3.2415 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=119 \[ -\frac{4 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}+\frac{153 (8 x+7) \sqrt{3 x^2+5 x+2}}{800 (2 x+3)^2}-\frac{153 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1600 \sqrt{5}} \]

[Out]

(153*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) - (4*(
2 + 5*x + 3*x^2)^(3/2))/(5*(3 + 2*x)^3) - (153*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1600*Sqr
t[5])

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Rubi [A]  time = 0.0642309, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \[ -\frac{4 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}+\frac{153 (8 x+7) \sqrt{3 x^2+5 x+2}}{800 (2 x+3)^2}-\frac{153 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1600 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(153*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) - (4*(
2 + 5*x + 3*x^2)^(3/2))/(5*(3 + 2*x)^3) - (153*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(1600*Sqr
t[5])

Rule 834

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

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)^5} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac{1}{20} \int \frac{\left (-\frac{123}{2}+39 x\right ) \sqrt{2+5 x+3 x^2}}{(3+2 x)^4} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac{4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}+\frac{153}{40} \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=\frac{153 (7+8 x) \sqrt{2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac{4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}-\frac{153 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{1600}\\ &=\frac{153 (7+8 x) \sqrt{2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac{4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}+\frac{153}{800} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{153 (7+8 x) \sqrt{2+5 x+3 x^2}}{800 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{20 (3+2 x)^4}-\frac{4 \left (2+5 x+3 x^2\right )^{3/2}}{5 (3+2 x)^3}-\frac{153 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{1600 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.050206, size = 119, normalized size = 1. \[ \frac{1}{20} \left (-\frac{16 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^4}+\frac{153 (8 x+7) \sqrt{3 x^2+5 x+2}}{40 (2 x+3)^2}+\frac{153 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{80 \sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((153*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(40*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^4 - (16*(2 +
5*x + 3*x^2)^(3/2))/(3 + 2*x)^3 + (153*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(80*Sqrt[5]))/20

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Maple [A]  time = 0.01, size = 153, normalized size = 1.3 \begin{align*} -{\frac{13}{320} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1}{10} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{153}{800} \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{153}{500} \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{153}{8000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{153\,\sqrt{5}}{8000}{\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{765+918\,x}{1000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}} \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)^5,x)

[Out]

-13/320/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(3/2)-1/10/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(3/2)-153/800/(x+3/2)^2*(
3*(x+3/2)^2-4*x-19/4)^(3/2)-153/500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-153/8000*(12*(x+3/2)^2-16*x-19)^(1/2)
+153/8000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+153/1000*(5+6*x)*(3*(x+3/2)^2-4
*x-19/4)^(1/2)

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Maxima [A]  time = 1.50443, size = 231, normalized size = 1.94 \begin{align*} \frac{153}{8000} \, \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{459}{800} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{20 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{4 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{5 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{153 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{200 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{153 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{200 \,{\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)^5,x, algorithm="maxima")

[Out]

153/8000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 459/800*sqrt(3*x^2 +
5*x + 2) - 13/20*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 4/5*(3*x^2 + 5*x + 2)^(3/
2)/(8*x^3 + 36*x^2 + 54*x + 27) - 153/200*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 153/200*sqrt(3*x^2 + 5*
x + 2)/(2*x + 3)

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Fricas [A]  time = 1.39306, size = 354, normalized size = 2.97 \begin{align*} \frac{153 \, \sqrt{5}{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 (1056 \, x^{3} + 5252 \, x^{2} + 9108 \, x + 4759\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{16000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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)^5,x, algorithm="fricas")

[Out]

1/16000*(153*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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*(1056*x^3 + 5252*x^2 + 9108*x + 4759)*sqrt(3*x^2 + 5*x + 2))/
(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)

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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}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, 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)**5,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(x
*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)

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Giac [A]  time = 1.19684, size = 247, normalized size = 2.08 \begin{align*} -\frac{3}{8000} \, \sqrt{5}{\left (44 \, \sqrt{5} \sqrt{3} + 51 \, \log \left (-\sqrt{5} \sqrt{3} + 4\right )\right )} \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) + \frac{153}{8000} \, \sqrt{5} \log \left ({\left | \sqrt{5}{\left (\sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{1600} \,{\left (\frac{5 \,{\left (\frac{2 \,{\left (\frac{65 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 24 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 25 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 132 \, \mathrm{sgn}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{8}{2 \, x + 3} + \frac{5}{{\left (2 \, x + 3\right )}^{2}} + 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)^5,x, algorithm="giac")

[Out]

-3/8000*sqrt(5)*(44*sqrt(5)*sqrt(3) + 51*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/(2*x + 3)) + 153/8000*sqrt(5)*log(ab
s(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 1/1600*(5*(2*(
65*sgn(1/(2*x + 3))/(2*x + 3) - 24*sgn(1/(2*x + 3)))/(2*x + 3) - 25*sgn(1/(2*x + 3)))/(2*x + 3) - 132*sgn(1/(2
*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)