3.2512 \(\int \frac{5-x}{(3+2 x)^2 (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{6 (47 x+37)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}-\frac{856 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{302 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) - (856*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (302*ArcTa
nh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sqrt[5])

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Rubi [A]  time = 0.0531627, antiderivative size = 94, 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 = {822, 806, 724, 206} \[ -\frac{6 (47 x+37)}{5 (2 x+3) \sqrt{3 x^2+5 x+2}}-\frac{856 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}+\frac{302 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) - (856*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (302*ArcTa
nh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*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 )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{209+282 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}-\frac{856 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{302}{25} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}-\frac{856 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}-\frac{604}{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{6 (37+47 x)}{5 (3+2 x) \sqrt{2+5 x+3 x^2}}-\frac{856 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{302 \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 = 0.0329341, size = 90, normalized size = 0.96 \[ -\frac{2 \left (6420 x^2+151 \sqrt{5} (2 x+3) \sqrt{3 x^2+5 x+2} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+14225 x+7055\right )}{125 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(7055 + 14225*x + 6420*x^2 + 151*Sqrt[5]*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqr
t[2 + 5*x + 3*x^2])]))/(125*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.01, size = 90, normalized size = 1. \begin{align*}{\frac{151}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{1070+1284\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{302\,\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}{\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)^2/(3*x^2+5*x+2)^(3/2),x)

[Out]

151/25/(3*(x+3/2)^2-4*x-19/4)^(1/2)-214/25*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)-302/125*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)^(1/2)

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Maxima [A]  time = 1.72983, size = 143, normalized size = 1.52 \begin{align*} -\frac{302}{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{1284 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{919}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13}{5 \,{\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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-302/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1284/25*x/sqrt(3*x^2
+ 5*x + 2) - 919/25/sqrt(3*x^2 + 5*x + 2) - 13/5/(2*sqrt(3*x^2 + 5*x + 2)*x + 3*sqrt(3*x^2 + 5*x + 2))

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Fricas [A]  time = 1.90001, size = 300, normalized size = 3.19 \begin{align*} \frac{151 \, \sqrt{5}{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\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 ) - 10 \,{\left (1284 \, x^{2} + 2845 \, x + 1411\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{125 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\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)^(3/2),x, algorithm="fricas")

[Out]

1/125*(151*sqrt(5)*(6*x^3 + 19*x^2 + 19*x + 6)*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)) - 10*(1284*x^2 + 2845*x + 1411)*sqrt(3*x^2 + 5*x + 2))/(6*x^3 + 19*x^2 + 19*x + 6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{12 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{3 x^{2} + 5 x + 2} + 18 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{12 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{3 x^{2} + 5 x + 2} + 18 \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)**(3/2),x)

[Out]

-Integral(x/(12*x**4*sqrt(3*x**2 + 5*x + 2) + 56*x**3*sqrt(3*x**2 + 5*x + 2) + 95*x**2*sqrt(3*x**2 + 5*x + 2)
+ 69*x*sqrt(3*x**2 + 5*x + 2) + 18*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(12*x**4*sqrt(3*x**2 + 5*x + 2) +
56*x**3*sqrt(3*x**2 + 5*x + 2) + 95*x**2*sqrt(3*x**2 + 5*x + 2) + 69*x*sqrt(3*x**2 + 5*x + 2) + 18*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{3}{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)^(3/2),x, algorithm="giac")

[Out]

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