3.2502 \(\int \frac{5-x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{73 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{13 \sqrt{3 x^2+5 x+2}}{10 (2 x+3)^2}+\frac{389 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{100 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(10*(3 + 2*x)^2) - (73*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (389*ArcTanh[(7 + 8
*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(100*Sqrt[5])

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Rubi [A]  time = 0.0608025, antiderivative size = 89, 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 = {834, 806, 724, 206} \[ -\frac{73 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{13 \sqrt{3 x^2+5 x+2}}{10 (2 x+3)^2}+\frac{389 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{100 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(10*(3 + 2*x)^2) - (73*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)) + (389*ArcTanh[(7 + 8
*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(100*Sqrt[5])

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)^3 \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{13 \sqrt{2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac{1}{10} \int \frac{-\frac{29}{2}+39 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac{73 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{389}{100} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac{73 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}-\frac{389}{50} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{10 (3+2 x)^2}-\frac{73 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{389 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{100 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0364021, size = 69, normalized size = 0.78 \[ \frac{1}{500} \left (-\frac{10 \sqrt{3 x^2+5 x+2} (292 x+503)}{(2 x+3)^2}-389 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*(503 + 292*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 - 389*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*
x + 3*x^2])])/500

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Maple [A]  time = 0.01, size = 74, normalized size = 0.8 \begin{align*} -{\frac{13}{40}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{73}{50}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{389\,\sqrt{5}}{500}{\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^(1/2),x)

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(1/2)-73/50/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-389/500*5^(1/2)*arcta
nh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

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Maxima [A]  time = 1.7313, size = 122, normalized size = 1.37 \begin{align*} -\frac{389}{500} \, \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{13 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{73 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{25 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-389/500*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/10*sqrt(3*x^2 + 5
*x + 2)/(4*x^2 + 12*x + 9) - 73/25*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 1.84967, size = 259, normalized size = 2.91 \begin{align*} \frac{389 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\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 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (292 \, x + 503\right )}}{1000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/1000*(389*sqrt(5)*(4*x^2 + 12*x + 9)*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*sqrt(3*x^2 + 5*x + 2)*(292*x + 503))/(4*x^2 + 12*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{8 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 54 x \sqrt{3 x^{2} + 5 x + 2} + 27 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{8 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 54 x \sqrt{3 x^{2} + 5 x + 2} + 27 \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)**3/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(8*x**3*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(3*x**2 + 5*x + 2) + 54*x*sqrt(3*x**2 + 5*x + 2) + 27
*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(8*x**3*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(3*x**2 + 5*x + 2) + 5
4*x*sqrt(3*x**2 + 5*x + 2) + 27*sqrt(3*x**2 + 5*x + 2)), x)

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Giac [B]  time = 1.17223, size = 278, normalized size = 3.12 \begin{align*} \frac{389}{500} \, \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{778 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 3551 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 13217 \, \sqrt{3} x + 4971 \, \sqrt{3} - 13217 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{50 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^3/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

389/500*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*s
qrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/50*(778*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 3551*sqrt(3
)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 13217*sqrt(3)*x + 4971*sqrt(3) - 13217*sqrt(3*x^2 + 5*x + 2))/(2*(sq
rt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^2