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

Optimal. Leaf size=64 \[ \frac {47 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10 \sqrt {5}}-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {806, 724, 206} \begin {gather*} \frac {47 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10 \sqrt {5}}-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (47*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(10*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]

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}+\frac {47}{10} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{5 (3+2 x)}-\frac {47}{5} \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}}{5 (3+2 x)}+\frac {47 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{10 \sqrt {5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 64, normalized size = 1.00 \begin {gather*} -\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {47 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{10 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) - (47*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(10*Sqr
t[5])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.46, size = 61, normalized size = 0.95 \begin {gather*} \frac {47 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{5 \sqrt {5}}-\frac {13 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) + (47*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(5*Sqrt[5])

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 80, normalized size = 1.25 \begin {gather*} \frac {47 \, \sqrt {5} {\left (2 \, x + 3\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 ) - 260 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{100 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/100*(47*sqrt(5)*(2*x + 3)*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)) - 260*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

________________________________________________________________________________________

giac [B]  time = 0.34, size = 127, normalized size = 1.98 \begin {gather*} \frac {1}{50} \, \sqrt {5} {\left (13 \, \sqrt {5} \sqrt {3} + 47 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {47 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{50 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {13 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}}{10 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*sqrt(5)*(13*sqrt(5)*sqrt(3) + 47*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) - 47/50*sqrt(5)*log(abs(sqrt
(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))/sgn(1/(2*x + 3)) - 13/10*sqrt(-8/(2*x +
3) + 5/(2*x + 3)^2 + 3)/sgn(1/(2*x + 3))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 53, normalized size = 0.83 \begin {gather*} -\frac {47 \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 )}{50}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{10 \left (x +\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13/10/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-47/50*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-1
9)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.29, size = 64, normalized size = 1.00 \begin {gather*} -\frac {47}{50} \, \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}}{5 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-47/50*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 13/5*sqrt(3*x^2 + 5*x
+ 2)/(2*x + 3)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^2\,\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)^2*(5*x + 3*x^2 + 2)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 12 x \sqrt {3 x^{2} + 5 x + 2} + 9 \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)**2/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(4*x**2*sqrt(3*x**2 + 5*x + 2) + 12*x*sqrt(3*x**2 + 5*x + 2) + 9*sqrt(3*x**2 + 5*x + 2)), x) - Int
egral(-5/(4*x**2*sqrt(3*x**2 + 5*x + 2) + 12*x*sqrt(3*x**2 + 5*x + 2) + 9*sqrt(3*x**2 + 5*x + 2)), x)

________________________________________________________________________________________