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

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

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Rubi [A]  time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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 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])

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*}

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Mathematica [A]  time = 0.03, size = 69, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.55, size = 66, normalized size = 0.74 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} (-292 x-503)}{50 (2 x+3)^2}+\frac {389 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{50 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-503 - 292*x)*Sqrt[2 + 5*x + 3*x^2])/(50*(3 + 2*x)^2) + (389*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))
])/(50*Sqrt[5])

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fricas [A]  time = 0.40, size = 95, normalized size = 1.07 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.32, size = 206, normalized size = 2.31 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 74, normalized size = 0.83 \begin {gather*} -\frac {389 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{500}-\frac {73 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{50 \left (x +\frac {3}{2}\right )}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{40 \left (x +\frac {3}{2}\right )^{2}} \end {gather*}

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]

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

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maxima [A]  time = 1.27, size = 90, normalized size = 1.01 \begin {gather*} -\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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^3\,\sqrt {3\,x^2+5\,x+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \left (- \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}}\right )\, dx \end {gather*}

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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