3.2430 \(\int \frac{(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^8} \, dx\)

Optimal. Leaf size=174 \[ -\frac{4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac{433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac{4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac{4663 (8 x+7) \sqrt{3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac{4663 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1600000 \sqrt{5}} \]

[Out]

(-4663*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800000*(3 + 2*x)^2) + (4663*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(60000
*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^7) - (433*(2 + 5*x + 3*x^2)^(5/2))/(1050*(3 + 2*x)^
6) - (4892*(2 + 5*x + 3*x^2)^(5/2))/(13125*(3 + 2*x)^5) + (4663*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/(1600000*Sqrt[5])

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Rubi [A]  time = 0.106746, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{4892 \left (3 x^2+5 x+2\right )^{5/2}}{13125 (2 x+3)^5}-\frac{433 \left (3 x^2+5 x+2\right )^{5/2}}{1050 (2 x+3)^6}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^7}+\frac{4663 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{60000 (2 x+3)^4}-\frac{4663 (8 x+7) \sqrt{3 x^2+5 x+2}}{800000 (2 x+3)^2}+\frac{4663 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1600000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4663*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(800000*(3 + 2*x)^2) + (4663*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(60000
*(3 + 2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^7) - (433*(2 + 5*x + 3*x^2)^(5/2))/(1050*(3 + 2*x)^
6) - (4892*(2 + 5*x + 3*x^2)^(5/2))/(13125*(3 + 2*x)^5) + (4663*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/(1600000*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) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^8} \, dx &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{1}{35} \int \frac{\left (-\frac{199}{2}+78 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}+\frac{\int \frac{\left (\frac{5887}{2}-1299 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx}{1050}\\ &=-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac{4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac{4663 \int \frac{\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{1500}\\ &=\frac{4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac{4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac{4663 \int \frac{\sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{40000}\\ &=-\frac{4663 (7+8 x) \sqrt{2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac{4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac{4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac{4663 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{1600000}\\ &=-\frac{4663 (7+8 x) \sqrt{2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac{4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac{4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}-\frac{4663 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{800000}\\ &=-\frac{4663 (7+8 x) \sqrt{2+5 x+3 x^2}}{800000 (3+2 x)^2}+\frac{4663 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{60000 (3+2 x)^4}-\frac{13 \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^7}-\frac{433 \left (2+5 x+3 x^2\right )^{5/2}}{1050 (3+2 x)^6}-\frac{4892 \left (2+5 x+3 x^2\right )^{5/2}}{13125 (3+2 x)^5}+\frac{4663 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{1600000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.162717, size = 157, normalized size = 0.9 \[ \frac{1}{35} \left (-\frac{4892 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac{433 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}-\frac{13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}+\frac{32641 \left (\frac{10 \sqrt{3 x^2+5 x+2} \left (864 x^3+2068 x^2+1572 x+371\right )}{(2 x+3)^4}-3 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )}{4800000}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-13*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^7 - (433*(2 + 5*x + 3*x^2)^(5/2))/(30*(3 + 2*x)^6) - (4892*(2 + 5*x +
3*x^2)^(5/2))/(375*(3 + 2*x)^5) + (32641*((10*Sqrt[2 + 5*x + 3*x^2]*(371 + 1572*x + 2068*x^2 + 864*x^3))/(3 +
2*x)^4 - 3*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]))/4800000)/35

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Maple [A]  time = 0.016, size = 253, normalized size = 1.5 \begin{align*} -{\frac{13}{4480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}-{\frac{433}{67200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{1223}{105000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{4663}{240000} \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{4663}{150000} \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{144553}{3000000} \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{135227}{1875000} \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{4663}{15000000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{23315+27978\,x}{1000000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{4663}{8000000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{4663\,\sqrt{5}}{8000000}{\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{676135+811362\,x}{3750000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \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)^8,x)

[Out]

-13/4480/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(5/2)-433/67200/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(5/2)-1223/105000/(
x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(5/2)-4663/240000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(5/2)-4663/150000/(x+3/2)^3
*(3*(x+3/2)^2-4*x-19/4)^(5/2)-144553/3000000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(5/2)-135227/1875000/(x+3/2)*(3*
(x+3/2)^2-4*x-19/4)^(5/2)+4663/15000000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-4663/1000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/
4)^(1/2)+4663/8000000*(12*(x+3/2)^2-16*x-19)^(1/2)-4663/8000000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+
3/2)^2-16*x-19)^(1/2))+135227/3750000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)

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Maxima [B]  time = 1.56757, size = 456, normalized size = 2.62 \begin{align*} \frac{144553}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{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{433 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{1050 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{4892 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{13125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{4663 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{15000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{4663 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{18750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{144553 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{750000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{13989}{500000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{4663}{8000000} \, \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{88597}{4000000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{135227 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}{750000 \,{\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)^8,x, algorithm="maxima")

[Out]

144553/1000000*(3*x^2 + 5*x + 2)^(3/2) - 13/35*(3*x^2 + 5*x + 2)^(5/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*
x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 433/1050*(3*x^2 + 5*x + 2)^(5/2)/(64*x^6 + 576*x^5 + 2160*x^4
+ 4320*x^3 + 4860*x^2 + 2916*x + 729) - 4892/13125*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*
x^2 + 810*x + 243) - 4663/15000*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 4663/18750*
(3*x^2 + 5*x + 2)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 144553/750000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9
) - 13989/500000*sqrt(3*x^2 + 5*x + 2)*x - 4663/8000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3)
+ 5/2/abs(2*x + 3) - 2) - 88597/4000000*sqrt(3*x^2 + 5*x + 2) - 135227/750000*(3*x^2 + 5*x + 2)^(3/2)/(2*x +
3)

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Fricas [A]  time = 1.80231, size = 556, normalized size = 3.2 \begin{align*} \frac{97923 \, \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 (191232 \, x^{6} + 2893088 \, x^{5} + 16376240 \, x^{4} + 55403520 \, x^{3} + 64140640 \, x^{2} + 15759118 \, x - 6554463\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{336000000 \,{\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)^(3/2)/(3+2*x)^8,x, algorithm="fricas")

[Out]

1/336000000*(97923*sqrt(5)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187
)*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*(191232*x^6
+ 2893088*x^5 + 16376240*x^4 + 55403520*x^3 + 64140640*x^2 + 15759118*x - 6554463)*sqrt(3*x^2 + 5*x + 2))/(128
*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)**(3/2)/(3+2*x)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.20279, size = 622, normalized size = 3.57 \begin{align*} \frac{4663}{8000000} \, \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{6267072 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 122207904 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 3852187808 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 18344551344 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 131374293680 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 134399090784 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 264419126976 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 1446858601104 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 6675760646156 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 5954681858370 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 10149146991914 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 3640765552263 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 2268672558411 \, \sqrt{3} x - 208833935688 \, \sqrt{3} + 2268672558411 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{16800000 \,{\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)^(3/2)/(3+2*x)^8,x, algorithm="giac")

[Out]

4663/8000000*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))) - 1/16800000*(6267072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^
13 + 122207904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 3852187808*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^11 + 18344551344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 131374293680*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^9 + 134399090784*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 264419126976*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^7 - 1446858601104*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 6675760646156*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2))^5 - 5954681858370*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 10149146991914*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^3 - 3640765552263*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 2268672558411*sqrt(3)
*x - 208833935688*sqrt(3) + 2268672558411*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)^7