[ \frac{-\frac{10 \left (83376 x^4+424938 x^3+792065 x^2+634312 x+181559\right )}{(2 x+3)^3 \sqrt{3 x^2+5 x+2}}-9867 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*(181559 + 634312*x + 792065*x^2 + 424938*x^3 + 83376*x^4))/((3 + 2*x)^3*Sqrt[2 + 5*x + 3*x^2]) - 9867*Sq
rt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1875

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Maple [A]  time = 0.008, size = 132, normalized size = 0.9 \begin{align*} -{\frac{349}{600} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{271}{75} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{3289}{250}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{5790+6948\,x}{125}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{3289\,\sqrt{5}}{625}{\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}{120} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}} \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]

-349/600/(x+3/2)^2/(3*(x+3/2)^2-4*x-19/4)^(1/2)-271/75/(x+3/2)/(3*(x+3/2)^2-4*x-19/4)^(1/2)+3289/250/(3*(x+3/2
)^2-4*x-19/4)^(1/2)-1158/125*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)-3289/625*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1
/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-13/120/(x+3/2)^3/(3*(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 2.30726, size = 304, normalized size = 2.11 \begin{align*} -\frac{3289}{625} \, \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{6948 \, x}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{8291}{250 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13}{15 \,{\left (8 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} - \frac{349}{150 \,{\left (4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} - \frac{542}{75 \,{\left (2 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt{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="maxima")

[Out]

-3289/625*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 6948/125*x/sqrt(3*x
^2 + 5*x + 2) - 8291/250/sqrt(3*x^2 + 5*x + 2) - 13/15/(8*sqrt(3*x^2 + 5*x + 2)*x^3 + 36*sqrt(3*x^2 + 5*x + 2)
*x^2 + 54*sqrt(3*x^2 + 5*x + 2)*x + 27*sqrt(3*x^2 + 5*x + 2)) - 349/150/(4*sqrt(3*x^2 + 5*x + 2)*x^2 + 12*sqrt
(3*x^2 + 5*x + 2)*x + 9*sqrt(3*x^2 + 5*x + 2)) - 542/75/(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 2))

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Fricas [A]  time = 1.88078, size = 409, normalized size = 2.84 \begin{align*} \frac{9867 \, \sqrt{5}{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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 \,{\left (83376 \, x^{4} + 424938 \, x^{3} + 792065 \, x^{2} + 634312 \, x + 181559\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3750 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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]

1/3750*(9867*sqrt(5)*(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)*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)) - 20*(83376*x^4 + 424938*x^3 + 792065*x^2 + 634312*x + 1
81559)*sqrt(3*x^2 + 5*x + 2))/(24*x^5 + 148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{48 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 837 x \sqrt{3 x^{2} + 5 x + 2} + 162 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{48 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 368 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 1160 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 1920 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1755 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 837 x \sqrt{3 x^{2} + 5 x + 2} + 162 \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(x/(48*x**6*sqrt(3*x**2 + 5*x + 2) + 368*x**5*sqrt(3*x**2 + 5*x + 2) + 1160*x**4*sqrt(3*x**2 + 5*x +
2) + 1920*x**3*sqrt(3*x**2 + 5*x + 2) + 1755*x**2*sqrt(3*x**2 + 5*x + 2) + 837*x*sqrt(3*x**2 + 5*x + 2) + 162*
sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(48*x**6*sqrt(3*x**2 + 5*x + 2) + 368*x**5*sqrt(3*x**2 + 5*x + 2) +
1160*x**4*sqrt(3*x**2 + 5*x + 2) + 1920*x**3*sqrt(3*x**2 + 5*x + 2) + 1755*x**2*sqrt(3*x**2 + 5*x + 2) + 837*x
*sqrt(3*x**2 + 5*x + 2) + 162*sqrt(3*x**2 + 5*x + 2)), x)

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Giac [B]  time = 1.17173, size = 373, normalized size = 2.59 \begin{align*} \frac{3289}{625} \, \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{6 \,{\left (4209 \, x + 2959\right )}}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{2 \,{\left (118356 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 851850 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 6938110 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 8824815 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 15944775 \, \sqrt{3} x + 3678471 \, \sqrt{3} - 15944775 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{1875 \,{\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 )}^{3}} \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]

3289/625*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*
sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 6/625*(4209*x + 2959)/sqrt(3*x^2 + 5*x + 2) - 2/1875*(118356
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 851850*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 6938110*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 8824815*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 15944775*sqrt(3)*x +
3678471*sqrt(3) - 15944775*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)^3