3.2442 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=167 \[ \frac{(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac{(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac{(26934 x+57845) \sqrt{3 x^2+5 x+2}}{12800 (2 x+3)}+\frac{177}{128} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{137111 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25600 \sqrt{5}} \]

[Out]

-((57845 + 26934*x)*Sqrt[2 + 5*x + 3*x^2])/(12800*(3 + 2*x)) + ((17051 + 13074*x)*(2 + 5*x + 3*x^2)^(3/2))/(96
00*(3 + 2*x)^3) + ((119 + 114*x)*(2 + 5*x + 3*x^2)^(5/2))/(80*(3 + 2*x)^5) + (177*Sqrt[3]*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/128 - (137111*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25600*S
qrt[5])

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Rubi [A]  time = 0.106052, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {810, 812, 843, 621, 206, 724} \[ \frac{(114 x+119) \left (3 x^2+5 x+2\right )^{5/2}}{80 (2 x+3)^5}+\frac{(13074 x+17051) \left (3 x^2+5 x+2\right )^{3/2}}{9600 (2 x+3)^3}-\frac{(26934 x+57845) \sqrt{3 x^2+5 x+2}}{12800 (2 x+3)}+\frac{177}{128} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{137111 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25600 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((57845 + 26934*x)*Sqrt[2 + 5*x + 3*x^2])/(12800*(3 + 2*x)) + ((17051 + 13074*x)*(2 + 5*x + 3*x^2)^(3/2))/(96
00*(3 + 2*x)^3) + ((119 + 114*x)*(2 + 5*x + 3*x^2)^(5/2))/(80*(3 + 2*x)^5) + (177*Sqrt[3]*ArcTanh[(5 + 6*x)/(2
*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/128 - (137111*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25600*S
qrt[5])

Rule 810

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

Rule 812

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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])

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]

Rubi steps

\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx &=\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac{1}{160} \int \frac{(437+462 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx\\ &=\frac{(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac{\int \frac{(-45914-53868 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{12800}\\ &=-\frac{(57845+26934 x) \sqrt{2+5 x+3 x^2}}{12800 (3+2 x)}+\frac{(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}-\frac{\int \frac{-725956-849600 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{102400}\\ &=-\frac{(57845+26934 x) \sqrt{2+5 x+3 x^2}}{12800 (3+2 x)}+\frac{(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac{531}{128} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{137111 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{25600}\\ &=-\frac{(57845+26934 x) \sqrt{2+5 x+3 x^2}}{12800 (3+2 x)}+\frac{(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac{531}{64} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{137111 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{12800}\\ &=-\frac{(57845+26934 x) \sqrt{2+5 x+3 x^2}}{12800 (3+2 x)}+\frac{(17051+13074 x) \left (2+5 x+3 x^2\right )^{3/2}}{9600 (3+2 x)^3}+\frac{(119+114 x) \left (2+5 x+3 x^2\right )^{5/2}}{80 (3+2 x)^5}+\frac{177}{128} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )-\frac{137111 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{25600 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.158243, size = 120, normalized size = 0.72 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (172800 x^5+4630848 x^4+21586808 x^3+41641148 x^2+37019838 x+12600183\right )}{(2 x+3)^5}+411333 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+531000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{384000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(12600183 + 37019838*x + 41641148*x^2 + 21586808*x^3 + 4630848*x^4 + 172800*x^5))/
(3 + 2*x)^5 + 411333*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 531000*Sqrt[3]*ArcTanh[(5
+ 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/384000

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Maple [B]  time = 0.013, size = 279, normalized size = 1.7 \begin{align*} -{\frac{13}{800} \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{131}{8000} \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{521}{15000} \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{9349}{300000} \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{57455+68946\,x}{125000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{11491}{62500} \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{31405+37686\,x}{60000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{21805+26166\,x}{16000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{177\,\sqrt{3}}{128}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{137111\,\sqrt{5}}{128000}{\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{137111}{500000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{137111}{240000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{137111}{128000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}} \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)^6,x)

[Out]

-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(7/2)-131/8000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(7/2)-521/15000/(x+3/
2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)-9349/300000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(7/2)+11491/125000*(5+6*x)*(3*(
x+3/2)^2-4*x-19/4)^(5/2)-11491/62500/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)+6281/60000*(5+6*x)*(3*(x+3/2)^2-4*x-
19/4)^(3/2)+4361/16000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+177/128*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-
19/4)^(1/2))*3^(1/2)+137111/128000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))-137111
/500000*(3*(x+3/2)^2-4*x-19/4)^(5/2)-137111/240000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-137111/128000*(12*(x+3/2)^2-16
*x-19)^(1/2)

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Maxima [B]  time = 1.56705, size = 401, normalized size = 2.4 \begin{align*} \frac{9349}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{131 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{521 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{1875 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{9349 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{75000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{6281}{10000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{11491}{240000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{11491 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{25000 \,{\left (2 \, x + 3\right )}} + \frac{13083}{8000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{177}{128} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{137111}{128000} \, \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{49891}{64000} \, \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)^6,x, algorithm="maxima")

[Out]

9349/100000*(3*x^2 + 5*x + 2)^(5/2) - 13/25*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 8
10*x + 243) - 131/500*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 521/1875*(3*x^2 + 5*x
+ 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9349/75000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 6281/10000*
(3*x^2 + 5*x + 2)^(3/2)*x - 11491/240000*(3*x^2 + 5*x + 2)^(3/2) - 11491/25000*(3*x^2 + 5*x + 2)^(5/2)/(2*x +
3) + 13083/8000*sqrt(3*x^2 + 5*x + 2)*x + 177/128*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 137
111/128000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 49891/64000*sqrt(3
*x^2 + 5*x + 2)

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Fricas [A]  time = 1.55739, size = 641, normalized size = 3.84 \begin{align*} \frac{531000 \, \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 411333 \, \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 (172800 \, x^{5} + 4630848 \, x^{4} + 21586808 \, x^{3} + 41641148 \, x^{2} + 37019838 \, x + 12600183\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{768000 \,{\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)^(5/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/768000*(531000*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x
+ 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 411333*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*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*(172800*x^5 +
4630848*x^4 + 21586808*x^3 + 41641148*x^2 + 37019838*x + 12600183)*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(-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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.30446, size = 549, normalized size = 3.29 \begin{align*} -\frac{137111}{128000} \, \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{177}{128} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{9}{64} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{27201072 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 316934472 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 4873277176 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 14374341276 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 80473660448 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 98380998102 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 236231795506 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 119385279741 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 103767800973 \, \sqrt{3} x + 13144069068 \, \sqrt{3} - 103767800973 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{38400 \,{\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)^(5/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

-137111/128000*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))) - 177/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 5*x + 2)) - 5)) - 9/64*sqrt(3*x^2 + 5*x + 2) - 1/38400*(27201072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9
+ 316934472*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 4873277176*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 +
14374341276*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 80473660448*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5
+ 98380998102*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 236231795506*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^3 + 119385279741*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 103767800973*sqrt(3)*x + 13144069068*sqrt(3
) - 103767800973*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