3.2444 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=154 \[ -\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{256000 \sqrt{5}} \]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(128000*(3 + 2*x)^2) - (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(9600*(3 +
2*x)^4) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(600*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x
)^7) - (47*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(256000*Sqrt[5])

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Rubi [A]  time = 0.0777246, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \[ -\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac{47 (8 x+7) \sqrt{3 x^2+5 x+2}}{128000 (2 x+3)^2}-\frac{47 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{256000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^8,x]

[Out]

(47*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(128000*(3 + 2*x)^2) - (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(9600*(3 +
2*x)^4) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(600*(3 + 2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x
)^7) - (47*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(256000*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 )^{5/2}}{(3+2 x)^8} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}+\frac{47}{10} \int \frac{\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac{47}{240} \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}}{9600 (3+2 x)^4}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}+\frac{47 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{6400}\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac{47 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{256000}\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}+\frac{47 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{128000}\\ &=\frac{47 (7+8 x) \sqrt{2+5 x+3 x^2}}{128000 (3+2 x)^2}-\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^4}+\frac{47 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{600 (3+2 x)^6}-\frac{13 \left (2+5 x+3 x^2\right )^{7/2}}{35 (3+2 x)^7}-\frac{47 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{256000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0847017, size = 154, normalized size = 1. \[ -\frac{13 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}+\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{600 (2 x+3)^6}-\frac{47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^4}+\frac{47 \left (\frac{10 \sqrt{3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{1280000} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^8,x]

[Out]

(-47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(9600*(3 + 2*x)^4) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(600*(3 +
2*x)^6) - (13*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2*x)^7) + (47*((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])]))/1280000

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Maple [B]  time = 0.019, size = 290, normalized size = 1.9 \begin{align*} -{\frac{47}{9600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{47}{6000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{987}{80000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{2867}{150000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{87373}{3000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{136535+163842\,x}{1250000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{27307}{625000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{6815+8178\,x}{600000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{235+282\,x}{160000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{47\,\sqrt{5}}{1280000}{\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{47}{5000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{47}{2400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{47}{1280000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{4480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^8,x)

[Out]

-47/9600/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(7/2)-47/6000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(7/2)-987/80000/(x+3/
2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-2867/150000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)-87373/3000000/(x+3/2)^2*(
3*(x+3/2)^2-4*x-19/4)^(7/2)+27307/1250000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)-27307/625000/(x+3/2)*(3*(x+3/2)
^2-4*x-19/4)^(7/2)-1363/600000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+47/160000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(
1/2)+47/1280000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-47/5000000*(3*(x+3/2)^2-4
*x-19/4)^(5/2)-47/2400000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-47/1280000*(12*(x+3/2)^2-16*x-19)^(1/2)-13/4480/(x+3/2)
^7*(3*(x+3/2)^2-4*x-19/4)^(7/2)

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Maxima [B]  time = 1.88252, size = 495, normalized size = 3.21 \begin{align*} \frac{87373}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{35 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{47 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{150 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{94 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{375 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{987 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{2867 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{18750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{87373 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{750000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1363}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{27307}{2400000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{27307 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{250000 \,{\left (2 \, x + 3\right )}} + \frac{141}{80000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{47}{1280000} \, \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{893}{640000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^8,x, algorithm="maxima")

[Out]

87373/1000000*(3*x^2 + 5*x + 2)^(5/2) - 13/35*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x
^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 47/150*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4
320*x^3 + 4860*x^2 + 2916*x + 729) - 94/375*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 8
10*x + 243) - 987/5000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 2867/18750*(3*x^2 +
5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 87373/750000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1363/1
00000*(3*x^2 + 5*x + 2)^(3/2)*x - 27307/2400000*(3*x^2 + 5*x + 2)^(3/2) - 27307/250000*(3*x^2 + 5*x + 2)^(5/2)
/(2*x + 3) + 141/80000*sqrt(3*x^2 + 5*x + 2)*x + 47/1280000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x
+ 3) + 5/2/abs(2*x + 3) - 2) + 893/640000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 1.37727, size = 559, normalized size = 3.63 \begin{align*} \frac{987 \, \sqrt{5}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\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 (1089792 \, x^{6} + 22620128 \, x^{5} + 81951440 \, x^{4} + 127557120 \, x^{3} + 100711840 \, x^{2} + 39981058 \, x + 6404247\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{53760000 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^8,x, algorithm="fricas")

[Out]

1/53760000*(987*sqrt(5)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)*l
og(-(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*(1089792*x^6 +
22620128*x^5 + 81951440*x^4 + 127557120*x^3 + 100711840*x^2 + 39981058*x + 6404247)*sqrt(3*x^2 + 5*x + 2))/(1
28*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.2643, size = 622, normalized size = 4.04 \begin{align*} -\frac{47}{1280000} \, \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{72512832 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 651952224 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 6898276448 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 8494566864 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 58878767920 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 326450774496 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 2207907445056 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 3147944405424 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 9314774279636 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 6492162811470 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 9472821206534 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3070624865553 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 1792565462541 \, \sqrt{3} x - 158637115728 \, \sqrt{3} + 1792565462541 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{2688000 \,{\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 )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^8,x, algorithm="giac")

[Out]

-47/1280000*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))) - 1/2688000*(72512832*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^1
3 + 651952224*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 6898276448*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
11 + 8494566864*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 58878767920*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2
))^9 - 326450774496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 2207907445056*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2))^7 - 3147944405424*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 9314774279636*(sqrt(3)*x - sqrt(3*x^
2 + 5*x + 2))^5 - 6492162811470*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 9472821206534*(sqrt(3)*x - sqr
t(3*x^2 + 5*x + 2))^3 - 3070624865553*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 1792565462541*sqrt(3)*x
- 158637115728*sqrt(3) + 1792565462541*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^7