3.2439 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=160 \[ -\frac{(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{192 (2 x+3)}+\frac{5 (3763-7854 x) \sqrt{3 x^2+5 x+2}}{1536}-\frac{199615 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3072 \sqrt{3}}+\frac{4295}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

(5*(3763 - 7854*x)*Sqrt[2 + 5*x + 3*x^2])/1536 + (5*(573 + 164*x)*(2 + 5*x + 3*x^2)^(3/2))/(192*(3 + 2*x)) - (
(29 + 2*x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^2) - (199615*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x
^2])])/(3072*Sqrt[3]) + (4295*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

________________________________________________________________________________________

Rubi [A]  time = 0.103746, antiderivative size = 160, 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 = {812, 814, 843, 621, 206, 724} \[ -\frac{(2 x+29) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}+\frac{5 (164 x+573) \left (3 x^2+5 x+2\right )^{3/2}}{192 (2 x+3)}+\frac{5 (3763-7854 x) \sqrt{3 x^2+5 x+2}}{1536}-\frac{199615 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3072 \sqrt{3}}+\frac{4295}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(3763 - 7854*x)*Sqrt[2 + 5*x + 3*x^2])/1536 + (5*(573 + 164*x)*(2 + 5*x + 3*x^2)^(3/2))/(192*(3 + 2*x)) - (
(29 + 2*x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^2) - (199615*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x
^2])])/(3072*Sqrt[3]) + (4295*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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)^3} \, dx &=-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{5}{64} \int \frac{(-274-328 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx\\ &=\frac{5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}+\frac{5}{512} \int \frac{(-8836-10472 x) \sqrt{2+5 x+3 x^2}}{3+2 x} \, dx\\ &=\frac{5 (3763-7854 x) \sqrt{2+5 x+3 x^2}}{1536}+\frac{5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{5 \int \frac{545832+638768 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{24576}\\ &=\frac{5 (3763-7854 x) \sqrt{2+5 x+3 x^2}}{1536}+\frac{5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{199615 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{3072}+\frac{21475}{256} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=\frac{5 (3763-7854 x) \sqrt{2+5 x+3 x^2}}{1536}+\frac{5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{199615 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{1536}-\frac{21475}{128} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=\frac{5 (3763-7854 x) \sqrt{2+5 x+3 x^2}}{1536}+\frac{5 (573+164 x) \left (2+5 x+3 x^2\right )^{3/2}}{192 (3+2 x)}-\frac{(29+2 x) \left (2+5 x+3 x^2\right )^{5/2}}{16 (3+2 x)^2}-\frac{199615 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{3072 \sqrt{3}}+\frac{4295}{256} \sqrt{5} \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.112831, size = 120, normalized size = 0.75 \[ \frac{-\frac{6 \sqrt{3 x^2+5 x+2} \left (1728 x^5-8544 x^4-14456 x^3-57292 x^2-290742 x-295719\right )}{(2 x+3)^2}-154620 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-199615 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{9216} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(-295719 - 290742*x - 57292*x^2 - 14456*x^3 - 8544*x^4 + 1728*x^5))/(3 + 2*x)^2 - 1
54620*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 199615*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt
[6 + 15*x + 9*x^2])])/9216

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 216, normalized size = 1.4 \begin{align*} -{\frac{13}{40} \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{83}{50} \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{859}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{545+654\,x}{64} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{32725+39270\,x}{1536}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{199615\,\sqrt{3}}{9216}\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{859}{96} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{4295}{256}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{4295\,\sqrt{5}}{256}{\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{415+498\,x}{100} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \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)^3,x)

[Out]

-13/40/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(7/2)+83/50/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)+859/200*(3*(x+3/2)^2-
4*x-19/4)^(5/2)-109/64*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-6545/1536*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-199
615/9216*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+859/96*(3*(x+3/2)^2-4*x-19/4)^(3/2)+42
95/256*(12*(x+3/2)^2-16*x-19)^(1/2)-4295/256*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/
2))-83/100*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.72102, size = 255, normalized size = 1.59 \begin{align*} \frac{39}{40} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{10 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{327}{32} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{83}{192} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{83 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{20 \,{\left (2 \, x + 3\right )}} - \frac{6545}{256} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{199615}{9216} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{4295}{256} \, \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{18815}{1536} \, \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)^3,x, algorithm="maxima")

[Out]

39/40*(3*x^2 + 5*x + 2)^(5/2) - 13/10*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 327/32*(3*x^2 + 5*x + 2)^(3
/2)*x + 83/192*(3*x^2 + 5*x + 2)^(3/2) + 83/20*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 6545/256*sqrt(3*x^2 + 5*x +
2)*x - 199615/9216*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 4295/256*sqrt(5)*log(sqrt(5)*sqrt
(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 18815/1536*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.41489, size = 478, normalized size = 2.99 \begin{align*} \frac{199615 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\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 ) + 154620 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\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 ) - 12 \,{\left (1728 \, x^{5} - 8544 \, x^{4} - 14456 \, x^{3} - 57292 \, x^{2} - 290742 \, x - 295719\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{18432 \,{\left (4 \, x^{2} + 12 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/18432*(199615*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 4
9) + 154620*sqrt(5)*(4*x^2 + 12*x + 9)*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)) - 12*(1728*x^5 - 8544*x^4 - 14456*x^3 - 57292*x^2 - 290742*x - 295719)*sqrt(3*x^2 + 5*x +
2))/(4*x^2 + 12*x + 9)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \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)**3,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2
)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27
), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-15*x**4*sqrt(
3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**
2 + 54*x + 27), x)

________________________________________________________________________________________

Giac [B]  time = 1.26707, size = 363, normalized size = 2.27 \begin{align*} -\frac{1}{1536} \,{\left (2 \,{\left (12 \,{\left (18 \, x - 143\right )} x + 2855\right )} x - 23731\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{4295}{256} \, \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{199615}{9216} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac{5 \,{\left (4214 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 15793 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 53551 \, \sqrt{3} x + 19053 \, \sqrt{3} - 53551 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{128 \,{\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 )}^{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)^3,x, algorithm="giac")

[Out]

-1/1536*(2*(12*(18*x - 143)*x + 2855)*x - 23731)*sqrt(3*x^2 + 5*x + 2) + 4295/256*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))) + 199615/9216*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 5/128*(4214*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 15793*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 53551*sqrt(3)*x +
19053*sqrt(3) - 53551*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)^2