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

Optimal. Leaf size=83 \[ -\frac {2 (2 x+3) (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}+\frac {184}{3} \sqrt {3 x^2+5 x+2}+2 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 83, 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 = {818, 640, 621, 206} \begin {gather*} -\frac {2 (2 x+3) (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}+\frac {184}{3} \sqrt {3 x^2+5 x+2}+2 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3 + 2*x)*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (184*Sqrt[2 + 5*x + 3*x^2])/3 + 2*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]

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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 818

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

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {2 (3+2 x) (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{3} \int \frac {239+276 x}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x) (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {184}{3} \sqrt {2+5 x+3 x^2}+6 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (3+2 x) (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {184}{3} \sqrt {2+5 x+3 x^2}+12 \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 (3+2 x) (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {184}{3} \sqrt {2+5 x+3 x^2}+2 \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 68, normalized size = 0.82 \begin {gather*} -\frac {4 x^2-6 \sqrt {9 x^2+15 x+6} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+398 x+358}{3 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(358 + 398*x + 4*x^2 - 6*Sqrt[6 + 15*x + 9*x^2]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/Sqrt[2 + 5
*x + 3*x^2]

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IntegrateAlgebraic [A]  time = 0.44, size = 74, normalized size = 0.89 \begin {gather*} 4 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )-\frac {2 \left (2 x^2+199 x+179\right ) \sqrt {3 x^2+5 x+2}}{3 (x+1) (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(179 + 199*x + 2*x^2)*Sqrt[2 + 5*x + 3*x^2])/(3*(1 + x)*(2 + 3*x)) + 4*Sqrt[3]*ArcTanh[Sqrt[2 + 5*x + 3*x^
2]/(Sqrt[3]*(1 + x))]

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fricas [A]  time = 0.41, size = 87, normalized size = 1.05 \begin {gather*} \frac {3 \, \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 ) - 2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x^{2} + 199 \, x + 179\right )}}{3 \, {\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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(3*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) - 2*sqrt
(3*x^2 + 5*x + 2)*(2*x^2 + 199*x + 179))/(3*x^2 + 5*x + 2)

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giac [A]  time = 0.22, size = 58, normalized size = 0.70 \begin {gather*} -2 \, \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 ({\left (2 \, x + 199\right )} x + 179\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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*((2*x + 199)*x + 179)/sqrt(3*x^2
+ 5*x + 2)

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maple [A]  time = 0.01, size = 96, normalized size = 1.16 \begin {gather*} -\frac {4 x^{2}}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {6 x}{\sqrt {3 x^{2}+5 x +2}}+2 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )-\frac {124}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {190 \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)^2/(3*x^2+5*x+2)^(3/2),x)

[Out]

-4/3/(3*x^2+5*x+2)^(1/2)*x^2-6/(3*x^2+5*x+2)^(1/2)*x-124/9/(3*x^2+5*x+2)^(1/2)-190/9*(6*x+5)/(3*x^2+5*x+2)^(1/
2)+2*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 = 1.60, size = 75, normalized size = 0.90 \begin {gather*} -\frac {4 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + 2 \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {398 \, x}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {358}{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)^2/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")

[Out]

-4/3*x^2/sqrt(3*x^2 + 5*x + 2) + 2*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 398/3*x/sqrt(3*x^2
+ 5*x + 2) - 358/3/sqrt(3*x^2 + 5*x + 2)

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

[Out]

-int(((2*x + 3)^2*(x - 5))/(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 \left (- \frac {51 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 \left (- \frac {8 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}}\right )\, dx - \int \frac {4 x^{3}}{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 {45}{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)**2/(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(-51*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) -
Integral(-8*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) -
Integral(4*x**3/(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) -
Integral(-45/(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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