3.2414 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=94 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^3}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{200 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400 \sqrt{5}} \]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(15*(3 + 2*x)^3) - (47*A
rcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(400*Sqrt[5])

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Rubi [A]  time = 0.0443192, antiderivative size = 94, 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 = {806, 720, 724, 206} \[ -\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{15 (2 x+3)^3}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{200 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^4,x]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(200*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(15*(3 + 2*x)^3) - (47*A
rcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(400*Sqrt[5])

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]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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 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) \sqrt{2+5 x+3 x^2}}{(3+2 x)^4} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^3}+\frac{47}{10} \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{200 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^3}-\frac{47}{400} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{200 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^3}+\frac{47}{200} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{200 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{15 (3+2 x)^3}-\frac{47 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{400 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.039096, size = 74, normalized size = 0.79 \[ \frac{\sqrt{3 x^2+5 x+2} \left (696 x^2+2758 x+1921\right )}{600 (2 x+3)^3}+\frac{47 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{400 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^4,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(1921 + 2758*x + 696*x^2))/(600*(3 + 2*x)^3) + (47*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2
+ 5*x + 3*x^2])])/(400*Sqrt[5])

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Maple [A]  time = 0.01, size = 132, normalized size = 1.4 \begin{align*} -{\frac{47}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{47}{125} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{47}{2000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{47\,\sqrt{5}}{2000}{\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 ) }+{\frac{235+282\,x}{250}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4,x)

[Out]

-47/200/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(3/2)-47/125/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-47/2000*(12*(x+3/2)
^2-16*x-19)^(1/2)+47/2000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+47/250*(5+6*x)*
(3*(x+3/2)^2-4*x-19/4)^(1/2)-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 1.51084, size = 182, normalized size = 1.94 \begin{align*} \frac{47}{2000} \, \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{141}{200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{50 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{47 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{50 \,{\left (2 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

47/2000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 141/200*sqrt(3*x^2 +
5*x + 2) - 13/15*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 47/50*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 +
12*x + 9) - 47/50*sqrt(3*x^2 + 5*x + 2)/(2*x + 3)

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Fricas [A]  time = 1.3881, size = 305, normalized size = 3.24 \begin{align*} \frac{141 \, \sqrt{5}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) + 20 \,{\left (696 \, x^{2} + 2758 \, x + 1921\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{12000 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/12000*(141*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*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)) + 20*(696*x^2 + 2758*x + 1921)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x +
27)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**4,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(x*sqrt(3*x**2 +
5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

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Giac [B]  time = 1.24516, size = 347, normalized size = 3.69 \begin{align*} -\frac{47}{2000} \, \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{1236 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 4830 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 90290 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 144945 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 287985 \, \sqrt{3} x - 69339 \, \sqrt{3} + 287985 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{600 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-47/2000*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*
sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/600*(1236*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 4830*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 90290*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 144945*sqrt(3)*(sqr
t(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 287985*sqrt(3)*x - 69339*sqrt(3) + 287985*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3