3.2428 \(\int \frac{(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=124 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac{141 (8 x+7) \sqrt{3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac{141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{32000 \sqrt{5}} \]

[Out]

(-141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(16000*(3 + 2*x)^2) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(400*(3 +
2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x
^2])])/(32000*Sqrt[5])

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Rubi [A]  time = 0.0616323, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \[ -\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac{141 (8 x+7) \sqrt{3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac{141 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{32000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(16000*(3 + 2*x)^2) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(400*(3 +
2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x
^2])])/(32000*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) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac{47}{10} \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac{141}{800} \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac{141 (7+8 x) \sqrt{2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac{141 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{32000}\\ &=-\frac{141 (7+8 x) \sqrt{2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac{141 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{16000}\\ &=-\frac{141 (7+8 x) \sqrt{2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac{141 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{32000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0826737, size = 128, normalized size = 1.03 \[ -\frac{83200 \left (3 x^2+5 x+2\right )^{5/2}-47 (2 x+3) \left (-30 (8 x+7) \sqrt{3 x^2+5 x+2} (2 x+3)^2+400 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}-3 \sqrt{5} (2 x+3)^4 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{160000 (2 x+3)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(83200*(2 + 5*x + 3*x^2)^(5/2) - 47*(3 + 2*x)*(-30*(3 + 2*x)^2*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2] + 400*(7 + 8*x
)*(2 + 5*x + 3*x^2)^(3/2) - 3*Sqrt[5]*(3 + 2*x)^4*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/(160
000*(3 + 2*x)^5)

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Maple [B]  time = 0.011, size = 211, normalized size = 1.7 \begin{align*} -{\frac{47}{1600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{47}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{1457}{20000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{1363}{12500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{47}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{705+846\,x}{20000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{141}{160000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{141\,\sqrt{5}}{160000}{\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{6815+8178\,x}{25000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{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)^(3/2)/(3+2*x)^6,x)

[Out]

-47/1600/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(5/2)-47/1000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1457/20000/(x+3
/2)^2*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1363/12500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+47/100000*(3*(x+3/2)^2-4*x-
19/4)^(3/2)-141/20000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+141/160000*(12*(x+3/2)^2-16*x-19)^(1/2)-141/160000*
5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+1363/25000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)
^(3/2)-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [B]  time = 1.51514, size = 325, normalized size = 2.62 \begin{align*} \frac{4371}{20000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{100 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{125 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{1457 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{5000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{423}{10000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{141}{160000} \, \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{2679}{80000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{1363 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{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)^(3/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

4371/20000*(3*x^2 + 5*x + 2)^(3/2) - 13/25*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 81
0*x + 243) - 47/100*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 47/125*(3*x^2 + 5*x + 2
)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1457/5000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 423/10000*sqrt(3
*x^2 + 5*x + 2)*x - 141/160000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2)
- 2679/80000*sqrt(3*x^2 + 5*x + 2) - 1363/5000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 1.6959, size = 410, normalized size = 3.31 \begin{align*} \frac{141 \, \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 (6336 \, x^{4} + 66616 \, x^{3} + 131516 \, x^{2} + 90126 \, x + 19031\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{320000 \,{\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)^(3/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/320000*(141*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*(6336*x^4 + 66616*x^3 + 131516*x^2 + 90126*x + 1
9031)*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{10 \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{23 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 - \int - \frac{10 x^{2} \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{3 x^{3} \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)**(3/2)/(3+2*x)**6,x)

[Out]

-Integral(-10*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(-23*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) - Integral(-10*x**2*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(3*x**3*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 = 1.18735, size = 485, normalized size = 3.91 \begin{align*} \frac{141}{160000} \, \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{146256 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 654456 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 415048 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 15455452 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 140042336 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 207568854 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 544555762 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 286352757 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 252454821 \, \sqrt{3} x - 31985676 \, \sqrt{3} + 252454821 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{16000 \,{\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)^(3/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

141/160000*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))) - 1/16000*(146256*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 65
4456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 415048*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 15455452*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 140042336*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 207568854*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 544555762*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 286352757*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 252454821*sqrt(3)*x - 31985676*sqrt(3) + 252454821*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