3.2505 \(\int \frac{5-x}{(3+2 x)^6 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=164 \[ -\frac{15891 \sqrt{3 x^2+5 x+2}}{6250 (2 x+3)}-\frac{1007 \sqrt{3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac{2321 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac{443 \sqrt{3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac{128381 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{50000 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^5) - (443*Sqrt[2 + 5*x + 3*x^2])/(500*(3 + 2*x)^4) - (2321*Sqrt[2 +
5*x + 3*x^2])/(1875*(3 + 2*x)^3) - (1007*Sqrt[2 + 5*x + 3*x^2])/(600*(3 + 2*x)^2) - (15891*Sqrt[2 + 5*x + 3*x^
2])/(6250*(3 + 2*x)) + (128381*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(50000*Sqrt[5])

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Rubi [A]  time = 0.143914, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ -\frac{15891 \sqrt{3 x^2+5 x+2}}{6250 (2 x+3)}-\frac{1007 \sqrt{3 x^2+5 x+2}}{600 (2 x+3)^2}-\frac{2321 \sqrt{3 x^2+5 x+2}}{1875 (2 x+3)^3}-\frac{443 \sqrt{3 x^2+5 x+2}}{500 (2 x+3)^4}-\frac{13 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^5}+\frac{128381 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{50000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^5) - (443*Sqrt[2 + 5*x + 3*x^2])/(500*(3 + 2*x)^4) - (2321*Sqrt[2 +
5*x + 3*x^2])/(1875*(3 + 2*x)^3) - (1007*Sqrt[2 + 5*x + 3*x^2])/(600*(3 + 2*x)^2) - (15891*Sqrt[2 + 5*x + 3*x^
2])/(6250*(3 + 2*x)) + (128381*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(50000*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)^6 \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{1}{25} \int \frac{\frac{25}{2}+156 x}{(3+2 x)^5 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}+\frac{1}{500} \int \frac{-\frac{2677}{2}-3987 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac{2321 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac{\int \frac{\frac{41237}{2}+55704 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx}{7500}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac{2321 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac{1007 \sqrt{2+5 x+3 x^2}}{600 (3+2 x)^2}+\frac{\int \frac{-\frac{179415}{2}-377625 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx}{75000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac{2321 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac{1007 \sqrt{2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac{15891 \sqrt{2+5 x+3 x^2}}{6250 (3+2 x)}+\frac{128381 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{50000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac{2321 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac{1007 \sqrt{2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac{15891 \sqrt{2+5 x+3 x^2}}{6250 (3+2 x)}-\frac{128381 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{25000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^5}-\frac{443 \sqrt{2+5 x+3 x^2}}{500 (3+2 x)^4}-\frac{2321 \sqrt{2+5 x+3 x^2}}{1875 (3+2 x)^3}-\frac{1007 \sqrt{2+5 x+3 x^2}}{600 (3+2 x)^2}-\frac{15891 \sqrt{2+5 x+3 x^2}}{6250 (3+2 x)}+\frac{128381 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{50000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.071525, size = 84, normalized size = 0.51 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (3051072 x^4+19313432 x^3+46092332 x^2+49233702 x+19918587\right )}{(2 x+3)^5}-385143 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{750000} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^6*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(19918587 + 49233702*x + 46092332*x^2 + 19313432*x^3 + 3051072*x^4))/(3 + 2*x)^5 -
385143*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/750000

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Maple [A]  time = 0.012, size = 137, normalized size = 0.8 \begin{align*} -{\frac{443}{8000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{2321}{15000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1007}{2400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{15891}{12500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{128381\,\sqrt{5}}{250000}{\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}{800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2),x)

[Out]

-443/8000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(1/2)-2321/15000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(1/2)-1007/2400/(
x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(1/2)-15891/12500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-128381/250000*5^(1/2)*a
rctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(1/2)

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Maxima [A]  time = 1.97035, size = 267, normalized size = 1.63 \begin{align*} -\frac{128381}{250000} \, \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}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{443 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{2321 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{1875 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1007 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{600 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{15891 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{6250 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-128381/250000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/25*sqrt(3*x
^2 + 5*x + 2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 443/500*sqrt(3*x^2 + 5*x + 2)/(16*x^4 +
96*x^3 + 216*x^2 + 216*x + 81) - 2321/1875*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1007/600*sqrt(
3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 15891/6250*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 1.8404, size = 435, normalized size = 2.65 \begin{align*} \frac{385143 \, \sqrt{5}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (3051072 \, x^{4} + 19313432 \, x^{3} + 46092332 \, x^{2} + 49233702 \, x + 19918587\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{1500000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/1500000*(385143*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*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*(3051072*x^4 + 19313432*x^3 + 46092332*x^2 +
49233702*x + 19918587)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{64 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 576 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 2160 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 4320 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 4860 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2916 x \sqrt{3 x^{2} + 5 x + 2} + 729 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{64 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 576 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 2160 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 4320 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 4860 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2916 x \sqrt{3 x^{2} + 5 x + 2} + 729 \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)**6/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(64*x**6*sqrt(3*x**2 + 5*x + 2) + 576*x**5*sqrt(3*x**2 + 5*x + 2) + 2160*x**4*sqrt(3*x**2 + 5*x +
2) + 4320*x**3*sqrt(3*x**2 + 5*x + 2) + 4860*x**2*sqrt(3*x**2 + 5*x + 2) + 2916*x*sqrt(3*x**2 + 5*x + 2) + 729
*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(64*x**6*sqrt(3*x**2 + 5*x + 2) + 576*x**5*sqrt(3*x**2 + 5*x + 2) +
2160*x**4*sqrt(3*x**2 + 5*x + 2) + 4320*x**3*sqrt(3*x**2 + 5*x + 2) + 4860*x**2*sqrt(3*x**2 + 5*x + 2) + 2916
*x*sqrt(3*x**2 + 5*x + 2) + 729*sqrt(3*x**2 + 5*x + 2)), x)

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Giac [B]  time = 1.27916, size = 485, normalized size = 2.96 \begin{align*} \frac{128381}{250000} \, \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{6162288 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 83190888 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 1461489304 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 4863585804 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 30365807072 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 40931011758 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 107175203674 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 58461317289 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 54344360217 \, \sqrt{3} x + 7303159752 \, \sqrt{3} - 54344360217 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{75000 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^6/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

128381/250000*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))) - 1/75000*(6162288*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9
+ 83190888*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 1461489304*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 +
4863585804*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 30365807072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 +
40931011758*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 107175203674*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
3 + 58461317289*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 54344360217*sqrt(3)*x + 7303159752*sqrt(3) - 5
4344360217*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)^5