3.2504 \(\int \frac{5-x}{(3+2 x)^5 \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{681 \sqrt{3 x^2+5 x+2}}{250 (2 x+3)}-\frac{41 \sqrt{3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac{86 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac{5771 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2000 \sqrt{5}} \]

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) - (86*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (41*Sqrt[2 + 5*x
+ 3*x^2])/(24*(3 + 2*x)^2) - (681*Sqrt[2 + 5*x + 3*x^2])/(250*(3 + 2*x)) + (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*
Sqrt[2 + 5*x + 3*x^2])])/(2000*Sqrt[5])

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Rubi [A]  time = 0.0972799, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \[ -\frac{681 \sqrt{3 x^2+5 x+2}}{250 (2 x+3)}-\frac{41 \sqrt{3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac{86 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac{5771 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{2000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) - (86*Sqrt[2 + 5*x + 3*x^2])/(75*(3 + 2*x)^3) - (41*Sqrt[2 + 5*x
+ 3*x^2])/(24*(3 + 2*x)^2) - (681*Sqrt[2 + 5*x + 3*x^2])/(250*(3 + 2*x)) + (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*
Sqrt[2 + 5*x + 3*x^2])])/(2000*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)^5 \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{1}{20} \int \frac{\frac{7}{2}+117 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{86 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac{1}{300} \int \frac{-\frac{1067}{2}-2064 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{86 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{41 \sqrt{2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac{\int \frac{\frac{5265}{2}+15375 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx}{3000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{86 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{41 \sqrt{2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac{681 \sqrt{2+5 x+3 x^2}}{250 (3+2 x)}+\frac{5771 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{2000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{86 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{41 \sqrt{2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac{681 \sqrt{2+5 x+3 x^2}}{250 (3+2 x)}-\frac{5771 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{1000}\\ &=-\frac{13 \sqrt{2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac{86 \sqrt{2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac{41 \sqrt{2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac{681 \sqrt{2+5 x+3 x^2}}{250 (3+2 x)}+\frac{5771 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{2000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0571851, size = 79, normalized size = 0.57 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (65376 x^3+314692 x^2+509668 x+279039\right )}{(2 x+3)^4}-17313 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{30000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(279039 + 509668*x + 314692*x^2 + 65376*x^3))/(3 + 2*x)^4 - 17313*Sqrt[5]*ArcTanh[
(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/30000

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Maple [A]  time = 0.012, size = 116, normalized size = 0.8 \begin{align*} -{\frac{13}{320}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{43}{300}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{41}{96}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{681}{500}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{5771\,\sqrt{5}}{10000}{\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)^5/(3*x^2+5*x+2)^(1/2),x)

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(1/2)-43/300/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(1/2)-41/96/(x+3/2)^2*(
3*(x+3/2)^2-4*x-19/4)^(1/2)-681/500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-5771/10000*5^(1/2)*arctanh(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 = 1.65185, size = 212, normalized size = 1.53 \begin{align*} -\frac{5771}{10000} \, \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}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{86 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{75 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{41 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{24 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{681 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{250 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5771/10000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/20*sqrt(3*x^2
+ 5*x + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 86/75*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27)
- 41/24*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 681/250*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 1.92231, size = 365, normalized size = 2.63 \begin{align*} \frac{17313 \, \sqrt{5}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (65376 \, x^{3} + 314692 \, x^{2} + 509668 \, x + 279039\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{60000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60000*(17313*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(65376*x^3 + 314692*x^2 + 509668*x + 279039)*sqrt(3*x^2 + 5*
x + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{32 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 810 x \sqrt{3 x^{2} + 5 x + 2} + 243 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{32 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 810 x \sqrt{3 x^{2} + 5 x + 2} + 243 \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)**5/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 720*x**3*sqrt(3*x**2 + 5*x + 2
) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x) - Integr
al(-5/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 720*x**3*sqrt(3*x**2 + 5*x + 2) + 10
80*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*sqrt(3*x**2 + 5*x + 2) + 243*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}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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