3.2511 \(\int \frac{5-x}{(3+2 x) (2+5 x+3 x^2)^{3/2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0382234, antiderivative size = 62, normalized size of antiderivative = 1., 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} \[ \frac{26 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{5 \sqrt{5}}-\frac{6 (47 x+37)}{5 \sqrt{3 x^2+5 x+2}} \]

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 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])

Rule 12

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

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) \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*}

Mathematica [A]  time = 0.0298696, size = 62, normalized size = 1. \[ -\frac{2 (141 x+111)}{5 \sqrt{3 x^2+5 x+2}}-\frac{26 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{5 \sqrt{5}} \]

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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Maple [A]  time = 0.006, size = 87, normalized size = 1.4 \begin{align*}{(5+6\,x){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}}+{\frac{13}{5}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{260+312\,x}{5}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{26\,\sqrt{5}}{25}{\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)/(3*x^2+5*x+2)^(3/2),x)

[Out]

(5+6*x)/(3*x^2+5*x+2)^(1/2)+13/5/(3*(x+3/2)^2-4*x-19/4)^(1/2)-52/5*(5+6*x)/(3*(x+3/2)^2-4*x-19/4)^(1/2)-26/25*
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 = 1.85472, size = 97, normalized size = 1.56 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.85001, size = 250, normalized size = 4.03 \begin{align*} \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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 - \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}}\, dx \end{align*}

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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Giac [A]  time = 1.19421, size = 126, normalized size = 2.03 \begin{align*} \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{align*}

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)