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

Optimal. Leaf size=62 \[ -\frac {2 (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {777, 621, 206} \begin {gather*} -\frac {2 (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}-\frac {2 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(3*Sqr
t[3])

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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 777

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

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {2}{3} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {2 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 53, normalized size = 0.85 \begin {gather*} -\frac {2}{9} \left (\frac {417 x+363}{\sqrt {3 x^2+5 x+2}}+\sqrt {3} \log \left (2 \sqrt {9 x^2+15 x+6}+6 x+5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((363 + 417*x)/Sqrt[2 + 5*x + 3*x^2] + Sqrt[3]*Log[5 + 6*x + 2*Sqrt[6 + 15*x + 9*x^2]]))/9

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IntegrateAlgebraic [A]  time = 0.37, size = 71, normalized size = 1.15 \begin {gather*} -\frac {2 \sqrt {3 x^2+5 x+2} (139 x+121)}{3 (x+1) (3 x+2)}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(121 + 139*x)*Sqrt[2 + 5*x + 3*x^2])/(3*(1 + x)*(2 + 3*x)) - (4*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1
+ x))])/(3*Sqrt[3])

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fricas [A]  time = 0.40, size = 81, normalized size = 1.31 \begin {gather*} \frac {\sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (139 \, x + 121\right )}}{9 \, {\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]

1/9*(sqrt(3)*(3*x^2 + 5*x + 2)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 6*sqrt(
3*x^2 + 5*x + 2)*(139*x + 121))/(3*x^2 + 5*x + 2)

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giac [A]  time = 0.21, size = 54, normalized size = 0.87 \begin {gather*} \frac {2}{9} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left (139 \, x + 121\right )}}{3 \, \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]

2/9*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*(139*x + 121)/sqrt(3*x^2 + 5*x
+ 2)

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maple [A]  time = 0.00, size = 79, normalized size = 1.27 \begin {gather*} \frac {2 x}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {2 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{9}-\frac {26}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {140 \left (6 x +5\right )}{9 \sqrt {3 x^{2}+5 x +2}} \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]

2/3/(3*x^2+5*x+2)^(1/2)*x-26/9/(3*x^2+5*x+2)^(1/2)-140/9*(6*x+5)/(3*x^2+5*x+2)^(1/2)-2/9*3^(1/2)*ln(1/3*(3*x+5
/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))

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maxima [A]  time = 1.23, size = 58, normalized size = 0.94 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {278 \, x}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {242}{3 \, \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]

-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 278/3*x/sqrt(3*x^2 + 5*x + 2) - 242/3/sqrt(3*x^2
+ 5*x + 2)

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mupad [B]  time = 0.38, size = 78, normalized size = 1.26 \begin {gather*} \frac {352\,x}{3\,\sqrt {3\,x^2+5\,x+2}}-\frac {6\,\left (35\,x+29\right )}{\sqrt {3\,x^2+5\,x+2}}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{9}+\frac {280}{3\,\sqrt {3\,x^2+5\,x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(352*x)/(3*(5*x + 3*x^2 + 2)^(1/2)) - (6*(35*x + 29))/(5*x + 3*x^2 + 2)^(1/2) - (2*3^(1/2)*log((5*x + 3*x^2 +
2)^(1/2) + (3^(1/2)*(3*x + 5/2))/3))/9 + 280/(3*(5*x + 3*x^2 + 2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {7 x}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {2 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {15}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \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(-7*x/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - I
ntegral(2*x**2/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - I
ntegral(-15/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x)

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