3.2416 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=144 \[ -\frac{87 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^3}-\frac{339 \left (3 x^2+5 x+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}+\frac{3159 (8 x+7) \sqrt{3 x^2+5 x+2}}{20000 (2 x+3)^2}-\frac{3159 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40000 \sqrt{5}} \]

[Out]

(3159*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20000*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(25*(3 + 2*x)^5) - (
339*(2 + 5*x + 3*x^2)^(3/2))/(500*(3 + 2*x)^4) - (87*(2 + 5*x + 3*x^2)^(3/2))/(125*(3 + 2*x)^3) - (3159*ArcTan
h[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40000*Sqrt[5])

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Rubi [A]  time = 0.0898709, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{87 \left (3 x^2+5 x+2\right )^{3/2}}{125 (2 x+3)^3}-\frac{339 \left (3 x^2+5 x+2\right )^{3/2}}{500 (2 x+3)^4}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{25 (2 x+3)^5}+\frac{3159 (8 x+7) \sqrt{3 x^2+5 x+2}}{20000 (2 x+3)^2}-\frac{3159 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{40000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^6,x]

[Out]

(3159*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20000*(3 + 2*x)^2) - (13*(2 + 5*x + 3*x^2)^(3/2))/(25*(3 + 2*x)^5) - (
339*(2 + 5*x + 3*x^2)^(3/2))/(500*(3 + 2*x)^4) - (87*(2 + 5*x + 3*x^2)^(3/2))/(125*(3 + 2*x)^3) - (3159*ArcTan
h[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40000*Sqrt[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)^6} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac{1}{25} \int \frac{\left (-\frac{105}{2}+78 x\right ) \sqrt{2+5 x+3 x^2}}{(3+2 x)^5} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac{339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}+\frac{1}{500} \int \frac{\left (\frac{2169}{2}-1017 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}}{25 (3+2 x)^5}-\frac{339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac{3159 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1000}\\ &=\frac{3159 (7+8 x) \sqrt{2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac{339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac{3159 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{40000}\\ &=\frac{3159 (7+8 x) \sqrt{2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac{339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}+\frac{3159 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{20000}\\ &=\frac{3159 (7+8 x) \sqrt{2+5 x+3 x^2}}{20000 (3+2 x)^2}-\frac{13 \left (2+5 x+3 x^2\right )^{3/2}}{25 (3+2 x)^5}-\frac{339 \left (2+5 x+3 x^2\right )^{3/2}}{500 (3+2 x)^4}-\frac{87 \left (2+5 x+3 x^2\right )^{3/2}}{125 (3+2 x)^3}-\frac{3159 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{40000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.116084, size = 146, normalized size = 1.01 \[ \frac{1}{25} \left (-\frac{87 \left (3 x^2+5 x+2\right )^{3/2}}{5 (2 x+3)^3}-\frac{339 \left (3 x^2+5 x+2\right )^{3/2}}{20 (2 x+3)^4}-\frac{13 \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}+\frac{3159 \left (\frac{10 \sqrt{3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{8000}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^6,x]

[Out]

((-13*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^5 - (339*(2 + 5*x + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) - (87*(2 + 5*x + 3
*x^2)^(3/2))/(5*(3 + 2*x)^3) + (3159*((10*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 + Sqrt[5]*ArcTanh[(-7 -
8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/8000)/25

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Maple [A]  time = 0.012, size = 174, normalized size = 1.2 \begin{align*} -{\frac{339}{8000} \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{87}{1000} \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{3159}{20000} \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{3159}{12500} \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{3159}{200000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{3159\,\sqrt{5}}{200000}{\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{15795+18954\,x}{25000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}} \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)^6,x)

[Out]

-339/8000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(3/2)-87/1000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(3/2)-3159/20000/(x+
3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(3/2)-3159/12500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-3159/200000*(12*(x+3/2)^2-
16*x-19)^(1/2)+3159/200000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+3159/25000*(5+
6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [A]  time = 1.52101, size = 286, normalized size = 1.99 \begin{align*} \frac{3159}{200000} \, \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{9477}{20000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{339 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{87 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{125 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{3159 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{5000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{3159 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{5000 \,{\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)^6,x, algorithm="maxima")

[Out]

3159/200000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 9477/20000*sqrt(3
*x^2 + 5*x + 2) - 13/25*(3*x^2 + 5*x + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 339/50
0*(3*x^2 + 5*x + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 87/125*(3*x^2 + 5*x + 2)^(3/2)/(8*x^3 + 3
6*x^2 + 54*x + 27) - 3159/5000*(3*x^2 + 5*x + 2)^(3/2)/(4*x^2 + 12*x + 9) - 3159/5000*sqrt(3*x^2 + 5*x + 2)/(2
*x + 3)

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Fricas [A]  time = 1.38794, size = 419, normalized size = 2.91 \begin{align*} \frac{3159 \, \sqrt{5}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (35136 \, x^{4} + 225816 \, x^{3} + 549516 \, x^{2} + 606326 \, x + 244331\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{400000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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)^6,x, algorithm="fricas")

[Out]

1/400000*(3159*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*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*(35136*x^4 + 225816*x^3 + 549516*x^2 + 606326*
x + 244331)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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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}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, 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)**6,x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x
) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729),
x)

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Giac [B]  time = 1.26287, size = 485, normalized size = 3.37 \begin{align*} -\frac{3159}{200000} \, \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{50544 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 682344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 11747352 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 40431852 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 269183136 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 388190654 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1077361162 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 615279657 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 592102521 \, \sqrt{3} x + 81620976 \, \sqrt{3} - 592102521 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{20000 \,{\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 )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-3159/200000*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))) + 1/20000*(50544*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 6
82344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 11747352*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 4043185
2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 269183136*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 388190654*
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1077361162*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 615279657*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 592102521*sqrt(3)*x + 81620976*sqrt(3) - 592102521*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)^5