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

Optimal. Leaf size=62 \[ \frac {26 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}-\frac {6 (47 x+37)}{5 \sqrt {3 x^2+5 x+2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 62, 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 = {822, 12, 724, 206} \begin {gather*} \frac {26 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}-\frac {6 (47 x+37)}{5 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (26*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(5*Sqrt
[5])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {6 (37+47 x)}{5 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int -\frac {13}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 \sqrt {2+5 x+3 x^2}}+\frac {26}{5} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 \sqrt {2+5 x+3 x^2}}-\frac {52}{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 {6 (37+47 x)}{5 \sqrt {2+5 x+3 x^2}}+\frac {26 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{5 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 62, normalized size = 1.00 \begin {gather*} -\frac {2 (141 x+111)}{5 \sqrt {3 x^2+5 x+2}}-\frac {26 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(111 + 141*x))/(5*Sqrt[2 + 5*x + 3*x^2]) - (26*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(5*S
qrt[5])

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IntegrateAlgebraic [A]  time = 0.39, size = 71, normalized size = 1.15 \begin {gather*} \frac {52 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{5 \sqrt {5}}-\frac {6 (47 x+37) \sqrt {3 x^2+5 x+2}}{5 (x+1) (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x)*Sqrt[2 + 5*x + 3*x^2])/(5*(1 + x)*(2 + 3*x)) + (52*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 +
x))])/(5*Sqrt[5])

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fricas [A]  time = 0.41, size = 95, normalized size = 1.53 \begin {gather*} \frac {13 \, \sqrt {5} {\left (3 \, x^{2} + 5 \, x + 2\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 ) - 30 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (47 \, x + 37\right )}}{25 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(13*sqrt(5)*(3*x^2 + 5*x + 2)*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)) - 30*sqrt(3*x^2 + 5*x + 2)*(47*x + 37))/(3*x^2 + 5*x + 2)

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giac [A]  time = 0.28, size = 93, normalized size = 1.50 \begin {gather*} \frac {26}{25} \, \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 {6 \, {\left (47 \, x + 37\right )}}{5 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

26/25*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*sqr
t(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 6/5*(47*x + 37)/sqrt(3*x^2 + 5*x + 2)

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maple [A]  time = 0.01, size = 87, normalized size = 1.40 \begin {gather*} -\frac {26 \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 )}{25}+\frac {6 x +5}{\sqrt {3 x^{2}+5 x +2}}+\frac {13}{5 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {52 \left (6 x +5\right )}{5 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*x+5)/(3*x^2+5*x+2)^(1/2)+13/5/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-52/5*(6*x+5)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)-26/2
5*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.36, size = 72, normalized size = 1.16 \begin {gather*} -\frac {26}{25} \, \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 {282 \, x}{5 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {222}{5 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-26/25*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 282/5*x/sqrt(3*x^2 + 5
*x + 2) - 222/5/sqrt(3*x^2 + 5*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{6 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 19 x \sqrt {3 x^{2} + 5 x + 2} + 6 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{6 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 19 x \sqrt {3 x^{2} + 5 x + 2} + 6 \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*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(x/(6*x**3*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(3*x**2 + 5*x + 2) + 19*x*sqrt(3*x**2 + 5*x + 2) + 6*
sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(6*x**3*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(3*x**2 + 5*x + 2) + 19
*x*sqrt(3*x**2 + 5*x + 2) + 6*sqrt(3*x**2 + 5*x + 2)), x)

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